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http://mathoverflow.net/feeds/question/79372
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Terminology for a partition of unity for an étale groupoid - MathOverflow most recent 30 from http://mathoverflow.net 2013-05-23T02:41:21Z http://mathoverflow.net/feeds/question/79372 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/79372/terminology-for-a-partition-of-unity-for-an-etale-groupoid Terminology for a partition of unity for an étale groupoid H. Shindoh 2011-10-28T09:20:40Z 2011-10-28T09:20:40Z <p>I would like to ask about terminology for a partition of unity for an étale groupoid.</p> <p>I am reading the lecture notes "Cohomology of Stacks" by Prof. Behrend. A partition of unity is defined in Definition 22. Let $X_1 \rightrightarrows X_0$ be a Lie groupoid satisfying the following conditions.</p> <p>1) The source map $s$ and the target map $t$ are both étale (i.e., induce isomorphisms on tangent spaces).</p> <p>2) The map $(s,t):X_1 \to X \times X$ is proper and unramified, with finite fibers (unramified means injective on tangent spaces).</p> <p>For such a Lie groupoid, a partition of unity is defined as follows.</p> <blockquote> <p>A partition of unity for the groupoid $X_1 \rightrightarrows X_0$ is an $\mathbb{R}$-valued $C^\infty$-function $\rho$ on $X_0$ with the property that $s^*\rho$ has proper support with respect to $t:X_1 \to X_0$ and $t_!s^* \rho \equiv 1$.</p> </blockquote> <p>a) What does "$\rho$ has proper support with respect to $t:X_1 \to X_0$" means?</p> <p>b) How do we define the operator $t_!$?</p> <p>I would be most grateful if you could tell me references on these terminology.</p>
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2013-05-23 02:41:25
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https://zbmath.org/?q=an:0756.73012
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# zbMATH — the first resource for mathematics
Global smooth solutions and asymptotic stability in one-dimensional nonlinear thermoelasticity. (English) Zbl 0756.73012
It is established a global existence theorem of nonlinear classical dynamic coupled thermoelasticity for a one-dimensional displacement- temperature initial-boundary value problem with homogeneous boundary conditions. The theorem asserts that for sufficiently “small” initial data there is a smooth and unique solution to the problem, and suitably defined norms of the solution vanish or are bounded as the time goes to infinity. The reviewer notes a number of slips and typographical errors, e.g.: 1. The field equations (0.3)-(0.4) are not postulated in a dimensionless form. Therefore, introducing a unit interval $$0<x<1$$ as a reference configuration makes a confusion. 2. For the same reason, the definition of the norm: $$| v|_{t,K,L}$$ (p. 4) involving the coefficient $$(1+s)^ K$$, when $$0\leq s\leq t$$, and $$t$$ is the time, makes another confusion. 3. The hypothesis (4.5), p. 22, in which a constant $$\sigma$$ is an upper bound for the three fields of different physical dimensions, can be hardly justified.
##### MSC:
74A15 Thermodynamics in solid mechanics 74B20 Nonlinear elasticity 35Q72 Other PDE from mechanics (MSC2000)
Full Text:
##### References:
[1] Adams, R. A.: Sobolev spaces. Academic Press (1975). · Zbl 0314.46030 [2] Carlson, D. E.: Linear thermoelasticity. Handbuch der Physik VIa/2, 297-346, Springer-Verlag (1972). [3] Chrzeszczyk, A.: Some existence results in dynamical thermoelasticity. Part I. Nonlinear case. Arch. Mech. 39 (6) (1987), 605-617. · Zbl 0682.73004 [4] Dafermos, C. M. & Hsiao, L.: Development of singularities in solutions of the equations of nonlinear thermoelasticity. Quart. Appl. Math. 44 (1986), 463-474. · Zbl 0661.35009 [5] Hrusa, W. J. & Tarabek, M. A.: On smooth solutions of the Cauchy problem in one-dimensional nonlinear thermoelasticity. Quart. Appl. Math. 47 (1989), 631-644. · Zbl 0692.73005 [6] Hrusa, W. J. & Messaoudi, S. A.: On formation of singularities in in one-dimensional nonlinear thermoelasticity. Arch. Rational Mech. Anal. 111 (1990), 135-151. · Zbl 0712.73023 · doi:10.1007/BF00375405 [7] Ikawa, M.: Mixed problems for hyperbolic equations of second order. J. Math. Soc. Japan 20 (1968), 580-608. · Zbl 0172.14304 · doi:10.2969/jmsj/02040580 [8] Jiang, S.: Global existence of smooth solutions in one-dimensional nonlinear thermoelasticity. Proc. Roy. Soc. Edinburgh 115A (1990), 257-274. · Zbl 0723.35044 [9] Jiang, S.: Global solutions of the Dirichlet problem in one-dimensional nonlinear thermoelasticity, Preprint 138, SFB 256, Univ. Bonn (1990). · Zbl 0723.35044 [10] Jiang, S. & Racke, R.: On some quasilinear hyperbolic-parabolic initial boundary value problems. Math. Meth. Appl. Sci. 12 (1990), 315-339. · Zbl 0706.35098 · doi:10.1002/mma.1670120404 [11] Kawashima, S.: Systems of a hyperbolic-parabolic composite type, with applications to the equations of magnetohydrodynamics. Thesis, Kyoto Univ. (1983). [12] Kawashima, S. & Okada, M.: Smooth global solutions for the one-dimensional equations in magnetohydrodynamics. Proc. Jap. Acad. 53, Ser. A (1982), 384-387. · Zbl 0522.76098 [13] Ponce, G. & Racke, R.: Global existence of solutions to the initial value problem for nonlinear thermoelasticity. J. Diff. Eqns. 87 (1990), 70-83. · Zbl 0725.35065 · doi:10.1016/0022-0396(90)90016-I [14] Racke, R.: Blow-up in nonlinear three-dimensional thermoelasticity. Math. Meth. Appl. Sci. 12 (1990), 267-273. · Zbl 0705.35081 · doi:10.1002/mma.1670120308 [15] Racke, R.: On the Cauchy problem in nonlinear 3-d thermoelasticity. Math. Z. 203 (1990), 649-682. · Zbl 0701.73002 · doi:10.1007/BF02570763 [16] Seeley, R. T.: Integral equations depending analytically on a parameter. Indag. Math. 24 (1962), 434-442. · Zbl 0106.08102 [17] Shibata, Y.: On the global existence of classical solutions of mixed problems forsome second order non-linear hyperbolic operators with dissipative term in the interior domain. Funk. Ekva. 25 (1982), 303-345. · Zbl 0524.35070 [18] Shibata, Y.: On the global existence of classical solutions of second order fully nonlinear hyperbolic equations with first order dissipation in the exterior domain. Tsukuba J. Math. 7 (1983), 1-68. · Zbl 0524.35071 [19] Shibata, Y. & Tsutsumi, Y.: On a global existence theorem of small amplitude solutions for nonlinear wave equations in an exterior domain. Math. Z. 191 (1986), 165-199. · Zbl 0592.35028 · doi:10.1007/BF01164023 [20] Slemrod, M.: Global existence, uniqueness, and asymptotic stability of classical smooth solutions in one-dimensional nonlinear thermoelasticity. Arch. Rational Mech. Anal. 76 (1981), 97-133. · Zbl 0481.73009 · doi:10.1007/BF00251248 [21] Zheng, S. & Shen, W.: Global solutions to the Cauchy problem of a class of quasilinear hyperbolic parabolic coupled systems. Sci. Sinica, Ser. A, 30 (1987), 1133-1149. · Zbl 0649.35013
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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2021-05-08 06:45:56
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https://mathinstitutes.org/highlights?page=6
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### Research Highlights
#### Waves, Kakeya sets, and Diophantine equations
MSRI - June 2017
Ciprian Demeter A central problem in Physics is to understand the complicated ways in which waves can interact with one another. The field of Harmonic Analysis grew out of the observation that any natural signal that is periodic in time can be built by superimposing simple waves with whole number frequencies. In geometric optics, quantum mechanics, and the study of water waves, we must...
#### SAMSI Brings Astronomers and Statisticians Together to Study Universe
SAMSI - June 2017
Contributed by: Jim Barrett, Ph.D. student, School of Physics & Astronomy, University of Birmingham, UK Maya Fishbach, Ph.D. student, Department of Astronomy and Astrophysics, University of Chicago, USA Bo Ning, Ph.D. candidate, Department of Statistics, North Carolina State University, USA Daniel Wysocki, Ph.D. student, School of Physics & Astronomy’s Astrophysical Sciences & Technology...
#### Visualizing PML
ICERM - September 2016
Visualizing PML David Dumas and François Guéritaud On the surface of a sphere, every simple closed curve (that is, a curve that starts and ends at the same point and which does not cross itself) forms the boundary of a disk, i.e. a contiguous region without holes. In this sense, the sphere has only one “type” of simple closed curve. In contrast, on a surface with a more complicated...
#### Special Year on Geometric Structures on 3-Manifolds
IAS - September 2016
During the 2015-16 academic year, the School of Mathematics conducted a special program on Geometric Structures on 3-Manifolds. The program was led by Distinguished Visiting Professor Ian Agol of the University of California at Berkeley. The theme of the program was classication of geometric structures on 3-manifolds. Twenty members took part in the program. Senior members included David Gabai,...
#### NET Maps
AIM - July 2016
It is human nature to try to classify things—that is, to sort them into organized types. Many of the central problems in mathematics are problems of classification of various types of related mathematical objects. The classification of finite groups, for example, was a landmark accomplishment of the last century, and the classification of manifolds continues to challenge topologists. The AIM...
#### Big Data meets Number Theory
ICERM - May 2016
Researchers from ICERM’s special semester “Computational Aspects of the Langlands Program” are creating new data-driven models for collaborative research in number theory, culminating in the May 10, 2016 official release of the L-functions and Modular Forms Database (LMFDB) at www.lmfdb.org. Computation is not new to number theory – in Babylon huge tablets of sines and cosines were created and...
#### Identifying Links Between the S&P500 and VIX Derivatives
IPAM - April 2016
By Andrew Papanicolaou The technique of volatility trading has been common practice since the 1970’s. Typically, a long (short) position in volatility included a long (short) position in options. In 2003 there came a more standardized way of trading volatility, as the VIX formula became the universally-accepted predictor of volatility. The VIX is a 30-day predictor of volatility given by a...
#### Overcoming the Curse of Dimensionality for Control Theory
IPAM - November 2015
Optimal control problems lead to Hamilton-Jacobi Bellman (HJB) differential equations in many space variables for finding the cost function to be optimized. This beautiful connection has not generally led to effective numerical methods because grid based solutions of partial differential equations in $$n$$ variables generally have memory requirements and complexity that is exponential in \( n...
#### Illumination and Security
MSRI - October 2015
ALEX WRIGHT AND KATHRYN MANN Imagine that you are in a room with walls made out of mirrors. The room may be very oddly shaped and have corridors and nooks, but all the walls are at mirrored planes. If you light a single lamp, must every point in the room be illuminated? Perhaps surprisingly, the answer to this problem is “no”, and the reasons are connected with deep mathematical problems whose...
#### Limits of Permutations
ICERM - October 2015
In a well-shuffled deck of cards, about half of the pairs of cards are out of order. Mathematically, we say that in a permutation $\pi$ of $[n]=\{1,2,\dots,n\}$ there are about $\frac12\binom{n}{2}$ inversions, that is, pairs $i<j$ for which $\pi(j)<\pi(i)$. Suppose we are interested in studying permutations for which the number...
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2022-05-29 00:11:58
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https://leanprover-community.github.io/archive/stream/113489-new-members/topic/list.20comprehensions.html
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## Stream: new members
### Topic: list comprehensions
#### Alexandre Rademaker (Jul 01 2020 at 01:43):
do we have any support for list comprehension in Lean? Something similar to Haskell with support to guards?
#### Bryan Gin-ge Chen (Jul 01 2020 at 01:53):
I don't think so, but you can use do notation:
def foo (n : ℕ) : list (list (ℕ × ℕ)) :=
do j <- list.range n,
return (do i <- [1,2],
return (i,j))
#eval foo 10
#### Alexandre Rademaker (Jul 01 2020 at 02:10):
Cool! I didn't know about the do notation. Thank you.
#### Alexandre Rademaker (Jul 01 2020 at 02:12):
The TPinL does not mention it.
#### Bryan Gin-ge Chen (Jul 01 2020 at 02:14):
There's some discussion about it in chapter 6 of the Hitchhiker's guide to logical verification: https://leanprover-community.github.io/learn.html#textbooks
Last updated: May 16 2021 at 21:11 UTC
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2021-05-16 21:28:09
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http://msp.org/apde/2017/10-4/p05.xhtml
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Vol. 10, No. 4, 2017
Recent Issues
The Journal Cover About the Cover Editorial Board Editors’ Interests About the Journal Scientific Advantages Submission Guidelines Submission Form Subscriptions Editorial Login Contacts Author Index To Appear ISSN: 1948-206X (e-only) ISSN: 2157-5045 (print)
Improving Beckner's bound via Hermite functions
Paata Ivanisvili and Alexander Volberg
Vol. 10 (2017), No. 4, 929–942
Abstract
We obtain an improvement of the Beckner inequality $\parallel f{\parallel }_{2}^{2}-\parallel f{\parallel }_{p}^{2}\le \left(2-p\right)\parallel \nabla f{\parallel }_{2}^{2}$ valid for $p\in \left[1,2\right]$ and the Gaussian measure. Our improvement is essential for the intermediate case $p\in \left(1,2\right)$, and moreover, we find the natural extension of the inequality for any real $p$.
Keywords
Poincaré inequality, log-Sobolev inequality, Sobolev inequality, Beckner inequality, Gaussian measure, log-concave measures, semigroups, Hermite polynomials, Hermite differential equation, confluent hypergeometric functions, Turán's inequality, error term in Jensen's inequality, phi-entropy, phi-Sobolev, F-Sobolev, phi-divergence, information theory, backwards heat, Monge–Amperè with drift, exterior differential systems
Mathematical Subject Classification 2010
Primary: 42B37, 52A40, 35K55, 42C05, 60G15
Secondary: 33C15, 46G12
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2017-08-23 23:08:32
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https://proofwiki.org/wiki/Definition:Finite_Subdivision
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# Definition:Subdivision (Real Analysis)/Finite
## Definition
Let $\left[{a \,.\,.\, b}\right]$ be a closed interval of the set $\R$ of real numbers.
Let $x_0, x_1, x_2, \ldots, x_{n - 1}, x_n$ be points of $\R$ such that:
$a = x_0 < x_1 < x_2 < \cdots < x_{n - 1} < x_n = b$
Then $\left\{{x_0, x_1, x_2, \ldots, x_{n - 1}, x_n}\right\}$ form a finite subdivision of $\left[{a \,.\,.\, b}\right]$.
### Normal Subdivision
$P$ is a normal subdivision of $\left[{a \,.\,.\, b}\right]$ if and only if:
the length of every interval of the form $\left[{x_i \,.\,.\, x_{i + 1} }\right]$ is the same as every other.
That is, if and only if:
$\exists c \in \R_{> 0}: \forall i \in \N_{< n}: x_{i + 1} - x_i = c$
## Also known as
Some sources use the term partition for this, but the latter term has an alternative and more general definition so it is probably better not to use it.
Some sources do not define the concept of infinite subdivision, and so simply refer to a finite subdivision as a subdivision.
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2019-09-16 04:34:24
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https://www.physicsforums.com/threads/linear-combination-proof-of-orthonormal-basis.837213/
|
# Linear Combination Proof of Orthonormal basis
Gold Member
## Homework Statement
Assume that $(|v_1>, |v_2>, |v_3>)$ is an orthonormal basis for V. Show that any vector in V which is orthogonal to $|v_3>$ can be expressed as a linear combination of $|v_1>$ and $|v_2>$.
## Homework Equations
Orthonormality conditions:
|v_i>*|v_j> = 0 if i≠j OR 1 if i=j.
## The Attempt at a Solution
[/B]
I dont know how to mathematically prove this obvious one.
I understand the definitions here.
Orthonormal basis implies that the set of three vectors lives in dimension three and are all orthogonal to one another. This means the |v_1> *|v_2> = 0.
If some vector is orthogonal to |v_3> that means |x> |v_3> = 0.
I am pretty sure we have to use sum notation for an inner product and orthonormality conditions to prove this statement.
Hints on starting out properly?
Here's the only lead I have:
|x> = c*|v_1> + t*|v_2>
as it is a linear combination of v_1 and v_2. But we don't want to assume the proof.
Mark44
Mentor
## Homework Statement
Assume that $(|v_1>, |v_2>, |v_3>)$ is an orthonormal basis for V. Show that any vector in V which is orthogonal to $|v_3>$ can be expressed as a linear combination of $|v_1>$ and $|v_2>$.
## Homework Equations
Orthonormality conditions:
|v_i>*|v_j> = 0 if i≠j OR 1 if i=j.
## The Attempt at a Solution
[/B]
I dont know how to mathematically prove this obvious one.
I understand the definitions here.
Orthonormal basis implies that the set of three vectors lives in dimension three and are all orthogonal to one another. This means the |v_1> *|v_2> = 0.
If some vector is orthogonal to |v_3> that means |x> |v_3> = 0.
I am pretty sure we have to use sum notation for an inner product and orthonormality conditions to prove this statement.
Hints on starting out properly?
Here's the only lead I have:
|x> = c*|v_1> + t*|v_2>
as it is a linear combination of v_1 and v_2. But we don't want to assume the proof.
It helps to think of the geometry here. Since ##v_1, v_2## and ##v_3## form a basis for V, then they span V. If ##v \in V## is orthogonal to ##v_3##, then it must lie in the two-dimensional subspace of V spanned by ##v_1## and ##v_2##; i.e., the plane that is spanned by these two vectors. At this point, I would draw a sketch, showing the plane of ##v_1## and ##v_2##, with ##v_3## sticking straight out of the plane. Recognizing that ##v_1## and ##v_2## lie in this plane, and are a basis for that two-D subspace of V, then v must be a linear combination of ##v_1## and ##v_2##. When you have a handle on the geometry, it should be easier to write a proof that fleshes out the details.
RJLiberator
HallsofIvy
Homework Helper
Since v1, v2, and v3 are a basis, any vector, v, can be written as v= av1+ bv2+ cv3. Now, take the inner product of both sides with v.
RJLiberator
Gold Member
Thanks guys, what you have stated I understand.
Taking the starting point that hallsofivy has gifted me, it's clear to see that c = 0, and a and b can be any numbers. This essentially is the proof.
HallsofIvy suggests taking the inner product of both sides with v.
LHS: <v|v> = |v|^2
RHS: <v|av_1> + <v|bv_2> + <v|cv_3>
a*<v|v_1>+b*<v|v_1>+c*<v|v_3> = |v|^2
I'm not really sure what this proves, is this not assuming the proof?
Mark44
Mentor
Thanks guys, what you have stated I understand.
Taking the starting point that hallsofivy has gifted me, it's clear to see that c = 0, and a and b can be any numbers. This essentially is the proof.
HallsofIvy suggests taking the inner product of both sides with v.
LHS: <v|v> = |v|^2
RHS: <v|av_1> + <v|bv_2> + <v|cv_3>
a*<v|v_1>+b*<v|v_1>+c*<v|v_3> = |v|^2
I'm not really sure what this proves, is this not assuming the proof?
What else are you given in this problem? There is information given that you aren't using.
Your last line should be an equation showing that v is a linear combination of ##v_1## and ##v_2##.
RJLiberator
Gold Member
Is it really as simple as stating that c must = 0 by the definition of orthogonal, and so low and behold, v must be a linear combination of $v_1$ and $v_2$ ?
Why do we need the inner product of both sides?
Can we not just jump immediately to that step? C = 0, so linear combination is seen.
Mark44
Mentor
Is it really as simple as stating that c must = 0 by the definition of orthogonal
No, there's nothing that says that c = 0.
What else are you given in this problem?
RJLiberator said:
, and so low and behold, v must be a linear combination of $v_1$ and $v_2$ ?
Why do we need the inner product of both sides?
It's convenient and helpful for reaching the conclusion.
RJLiberator said:
Can we not just jump immediately to that step? C = 0, so linear combination is seen.
No.
RJLiberator
Gold Member
What else are you given in this problem?
Is the piece of information that you are alluding to the fact that the set is an orthonormal basis?
Gold Member
Okay, so after taking the inner product, it is clear to see that the inner product of <v|v_3> = 0 because of the definition of orthogonality.
And now we have
<v|v> = a<v|v_1>+b<v|v_2>
Which shows that it is a linear combination of the other two vectors.
HallsofIvy
Homework Helper
Actually, I miswrote. I meant to say "take the dot product of each side with v1, v2, and v3 in turn".
RJLiberator
Gold Member
Ah, now this is the grove I needed.
Taking the dot product in turn shows what |v_i> * |v> is clearly due to the orthogonal and orthonormal rules.
It can be seen that |v_3>|v> must be 0 due to definition of orthogonality and so c=0.
|v_2>|v> = b
|v_1>|v> = a
and so the answer seems clear. :D
With c = 0, all you need to do is go back to basis, plug it in and we have a linear combination representation of v.
Mark44
Mentor
Ah, now this is the grove I needed.
Taking the dot product in turn shows what |v_i> * |v> is clearly due to the orthogonal and orthonormal rules.
It can be seen that |v_3>|v> must be 0 due to definition of orthogonality
More to the point, ##v_3 \cdot v = 0##, because this is given.
RJLiberator said:
and so c=0.
|v_2>|v> = b
|v_1>|v> = a
and so the answer seems clear. :D
With c = 0, all you need to do is go back to basis, plug it in and we have a linear combination representation of v.
A simpler proof:
##v = av_1 + bv_2 + cv_3##, since ##\{v_1, v_2, v_3\}## is a basis for V
##\Rightarrow cv_3 = v - av_1 - bv_2##
##\Rightarrow cv_3 \cdot v_3 = v \cdot v_3 - av_1 \cdot v_3 - bv_2 \cdot v_3##
##\Rightarrow c = 0 - 0 - 0 = 0## Can you give a justification for each of the four dot products?
Hence ##v = av_1 + bv_2##, as required.
RJLiberator
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2021-12-08 23:18:15
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http://courses.csail.mit.edu/6.854/16/Notes/n9-advanced_flow_algorithms.html
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\documentclass{article} \usepackage{me} \setlength{\parindent}{0pt}
\begin{center} Lecture 9: Advanced Max Flow Algorithm \end{center}
## Different Variants of the Max Flow Problem
So far, we were focused on the “canonical” maximum flow problem formulation. In some of the applications though, one needs to solve different variants of this problem. Amazing thing about the maximum flow problem is that many of these, seemingly, more general variants can be reduced to the “canonical” problem.
Some examples:
• (Multiple source-sinks maximum flow) We have multiple sources $s_1, \ldots, s_k$ and multiple sinks $t_1, \ldots, t_{k'}$ and just want to push as much flow as possible between them.
• Reduces to canonical maximum flow by adding a “super-source” $s$ and “super-sink” $t$ and arcs with unbounded (or just $nU$) capacity from $s$ to each $s_i$ and from each $t_i$ to $t$.
• Computing maximum flow in the new graph corresponds directly to solving the multiple source-sink version.
• (Multiple source-sinks maximum flow with demands) Again, we have multiple sources and sinks, but now we want each source $s_i$ to supply a prescribe amount $d_i^+$ of flow and each sink $t_j$ to demand a prescribed amount $d_j^-$ of flow. (Assuming that $\sum_i d_i^+ = \sum_j d_j^-$.)
• Do the same reduction as before but make the capacity of the edge connecting $s$ to $s_i$ be $d_i^+$ and the capacity of the edge connecting $t_j$ to $t$ be $d_j^-$.
• Send flow $f'$ along the lower bounds.
• This makes them be satisfied but creates new residual arcs and demands.
• Solve the resulting multiple source-sink with demands in the residual graph, obtaining a solution $\hat{f}$.
• The final solution is the flow $f:=f'+\hat{f}$. Note: that we are using linearity of the flows here. That is, that adding a feasible flow $f'$ in a graph to a flow that is feasible in the residual graph of $f'$ gives us a feasible flow in the original graph.
• (Bipartite matching problem) Let $G=(V,E)$ be an undirected graph that is bipartite, i.e., $V=S\cup T$, such that $S\cap T=\emptyset$ and $E\subseteq S\times T$. We want to find a maximum cardinality matching $M$ in $G$. That is, a subset $M$ of edges such that no two edges in it share an endpoint.
• Setup a graph $G'$ which is a copy of $G$ such that: we make each original edge have capacity $1$ and be oriented towards the vertex in $T$ (important!), and then we add a super-source $s$ (resp. super-sink $t$) that is connected via directed unit-capacity edges to each vertex in $S$ (resp. there is a directed unit-capacity edge from each vertex in $T$ to $t$).
• One can show that in the maximum integral $s$-$t$ flow in this graph (which we know always exists) all the original edges that flow non-zero amount of flow form a maximum matching.
• The above shows us that one can reduce bipartite matching problem to the maximum flow problem. Can one go the other way too? Yes! (But we will not cover that.)
• (Vertex capacities) Find a maximum $s$-$t$ flow in a graph where we do not have any edge capacities but we cap the total flow flowing through each vertex $v$ at $u_v$.
• Transform the graph by splitting every vertex $v$ into two vertices $v_{in}$ and $v_{out}$, connecting all the original incoming edges to $v_{in}$ and all the original outgoing edges to $v_{out}$, and then adding an edge $(v_{in}, v_{out})$ with capacity $u_v$.
• Finding an maximum $s$-$t$ flow in this new graph (with respect to edge capacities) gives us the maximum $s$-$t$ flow in the original graph that is feasible with respect to vertex capacities.
## Strongly Polynomial Max Flow Algorithm
• So far, our crowning achievement was an $O(m^2 \log U)$ maximum flow algorithm that we got via scaling. Still, even though this running time is polynomial in the size of our graph and the representation of its capacities, i.e., weakly polynomial, one could aim for strongly polynomial running time. That is, a running time that does not depend at all on $U$, just on the number of vertices and edges of the graph. (We assume here that elementary arithmetic operations on numbers can be performed in $O(1)$ time.)
• How to get such an algorithm?
• Recall: Our algorithms so far were “primal” greedy. That is, they applied a greedy strategy in which we improve our current solution in each step and then lower bounded the progress made by each such step in terms of the value of the max flow.
• Intuition: As the value of the max flow directly depends on the value of our capacities, this kind of “primal” greedy approach is inherently incapable of delivering strongly polynomial bounds.
$\Rightarrow$ We need a different way to measure our progress towards optimality.
• Recall: The way we certified the optimality of our current flow $f$ was by looking whether $s$ and $t$ are connected in $G_f$. If not, $f$ was already a max flow; otherwise, it was not optimal (and the $s$-$t$ path enables us to improve it).
• Can we make this yes/no statement quantitative to also be able to differentiate between the flows that are “almost maximum” and the ones that might be “far” from being maximum?
• Idea: Let us consider the $s$-$t$ distance $d_f(s,t)$ in the residual graph $G_f$. (We assume here that each edge with positive residual capacity has length of one, and every edge with zero residual capacity has length $+\infty$.)
• Note: Any feasible flow in $G_f$ has to have a fixed total volume (at most $m$, if all capacities are $1$).
$\Rightarrow$ If $s$ and $t$ far apart, not much flow can “fit” in network, as each flow path has to “utilize” a lot of volume. (Think: an $s$-$t$ path graph vs. a graph consisting of multiple parallel $(s,t)$ edges.)
• Key observation: If $d_f(s,t)\geq n$, $s$ and $t$ are disconnected in $G_f$.
$\Rightarrow$ $f$ is already a maximum flow.
Our New Goal: Design an augmenting path based algorithm that aims to increase the $s$-$t$ distance $d_f(s,t)$ in the residual graph. (Instead of trying to directly push as much flow "as possible” in each iteration - which was the case before in the maximum bottleneck capacity augmenting path algorithm.)
• What are the best augmenting paths to use if we are interested in increasing $d_f(s,t)$?
• Intuition: Shortest (augmenting) paths are the "obstacles" to $d_f(s,t)$ being large. So, let's try to "destroy" them.
$\Rightarrow$ How to "destroy" an augmenting path in $G_f$? Augment the flow using it!
• Note: Finding the shortest augmenting path correspond to running BFS in the residual graph. So, it take $O(m)$ time. (The same as the "normal" augmenting path search.)
• Challenge: How to ensure that this augmentation does not introduce new shortest paths in $G_f$? In principle, this might even reduce $d_f(s,t)$ instead of increasing it.
• Fortunately, this cannot be the case. But we need to prove that!
• Lemma: For any vertex $v$, if $d(v)$ (resp. $d'(v)$ is the distance from $s$ to $t$ in the residual graph before (resp. after) augmenting the flow along some shortest augmenting path, then $d(v)\leq d'(v)$.
• So, the distance from $s$ does not decrease not only for $t$ but for every vertex. (Note that the proof relies on establishing this stronger claim - it is one of the examples when sometime proving a stronger claim is actually easier.)
• By symmetry, one can argue that the distance from any vertex $v$ to $t$ is non-decreasing as well.
• Proof:
• Assume for the sake of contradiction that this is not the case.
• Let $A$ be the (non-empty) set of vertices $u$ for which $d(u)>d'(u)$. Take $v$ to be the vertex with minimal $d'(v)$ among all vertices in $A$.
• Let $P'$ be the shortest $s$-$v$ path in the residual graph after augmentation, and let $w$ be the vertex preceding $v$ on this path. (Note that we cannot have that $v=s$, so such path and $w$ exist.)
• Note $d'(v)=d'(w)+1$. Moreover, we must have that $d(w)\leq d'(w)$ as otherwise $w\in A$ and $d'(w)< d'(v)$, which would contradict minimality of $v$.
• We claim that the last edge of $P'$, i.e., $(w,v)$, had zero residual capacity before augmentation by $P$. Otherwise, we would have that $d(v)\leq d(w)+1 \leq d'(w)+1 = d'(v),$ and thus contradict the fact that $v\in A$.
• The only way for the edge $(w,v)$ to have non-zero residual capacity after augmentation by $P$ would be if the edge $(v,w)$ belonged to $P$.
• But $P$ was the shortest path before the augmentation.
$\Rightarrow$ $d(w)=d(v)+1$.
• However, all of that means that $d(v)=d(w)-1\leq d'(w)-1=d'(v)-2\leq d'(v),$ which contradicts our assumption that $v\in A$.
• So, augmenting the flow using shortest paths indeed does not make things worse. But does it make them better?
• Yes! But, again, we need to prove that.
• Lemma: We have at most $\frac{mn}{2}$ shortest path flow augmentations before $d_f(s,t)\geq n$ (and thus $f$ is the maximum flow).
• Proof:
• Each flow augmentation saturates at least one "bottlenecking" edge $(u,v)$.
• Before this edge is used saturated again in some subsequent flow augmentation, we must have pushed some flow via an augmenting path that contained the opposite edge $(v,u)$.
• Let $d(w)$ be the distance of $s$ to $w$ in the residual graph just before the first saturation of $(u,v)$, and let $d'(w)$ be the corresponding distance just before the flow was pushed along $(v,u)$.
• As we always use shortest paths to augmenting flow we need to have that $d(v)=d(u)+1$ and $d'(u)=d'(v)+1$.
• But, by the lemma we proved above, we know that $d(v)\leq d'(v)$.
$\Rightarrow$ We thus need to have that $d'(u)=d'(v)+1\geq d(v)+1 = d(u)+2.$
$\Rightarrow$ The distance from $s$ to $u$ had to increased by at least $2$ by the time the edge $(u,v)$ can again be saturated by some augmenting path.
$\Rightarrow$ $(u,v)$ can be saturated at most $\frac{n}{2}$ times before the $s$-$u$ distance becomes $\geq n$ and thus $d_f(s,t)\geq n$ as well.
• Each edge can be saturated at most $\frac{n}{2}$ times. So, there is at most $\frac{mn}{2}$ saturations and thus flow augmentations.
• Total running time: $O(m^2 n)$. Strongly polynomial!
• Note: In this analysis we have no way of lower bounding how much flow a particular flow augmentation pushed. We just can argue that over all the $\frac{mn}{2}$ augmentations we managed to push the whole max flow value (no matter how large it was!). This is an important feature of so-called primal-dual algorithms. (Will get back to this later on.)
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2018-01-16 21:07:58
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https://cugelaqucetykula.wrcch2016.com/tube-domains-and-the-cauchy-problem-book-2029mz.php
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# Tube domains and the Cauchy problem by S. G. Gindikin
Written in English
## Subjects:
• Cauchy problem.,
• Differential operators.
Edition Notes
Includes bibliographical references (p. 125-127) and indexes.
## Book details
Classifications The Physical Object Statement Simon Gindikin ; [translated from the Russian by Senya Shlosman ; translation edited by Sergei Gelfand]. Series Translations of mathematical monographs ;, v. 111 LC Classifications QA377 .G5313 1992 Pagination v, 132 p. ; Number of Pages 132 Open Library OL1716909M ISBN 10 0821845667 LC Control Number 92019406
Tube domains and the Cauchy problem. [S G Gindikin] Home. WorldCat Home About WorldCat Help. Search. Search for Library Items Search for Lists Search for Contacts Search for a Library.
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2021-08-01 23:22:06
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https://chat.stackexchange.com/transcript/41/2021/10/27
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3:54 AM
I have an unrelated question. In that IF statement, I want to say if there exists a v in \hat{V} such that ... and then I used the value of m that I expanded from v
I am working on my thesis, is it clear? if not how to express it more clearly
4:10 AM
@Skillmon He only pronounced the infection already having existed in the team :) For one, these little fellas predate him:
5
4
3 hours later…
6:55 AM
@yo' I spot an Analândia duck ;-)
@UlrikeFischer oh definitely :) there's more @PauloCereda ducks, but the Overleaf ones are not of them :)
7:27 AM
I will just leave this video of a bodyguard cat here: youtu.be/XqrZo7zU-oc (it is a meme video, 15s, no idea where the original is from)
8:20 AM
@yo' <3 <3 <3
@UlrikeFischer ooh
@yo' ooh quack :)
@DavidCarlisle ooh geometry
8:41 AM
8:58 AM
[2/4, 00:02/00:08] update: huawei [585k] (60501 -> 60873) ... done
ooh modem updated
9:39 AM
I have a query on using syntax highlighting. Is it okay to use syntax highlighting of source code snippets in academia, in scientific papers? Or should it be kept in the black text?
Where can I ask this query? Can anyone suggest any SE site or chatroom?
@raf if by syntax highlighting you mean colouring your code, I am inclined to say this is not a good idea. :) But bold and italics are usually acceptable.
@PauloCereda yes, I meant "colouring the code"
@raf Ah, in that case I'd not encourage you to do it. :)
But that's just me, of course. :)
and what about showing the line numbers?
Actually, this is the style and color I was thinking of using:
https://www.overleaf.com/learn/latex/Code_listing#Code_styles_and_colours
@raf I like line numbers. :) But I usually use them when it's relevant to the context, e.g, mentioning something that happens at line 4, otherwise lines might not make sense. :)
9:49 AM
In my context, I won't need to explain the code, I just like put them in my appendix.
so, do you recommend using the default style of listings?
or can you please show me a short mwe which is acceptable to you?
I just found this: arxiv.org/abs/2001.11334
10:04 AM
The reason for my query: Next month, I have a contest, where I need to write a formal scientific paper on a given physics problem. And, I am gonna put my used source code snippets in the appendix of my paper. In this context, I am confused if I should keep the syntax coloring or not!
@raf Talk to the organisers to see of there is a style guide you have to follow. (if there is none, make sure your paper is still accessible to people who are colour blind or print it in black and white)
@raf ^^ what @samcarter said. :) I usually favour bold/italics instead of colours, but again, that's just my style. One thing to keep in mind is whether colours could distract readers. If you think it wouldn't, go for it. :)
@samcarter no, the organizers haven't given the style guide to follow.
Thank you, @samcarter and @PauloCereda
Considering your opinions, I have decided to keep black and white.
10:32 AM
@raf I don't know how formal your contest is, but I wouldn't be surprised of some/most participants would just use their IDE to print the code as pdf and include this in their appendix ...
@samcarter I agree. There's a chance that many participants would do that. because no specific guideline is provided. But I want to keep the paper format formal as much as possible.
@raf my thesis had some code listings, and one of the reasons for keeping it b/w (besides the ones I mentioned before) is that printing a coloured page would cost me a lot more, so there's that. :)
@PauloCereda I see. In my case, I won't to print the paper. Just need to submit it online.
In this same context, do you suggest avoiding \hypersetup{colorlinks=true} in my paper?
10:51 AM
@raf I'd rather not comment on using hyperref. :)
@PauloCereda :)
why?
11:12 AM
@raf do you submit to a journal?
@Skillmon no, it's for a contest.
but the participants are told to produce a formal scientific paper.
@raf then you shouldn't use colorlinks. I don't recall ever seeing a scientific paper with coloured links (because most of the time those are also for print).
Okay, thank you. @Skillmon
Actually, reading this paper (arxiv.org/abs/2001.11334) in favor of using syntax highlighting made me confused today.
11:31 AM
@Skillmon Different worlds :) I would say most of the journals I read use coloured links, e.g. A&A, MNRAS, ApJ
@samcarter well, different worlds indeed :)
@Skillmon MDPI uses blue, bold links for citations (that jump to a citation) and internal and external links. It's quite discrete and probably almost unnoticeable in print, but quite pleasant (IMHO) when read onscreen.
...if the ever print, which I don't know.
@raf looking back to my bachelors and masters thesis, the printed code included syntax highlighting (coloured), but I'd never use colour for text highlighting, there I'd use different fonts or different shapes.
I wanted to use:
\hypersetup{
citecolor=blue,
urlcolor=blue,
}
@raf maybe pick a darker blue not the pure one. Something like blue!65!black or similar. The pure blue looks a bit too bright for text, imho.
2
11:41 AM
ooh blue da ba dee da ba da
Great, now that song is stuck inside my head
@Skillmon :) okay.
11:53 AM
@Skillmon thank you. It looks better now. I am not aware of the syntax blue!65!black. What does it mean by '!65!'?
12:04 PM
@raf that is the xcolor syntax. And you could make a guess what it means if you combine blue and black through a number.
@UlrikeFischer oh okay. Thank you.
12:54 PM
@raf (I'm pro syntax colouring) I'm not sure if I would base a decision about using syntax colouring on that paper. It does not really give any reasons that syntax colouring is beneficial other than just saying it is good, no data or studies to support their claims and in their section about addressing the problems, they give some suggestions what can be done, but fail to show that these actually solve the problem.
@samcarter thank you, @samcarter
Also, keep in mind to use a colorscheme that reads well on most devices, as well as printing (and also covers colour blindness and other colour technicalities).
1:15 PM
@PauloCereda I think there was some package released a couple of months ago providing colours matching some of those technicalities (high enough contrast, etc.)
@Skillmon ooh that's cool!
2:08 PM
@DavidCarlisle The package author suggested that, apparently: tex.stackexchange.com/a/207313/134574
@PhelypeOleinik well it's not my package so not my fault
@DavidCarlisle Aye to that
@raf look at the journal you plan to publish in, does it use colour?
:59458861 the typewriter used for my thesis didn't do colour
yesterday, by David Carlisle
@samcarter yes hex colours are banned primary colours are what you need
2:28 PM
@PauloCereda but I don't remember the name :(
Great news: svg2tikz now works with inkscape 1.x github.com/xyz2tex/svg2tikz/pull/61
2:40 PM
@DavidCarlisle You should have gotten some coloured pencils for your wife
2
Jul 4 '13 at 17:09, by David Carlisle
@percusse In my thesis, I just got my wife to draw in the big brackets with a pen, she had a steadier hand than me.
@DavidCarlisle Solve a hack with a yet another hack :-)
@PhelypeOleinik well when in six years there will be a question with % HACK % HACK % HACK % HACK % HACK % HACK % HACK % HACK % HACK % HACK % we will be able to identify the source ;-)
@UlrikeFischer The "Proceed at your own risk" exempts me from any blame, so that's fine :)
@PhelypeOleinik Hic sunt leones
@UlrikeFischer Yup seems right
@UlrikeFischer Also Overleaf templates
@JosephWright Like that, except lions don't spread as fast as bad code
2:52 PM
@JosephWright ooh a lion
@UlrikeFischer Finding Nemo's seagulls meet coding. :)
@PauloCereda 🦁🦁🦁🦁🦁
@egreg ooh-nicode
@PauloCereda @yo' Feature idea for overleaf: if a users creates a new project from a template, could you automatically add a comment with the url to the template? This would make it easier to answer questions when the user just says "I'm using THE template from overleaf"
6
@samcarter Good idea
@samcarter Or at least some kind of ID: OL-2021-001, etc.
@JosephWright yes, it doesn't have to be the link, just some identification which of the trillion cv.cls it is
2:59 PM
@samcarter might be a no-no (it's not that easy to modify the code reliably at that stage of project creation), but I'll see what can be done while work on other gallery improvements :)
@yo' If not at the stage of project creation, then maybe at the template creation level?
@samcarter good point! Still there are non-easy things: It will at least add a line of code into the project...
@yo' <3
@samcarter I'm not that generous
@DavidCarlisle :( There are even duck themed pencils: im01.itaiwantrade.com/a45753d7-c0f0-49e6-9f34-9a7c98d4f88a/…
3:05 PM
@samcarter ooh
@samcarter ideal for writing out menus, no doubt
If I have a C program that compiles and I change it so it doesn't compile, can I blame @UlrikeFischer?
@DavidCarlisle oh no
@DavidCarlisle #define BLAME Ulrike
@DavidCarlisle Correct the code and then add a comment "blame Ulrike". That should allow it to compile again.
@UlrikeFischer thanks, seems to work
3:54 PM
@Skillmon Do you mean ninecolors? There also is pgfplots' colorbrewer library..
4:32 PM
@schtandard ooh
@UlrikeFischer @MarcelKrüger is this expected? tex.stackexchange.com/a/620414/1090
@DavidCarlisle well the mapping is needed, but why does the document disables it?
@DavidCarlisle ah, because of the \newfontfamily it is not enabled by default.
@UlrikeFischer ah not polyglossia, I just tried without, I'll remove that comment
@DavidCarlisle yes, and better use Ligatures=TeX, then it works also without lualatex without warning.
@UlrikeFischer but I thought fontspec added this by default
4:46 PM
@DavidCarlisle well it can't add to all fonts, then it would affect typewriter too. So imho it is only the default for fonts you declare with \setmainfont and \setsansfont.
@UlrikeFischer ah
3
@samcarter colored circuit with ninecolors ;-) (told ya...)
@UlrikeFischer so I put a knowledgable comment under my answer giving the impression I'd ever seen the fontspec documentation
@Rmano ooh
@DavidCarlisle ooh
5:16 PM
@PauloCereda do you think we should flag the answer to this as spam? tex.stackexchange.com/q/620405/1090
@DavidCarlisle ooh
5:29 PM
@schtandard yes, I think it was ninecolors, thank you!
@DavidCarlisle maybe not spam, but I agree, that answer is not particularly helpful. The only worse possibility now would be someone recommending the use of a bloated OS like Emacs.
Oh no my favourite candidate disappeared from the election :(
2 hours later…
8:12 PM
@Rmano sooooo pretty!
12
(comma correctly placed :) )
@samcarter congratulations
@DavidCarlisle thanks :)
@samcarter -- Whee!
8:27 PM
@barbarabeeton <3
@samcarter wow! Congratulations!
@Rmano thx!
2 hours later…
10:05 PM
3
^^^ some ducks are celebrating
10:18 PM
@UlrikeFischer Congratulations :) Will tomorrow be a city-wide holiday to recuperate from the celebrations?
10:34 PM
@samcarter they will probably need 5 days ;-)
@UlrikeFischer sounds more fun than 0 days :)
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2021-12-09 08:27:43
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http://www.elib.vn/de-thi-olympic-tieng-anh-lop-10-vong-33-de-so-1-nam-2014-467017.html
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# Đề thi Olympic Tiếng Anh lớp 10 vòng 33 đề số 1 năm 2014
Chia sẻ: Minh Minh | Ngày: | 1 đề thi
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## Mô tả BST Đề thi Olympic Tiếng Anh lớp 10 vòng 33 đề số 1 năm 2014
Xin giới thiệu đến các bạn học sinh và quý thầy cô giáo Đề thi Olympic Tiếng Anh lớp 10 vòng 33 đề số 1 năm 2014. Thông qua đề thi này, các em học sinh sẽ dễ dàng từng bước tiếp cận với dạng đề thi từ đó rút kinh nghiệm cho kĩ năng giải đề thi Olympic Tiếng Anh của mình. Chúc các em học sinh và quý thầy cô thành công!
LIKE NẾU BẠN THÍCH BỘ SƯU TẬP
## Tóm tắt Đề thi Olympic Tiếng Anh lớp 10 vòng 33 đề số 1 năm 2014
Dưới đây là đoạn tài liệu được trích trong BST Đề thi Olympic Tiếng Anh lớp 10 vòng 33 đề số 1 năm 2014:
Chọn câu trả lời tốt nhất cho những câu hỏi dưới đây.
1. Julie does not want to spend more than $300 on ice skating. Her skates will cost$42, her lessons will cost a total of $56, and the practice time will cost$7.50 per hour. Which inequality should Julie use to determine the maximum number of hours, h, she can practice without spending more than $300? A. 42 + 56 + 7.50h ≤ 100 B. 56 + 7.50h < 300 C. 7.50h − 42 − 56 ≤ 300 D. 42 + 7.50h < 300 E. 42 + 56 + 7.50h ≤ 300 2. Daniel cut the corner off a cube as shown in the diagram above. Points A, B and C are the midpoints of the edges of the cube. What type of threedimensional figure has been cut off? A. cone B. cube C. triangular prism D. triangular pyramid E. None of them 3. Pippi calculates her total earnings for the month with the equation E = 15m + 5b, where E is the total number of dollars she earns, m is the number of lawns she mows, and b is the number of hours she baby-sits. If Pippi mows 6 lawns, how many hours must she baby-sit to earn a total of$200?
A. 10 B. 20 C. 45 D. 40 E. 22
4. George wants to conduct a survey to determine the types of music that the students want at a school dance. Which sample population should George survey to represent the entire student body?
A. survey the teachers
B. survey the captain from each sports team
C. randomly survey two people from each homeroom class
D. randomly survey 50 people from the freshman class
E. randomly survey 50 people from the junior class
5. Gene has a cylinder with radius 4 inches and height 2 inches. He cut the cylinder in half along the length of the diameter, as shown in the diagram above. What is the area of the shaded cross-section?
A. 48π square inches
B. 8 square inches
C. 24π square inches
D. 12 square inches
E. 16 square inches
6. A system of equations is shown above. What is the solution to the system of equations?
A. x = 7, y = –1 B. x = 3, y = 5
C. x = 19, y = 1 D. x = 1, y = 1
E. x = 19, y = 2
7. Carlos and Tiesha empty a bag of 100 colored candies and count the number of each color, as shown in the chart shown above. They return all the candies to the bag and shake the bag. Carlos removes 5 candies, 2 of which are blue. Tiesha then pulls out one candy. What is the probability that Tiesha pulls out a blue candy?
A. 6.67% B. 19.9% C. 15.00% D. 15.79% E. 13.68%
8. Which of these represents the graph of the equation −3x + 4y = −12?
A. B.
C. D.
E. none of them
9. Alanis is moving and needs to pack two mirrors. The larger mirror fits in a box that is 18 inches wide by 20 inches long. Her smaller mirror is similar in proportion to the larger mirror. Alanis determines that the width of the smaller box needs to be a minimum of 9 inches. What should be the minimum length of the box to hold the smaller mirror?
A. 2 inches B. 6 inches C. 9 inches
D. 10 inches E. 12 inches
10. The table shows the number of people who speak each of the six most common languages of the world. Which type of graph is appropriate to display the data in the table?
A. bar graph
B. box-and-whisker plot
C. line graph
D. scatterplot
E. pie
Hãy tham khảo đầy đủ Đề thi Olympic Tiếng Anh lớp 10 vòng 33 đề số 1 năm 2014 bằng cách đăng nhập thư viện và download BST nhé!
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2018-02-24 16:13:50
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http://math-doc.ujf-grenoble.fr/cgi-bin/sps?kwd=Fractional+Brownian+motion&kwd_op=contains
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Browse by: Author name - Classification - Keywords - Nature
6 matches found
XXV: 33, 407-424, LNM 1485 (1991)
ROSEN, Jay S.
Second order limit laws for the local times of stable processes (Limit theorems)
Using the method of moments, a central limit theorem is established for the increments $L^x_t-L^0_t$ of the local times of a symmetric $\beta$-stable process ($\beta>1$). The limit law is that of a fractional Brownian sheet, with Hurst index $\beta-1$, time-changed via $L_t^0$ in its time variable
Comment: Another proof due to Eisenbaum 3120 uses Dynkin's isomorphism. Ray-Knight theorems for these local times can be found in Eisenbaum-Kaspi-Marcus-Rosen-Shi Ann. Prob. 28 (2000). A good reference on this subject is Marcus-Rosen, Markov Processes, Gaussian Processes, and Local Times, Cambridge University Press (2006)
Keywords: Local times, Stable processes, Method of moments, Fractional Brownian motion, Brownian sheet
Nature: Original
Retrieve article from Numdam
XXIX: 26, 266-289, LNM 1613 (1995)
EISENBAUM, Nathalie
Une version sans conditionnement du théorème d'isomorphisme de Dynkin (Limit theorems)
After establishing an unconditional version of Dynkin's isomorphism theorem, the author applies this theorem to give a new proof of Ray-Knight theorems for Brownian local times, and also to give another proof to limit theorems due to Rosen 2533 concerning the increments of the local times of a symmetric $\beta$-stable process for $\beta>1$. Some results by Marcus-Rosen (Proc. Conf. Probability in Banach Spaces~8, Birkhäuser 1992) on Laplace transforms of the increments of local time are extended
Comment: A general reference on the subject is Marcus-Rosen, Markov Processes, Gaussian Processes, and Local Times, Cambridge University Press (2006)
Keywords: Stable processes, Local times, Central limit theorem, Dynkin isomorphism, Fractional Brownian motion, Brownian sheet
Nature: Original
Retrieve article from Numdam
XXXI: 20, 216-224, LNM 1655 (1997)
EISENBAUM, Nathalie
Théorèmes limites pour les temps locaux d'un processus stable symétrique (Limit theorems)
Using Dynkin's isomorphism, a central-limit type theorem is derived for the local times of a stable symmetric process of index $\beta$ at a finite number $n$ of levels. The limiting process is expressed in terms of a fractional, $n$-dimensional Brownian sheet with Hurst index $\beta-1$. The case when $n=1$ is due to Rosen 2533, and, for Brownian local times, to Yor 1709
Comment: This kind of result is now understood as a weak form of theorems à la Ray-Knight, describing the local times of a stable symmetric process: see Eisenbaum-Kaspi-Marcus-Rosen-Shi Ann. Prob. 28 (2000) for a Ray-Knight theorem involving fractional Brownian motion. Marcus-Rosen, Markov Processes, Gaussian Processes, and Local Times, Cambridge University Press (2006) is a general reference on the subject
Keywords: Stable processes, Local times, Central limit theorem, Dynkin isomorphism, Fractional Brownian motion, Brownian sheet
Nature: Original
Retrieve article from Numdam
XLIII: 01, 3-70, LNM 2006 (2011)
PICARD, Jean
Representation formulae for the fractional Brownian motion (Theory of fractional Brownian motion)
Keywords: Fractional Brownian motion, Brownian motion
Nature: Original, Survey
XLIII: 08, 215-219, LNM 2006 (2011)
PRATELLI, Maurizio
A Remark on the $1/H$-variation of the Fractional Brownian Motion (Theory of fractional Brownian motion)
Keywords: Fractional Brownian motion, $p$-variation, Ergodic theorem
Nature: Exposition
XLIII: 09, 221-239, LNM 2006 (2011)
MAROUBY, Matthieu
Simulation of a Local Time Fractional Stable Motion (Theory of fractional Brownian motion, Theory of stable processes)
Keywords: Stable processes, Self-similar processes, Shot noise series, Local times, Fractional Brownian motion, Simulation
Nature: Original
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2013-05-19 06:42:28
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https://popl22.sigplan.org/details/POPL-2022-popl-research-papers/11/The-Decidability-and-Complexity-of-Interleaved-Bidirected-Dyck-Reachability
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Wed 19 Jan 2022 15:30 - 15:55 at Salon I - Algorithmic Verification 1 Chair(s): Qirun Zhang
Dyck reachability is the standard formulation of a large domain of static analyses, as it achieves the sweet spot between precision and efficiency, and has thus been studied extensively. Interleaved Dyck reachability (denoted $D_k\odot D_k$) uses two Dyck languages for increased precision (e.g., context and field sensitivity) but is well-known to be undecidable. As many static analyses yield a certain type of bidirected graphs, they give rise to interleaved bidirected Dyck reachability problems. Although these problems have seen numerous applications, their decidability and complexity has largely remained open. In a recent work, Li \textit{et al}. made the first steps in this direction, showing that (i)~$D_1\odot D_1$ reachability (i.e., when both Dyck languages are over a single parenthesis and act as counters) is computable in $O(n^7)$ time, while (ii)~$D_k\odot D_k$ reachability is NP-hard. However, despite this recent progress, most natural questions about this intricate problem are open.
In this work we address the decidability and complexity of all variants of interleaved bidirected Dyck reachability. First, we show that $D_1\odot D_1$ reachability can be computed in $O(n^3\cdot \alpha(n))$ time, significantly improving over the existing $O(n^7)$ bound. Second, we show that $D_k\odot D_1$ reachability (i.e., when one language acts as a counter) is decidable, in contrast to the non-bidirected case where decidability is open. We further consider $D_k\odot D_1$ reachability where the counter remains linearly bounded. Our third result shows that this bounded variant can be solved in $O(n^2\cdot \alpha(n))$ time, while our fourth result shows that the problem has a (conditional) quadratic lower bound, and thus our upper bound is essentially optimal. Fifth, we show that full $D_k\odot D_k$ reachability is undecidable. This improves the recent NP-hardness lower-bound, and shows that the problem is equivalent to the non-bidirected case. Our experiments on standard benchmarks show that the new algorithms are very fast in practice, offering many orders-of-magnitude speedups over previous methods.
#### Wed 19 JanDisplayed time zone: Eastern Time (US & Canada) change
15:05 - 16:20 Algorithmic Verification 1POPL at Salon I Chair(s): Qirun Zhang Georgia Institute of Technology 15:0525mResearch paper Efficient Algorithms for Dynamic Bidirected Dyck-ReachabilityRemotePOPLYuanbo Li Georgia Institute of Technology, Kris Satya Georgia Institute of Technology, Qirun Zhang Georgia Institute of Technology DOI Media Attached 15:3025mResearch paper The Decidability and Complexity of Interleaved Bidirected Dyck ReachabilityRemotePOPLAdam Husted Kjelstrøm Aarhus University, Andreas Pavlogiannis Aarhus University DOI Media Attached 15:5525mResearch paper Subcubic Certificates for CFL ReachabilityRemotePOPLDmitry Chistikov University of Warwick, Rupak Majumdar MPI-SWS, Philipp Schepper CISPA DOI Media Attached
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2022-09-27 04:27:40
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http://www.ams.org/mathscinet-getitem?mr=822452
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MathSciNet bibliographic data MR822452 57M99 Klassen, Eric An open book decomposition for ${\bf R}{\rm P}^2\times S^1$${\bf R}{\rm P}^2\times S^1$. Proc. Amer. Math. Soc. 96 (1986), no. 3, 523–524. Article
For users without a MathSciNet license , Relay Station allows linking from MR numbers in online mathematical literature directly to electronic journals and original articles. Subscribers receive the added value of full MathSciNet reviews.
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2017-04-27 22:29:01
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https://encyclopediaofmath.org/wiki/Vietoris_homology
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# Vietoris homology
One of the first homology theories (cf. Homology theory) defined for the non-polyhedral case. It was first considered by L.E.J. Brouwer in 1911 (for the case of the plane), after which the definition was extended in 1927 by L. Vietoris to arbitrary subsets of Euclidean (and even metric) spaces.
An (ordered) $n$- dimensional $\epsilon$- simplex $t ^ {n}$ of a subset $A$ of a metric space $X$ is defined as an ordered subset $( e _ {0} \dots e _ {n} )$ in $A$ subject to the condition $\mathop{\rm diam} \{ e _ {0} \dots e _ {n} \} < \epsilon$. The $\epsilon$- chains of $A$ are then defined for a given coefficient group $G$ as formal finite linear combinations $\sum g _ {i} t _ {i}$ of $\epsilon$- simplices $t _ {i} ^ {n}$ with coefficients $g _ {i} \in G$. The boundary of an $\epsilon$- simplex $t ^ {n} = ( e _ {0} \dots e _ {n} )$ is defined as follows: $\Delta t ^ {n} = \sum _ {i} (- 1) ^ {i} ( e _ {0} \dots {\widehat{e} } _ {i} \dots e _ {n} )$; this is an $\epsilon$- chain. By linearity, the boundary of any $\epsilon$- chain is defined and $\epsilon$- cycles are defined as $\epsilon$- chains with zero boundary. An $\epsilon$- chain $x ^ {n}$ of a set is $\eta$- homologous to zero in $A$( the notation is $x ^ {n} \sim 0$) if $x ^ {n} = \Delta y ^ {n+ 1 }$ for a certain $\eta$- chain $y ^ {n+ 1 }$ in $A$.
A true cycle of a set $A$ is a sequence $z ^ {n} = \{ z _ {1} ^ {n} \dots z _ {k} ^ {n} ,\dots \}$ in which $z _ {k} ^ {n}$ is an $\epsilon _ {k}$- cycle in $A$ and $\epsilon _ {k} \rightarrow 0$( $k \rightarrow \infty$). The true cycles form a group, $Z ^ {n} ( A, G)$. A true cycle $z$ is homologous to zero in $A$ if for any $\epsilon > 0$ there exists an $N$ such that all $z _ {k} ^ {n}$ for $k \geq N$ are $\epsilon$- homologous to zero in $A$. One denotes by $\Delta ^ {n} ( A, G)$ the quotient group of the group $Z ^ {n} ( A, G)$ by the subgroup $H ^ {n} ( A, G)$ of cycles that are homologous to zero.
A cycle $z$ is called convergent if for any $\epsilon > 0$ there exists an $N$ such that any two cycles $z _ {k} ^ {n}$, $z _ {m} ^ {n}$ are mutually $\epsilon$- homologous in $A$ if $k, m \geq N$. The group of convergent cycles is denoted by $Z _ {c} ^ {n} ( A, G)$. Let $\Delta _ {c} ^ {n} ( A, G) = Z _ {c} ^ {n} ( A, G) / H _ {c} ^ {n} ( A, G)$ be the corresponding quotient group.
A cycle $z$ has compact support if there exists a compact set $F \subseteq A$ such that all the vertices of all simplices of all cycles $z _ {k} ^ {n}$ lie in $F$. One similarly modifies the concept of a cycle being homologous to zero by requiring the presence of a compact set on which all the homology-realizing chains lie; convergent cycles with compact support can thus be defined. By denoting with a subscript $k$ the transition to cycles and homology with compact support, one obtains the groups $\Delta _ {k} ^ {n} ( A, G)$ and $\Delta _ {ck} ^ {n} ( A, G)$. The latter group is known as the Vietoris homology group. If the polyhedron is finite, the Vietoris homology groups coincide with the standard homology groups.
Relative homology groups $\Delta ^ {n} ( A, B, G)$, $\Delta _ {c} ^ {n} ( A, B, G)$, $\Delta _ {k} ^ {n} ( A, B, G)$, $\Delta _ {ck} ^ {n} ( A, B, G)$ modulo a subset $B \subseteq A$ are also defined. An $\epsilon$- cycle of the set $A$ modulo $B$ is any $\epsilon$- chain $x ^ {n}$ in $A$ for which the chain $\Delta x ^ {n}$ lies in $B$. In a similar manner, an $\epsilon$- cycle $x ^ {n}$ modulo $B$ is $\eta$- homologous modulo $B$ to zero in $A$ if $x ^ {n} = \Delta y ^ {n+} 1 + w ^ {n}$, where $y ^ {n+} 1$ and $w ^ {n}$ are $\eta$- chains in $A$, while the chain $w ^ {n}$ lies in $B$.
#### References
[1] P.S. Aleksandrov, "An introduction to homological dimension theory and general combinatorial topology" , Moscow (1975) (In Russian)
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2022-08-09 22:16:30
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https://math.stackexchange.com/questions/2245232/how-to-apply-de-finettis-exchange-theorem
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# How to apply De Finetti's exchange theorem?
I am working with the following integral :
$$I(u_1,\ldots,u_M)=\sum_{\sigma\in S_M}\bigg(\prod_{i=1}^j\int_{u_{\sigma(i)}}^1\mathrm{d}y_i\bigg)\bigg(\prod_{l=j+1}^M\int_0^{u_{\sigma(l)}}\mathrm{d}y_l\bigg)n(y_1\ldots y_M)\\ = \sum_{\sigma\in S_M}\bigg(\prod_{i=1}^j\int_{u_{\sigma(i)}}^1\mathrm{d}y_i\bigg)\prod_{l=j+1}^M\bigg(\int_0^1\mathrm{d}y_l-\int_0^{u_{\sigma(l)}}\mathrm{d}y_l\bigg)n(y_1\ldots y_M)$$
Where $M$ is an integer (after all the calculations I would like to take $M\to\infty$ $S_M$ is the set of permutations of $\{1,\ldots , M\}$, $0\leq j\leq M$ and $u_i\in [0;1] \ \forall i\in\{1,\ldots , M\}$ and $n$ is a function that is symmetric over exchange of its arguments.
Provided I divide this by a normalization constant, I believe can see this as some sort of probability over the $u_i$ variables. For instance if $n$ is separable, i.e. $n(y_1,\ldots,y_M)=\prod_i n_i(y_i)$ then by writing (up to normalization) :
$$p_i\sim\int_{u_{\sigma(i)}}^1 \mathrm{d}y_i n_i(y_i)$$
the integral becomes something like :
$$I \sim \sum_{\sigma\in S_M}\prod_{i=1}^j p_i \prod_{l=j+1}^M(1-p_j)$$
And looks like the sum of bernoulli trials with different probabilities for each trial (here they would be independent), and I am summing over all the probabilities
However all I have over my probability distribution is not independence, but merely exchangeability over the $u_i$ variables. I did some research and came upon De Finetti's exchange theorem for exchangeable probability distributions. I didn't know the theorem previously and have no clue of how to apply it in this case.
Would there be a way to write this integral in the large $M$ limit, knowing what $n$ looks like, and by saying that the $u_i$'s are distributed along some distribution $g$ ?
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2019-05-22 12:42:14
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http://gmatclub.com/forum/can-someone-rate-my-awa-138306.html
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# Can someone rate my AWA?
Author Message
Intern
Joined: 03 Sep 2012
Posts: 2
Followers: 0
Kudos [?]: 0 [0], given: 1
Can someone rate my AWA? [#permalink] 03 Sep 2012, 08:21
"The following appeared in an announcement issued by the publisher of The Mercury, a weekly newspaper.“Since a competing lower-priced newspaper, The Bugle, was started five years ago, The Mercury’s circulation has declined by 10,000 readers. The best way to get more people to read The Mercury is to reduce its price below that of The Bugle, at least until circulation increases to former levels. The increased circulation of The Mercury will attract more businesses to buy advertising
space in the paper.”
------------------------------------------
The publisher has stated that the decrease of Mercury's subscription over Bugle is primarily due to the price difference. Moreover, The mercury newspaper has seen a decline of 10,000 subscriptions over the last five years. The publisher has also stated that, decreasing Mercury's price lower than Bugle would increase sales which helps in more advertising space and revenue. I find the reasoning provided by the publisher flawed and i explain the reasons in the following paragraphs.
Firstly, the article assumed that the decline is primarily due to the lower price offered by the Bugle than Mercury News. I find this reason quite flawed because, Bulge news paper could be publishing completely different news content than mercury and targeting different sections of the user groups. Moreover Bugle could be publishing less number of pages compared to mercury which has more pages which increase the production cost. It is also not stated if price was the only differentiating factor between the two news papers. This assumption is wrong.
Secondly, the article has not provided any data on the user subscriptions between the two news papers. If the subscriptions of Mercury news is in millions compared to Bugle which is if lower than Mercury than 10,000 subscriptions may not be a huge decline. However mercury news has to analyze on the decline. This could be due to a new editorial started by Bugle Or a new sports editions or some other factors.
Thirdly, there is no evidence of any increase in businesses spending more money on advertising. Had the publisher provide any evidence through research statistics or some sort reducing the price for mercury news may be a hasty decision.
Finally, had the article provided more information on type of readers between the two news papers, any external factors influencing the decline of Mercury news or another data, mercury news should not reduce the price. Therefore, the announcement issued by the publisher is seriously flawed.
Appreciate the help.
Regards,
Vik
Kaplan GMAT Instructor
Joined: 25 Aug 2009
Posts: 644
Location: Cambridge, MA
Followers: 73
Kudos [?]: 215 [1] , given: 2
Re: Can someone rate my AWA? [#permalink] 03 Sep 2012, 10:54
1
KUDOS
Expert's post
bvik wrote:
"The following appeared in an announcement issued by the publisher of The Mercury, a weekly newspaper.“Since a competing lower-priced newspaper, The Bugle, was started five years ago, The Mercury’s circulation has declined by 10,000 readers. The best way to get more people to read The Mercury is to reduce its price below that of The Bugle, at least until circulation increases to former levels. The increased circulation of The Mercury will attract more businesses to buy advertising
space in the paper.”
------------------------------------------
The publisher has stated that the decrease of Mercury's subscription over Bugle is primarily due to the price difference. Moreover, The mercury newspaper has seen a decline of 10,000 subscriptions over the last five years. The publisher has also stated that, decreasing Mercury's price lower than Bugle would increase sales which helps in more advertising space and revenue. I find the reasoning provided by the publisher flawed and i explain the reasons in the following paragraphs.
Firstly, the article assumed that the decline is primarily due to the lower price offered by the Bugle than Mercury News. I find this reason quite flawed because, Bulge news paper could be publishing completely different news content than mercury and targeting different sections of the user groups. Moreover Bugle could be publishing less number of pages compared to mercury which has more pages which increase the production cost. It is also not stated if price was the only differentiating factor between the two news papers. This assumption is wrong.
Secondly, the article has not provided any data on the user subscriptions between the two news papers. If the subscriptions of Mercury news is in millions compared to Bugle which is if lower than Mercury than 10,000 subscriptions may not be a huge decline. However mercury news has to analyze on the decline. This could be due to a new editorial started by Bugle Or a new sports editions or some other factors.
Thirdly, there is no evidence of any increase in businesses spending more money on advertising. Had the publisher provide any evidence through research statistics or some sort reducing the price for mercury news may be a hasty decision.
Finally, had the article provided more information on type of readers between the two news papers, any external factors influencing the decline of Mercury news or another data, mercury news should not reduce the price. Therefore, the announcement issued by the publisher is seriously flawed.
Appreciate the help.
Regards,
Vik
Hi Vik,
There is a core of a solid essay here, but it needs a lot of polish. Take a look at my advice, and some higher-scoring essays on this site, and I look forward to seeing your next attempt!
Regards,
_________________
Eli Meyer
Kaplan Teacher
http://www.kaptest.com/GMAT
Prepare with Kaplan and save $150 on a course! Kaplan Reviews Intern Joined: 03 Sep 2012 Posts: 2 Followers: 0 Kudos [?]: 0 [0], given: 1 Re: Can someone rate my AWA? [#permalink] 04 Sep 2012, 08:13 Eli, Thanks much for the reply and appreciate your help. I've written the essay again and would appreciate your inputs. ------------------------------------------------------------ The publisher has stated that Mercury news readership has declined by 10,000 readers over the last 5years. It is also stated that this is due to the lower priced newspaper, The Bugle. It is assumed that the main reason for this decline is due to the price difference between the two papers. Hence it is decided to decrease the price of Mercury news, which will help increase circulations to previous levels and hope to attract more businesses to buy advertising space in the paper. There are certain questionable assumptions which i find logically flawed. First, the publisher indicated that the decline is primarily due to the price difference between the two news papers and no evidence has been provided to support this. The decline could be due to various factors other than the price such as lack of mobile site to read mercury news which is required to access information faster these days. So, deceasing the price may not meet their long term goals of increase circulations. Secondly, the publisher hasn't provided any evidence on the ways mercury news advertise the news paper to attract more readers. Being a new newspaper, The Bugle could be aggressively marketing the paper with additional resources in their sales and marketing team which could have caused certain readers to opt for Bugle and hence the decline. So decreasing the price to attract more customers seems logically flawed. Finally, The technology has been rapidly chaining these days and newspaper industry has seen shape decline in the publications. If mercury news failed to adopt the technology changes into their businesses, its a major concern in the decline. Moreover, the decline could be due to readers adopting different ways to get to the news real time by using various sources like twitter, mobile news application and news feeds. Therefore, in order to make this more compelling, the publisher should provide more information on the assumptions discussed above. Thus, decreasing the price to increase circulations to former levers is flawed. ---------------------- Thanks, Vik Kaplan GMAT Instructor Joined: 25 Aug 2009 Posts: 644 Location: Cambridge, MA Followers: 73 Kudos [?]: 215 [0], given: 2 Re: Can someone rate my AWA? [#permalink] 04 Sep 2012, 21:53 Expert's post bvik wrote: Eli, Thanks much for the reply and appreciate your help. I've written the essay again and would appreciate your inputs. ------------------------------------------------------------ The publisher has stated that Mercury news readership has declined by 10,000 readers over the last 5years. It is also stated that this is due to the lower priced newspaper, The Bugle. It is assumed that the main reason for this decline is due to the price difference between the two papers. Hence it is decided to decrease the price of Mercury news, which will help increase circulations to previous levels and hope to attract more businesses to buy advertising space in the paper. There are certain questionable assumptions which i find logically flawed. First, the publisher indicated that the decline is primarily due to the price difference between the two news papers and no evidence has been provided to support this. The decline could be due to various factors other than the price such as lack of mobile site to read mercury news which is required to access information faster these days. So, deceasing the price may not meet their long term goals of increase circulations. Secondly, the publisher hasn't provided any evidence on the ways mercury news advertise the news paper to attract more readers. Being a new newspaper, The Bugle could be aggressively marketing the paper with additional resources in their sales and marketing team which could have caused certain readers to opt for Bugle and hence the decline. So decreasing the price to attract more customers seems logically flawed. Finally, The technology has been rapidly chaining these days and newspaper industry has seen shape decline in the publications. If mercury news failed to adopt the technology changes into their businesses, its a major concern in the decline. Moreover, the decline could be due to readers adopting different ways to get to the news real time by using various sources like twitter, mobile news application and news feeds. Therefore, in order to make this more compelling, the publisher should provide more information on the assumptions discussed above. Thus, decreasing the price to increase circulations to former levers is flawed. ---------------------- Thanks, Vik This is an improvement, so good job! But it still needs work. Multiple typos ("rapidly chaining," "former levers") are problematic. Also, your second-to-last sentence is not convicning. You must be more specific about how, precisely, the author could be strengthened to get a top tier score! Eli _________________ Eli Meyer Kaplan Teacher http://www.kaptest.com/GMAT Prepare with Kaplan and save$150 on a course!
Kaplan Reviews
Re: Can someone rate my AWA? [#permalink] 04 Sep 2012, 21:53
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2015-08-27 20:50:33
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https://sd18spring.github.io/assignments/mini-project-1-gene-finder.html
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# Mini Project 1: Gene Finder
Due: 1:30 pm, Fri 9 Feb
## Preliminaries
### Computing Skills Emphasized
• Modular design
• Unit testing
• Loops
• Functions
• Conditionals
• String processing
• [for the “going beyond” part] list comprehensions.
### Biology Skills Emphasized
• Gene detection in arbitrary DNA sequences (also known as “ab initio gene finding”)
• Understanding protein coding
• Using Protein BLAST and interpreting its results
• [for the “going beyond” part] Learning how to read research papers, regulatory mechanisms for protein synthesis
### Acknowledgments
This assignment was originally created by Professors Ran Libeskind-Hadas, Eliot C. Bush, and their collaborators at Harvey Mudd. Special thanks to Ran for allowing us to use this assignment and adapt it for this course.
### Some Notes on the Assignment Structure
In contrast to some of the mini-projects that will come later in this course, this project is much more scaffolded. This is by design. One of the principal goals of this course is to teach you how to write good code. Good code can mean lots of things: fast code, readable code, debuggable code, modular code, etc. One of the best ways to learn how to design and structure your own code is to see examples of well-designed software. In this assignment, you will have the opportunity to see a good example of modular code design. By modular we mean that the functions and their interactions have been carefully designed to create a concise, readable, and maintainable program.
SoftDes is one of those courses where there is a wide range of prior- programming experience level among students. This class is designed to provide the best possible learning experience for all, therefore, if you are one of those folks that is on the higher end of this range, you will want to take advantage of the Going Beyond extension for this assignment. This extension is not worth any extra points, but it will quite interesting (and isn’t knowledge its own reward?!?). The extension is focused around exploring both computational and biological content.
## Motivation and Introduction
Computational approaches for analyzing biological data have revolutionized many subfields of biology. This impact has been so large that a new term has been coined to describe this type of interdisciplinary fusion of biology, computation, math, and statistics: bioinformatics. The rate at which bioinformatics is growing as a field is staggering. New Ph.D. granting departments are being formed annually, millions of dollars in research grants are awarded to push the boundaries of the field, and new companies are springing up around these techniques.
There are many grand challenge problems in the field of bioinformatics. In this assignment we will be exploring one of these problems called “gene prediction”. In the “gene prediction” problem, a computer program must take a sequence of DNA as input and output a list of the regions of the DNA that are likely to code for proteins.
Gene prediction is one of the most important and alluring problems in computational biology. Its importance comes from the inherent value of the set of protein-coding genes for other analysis. Its allure is based on the apparently simple rules that the transcriptional machinery uses: strong, easily recognizable signals within the genome such as open reading frames, consensus splice sites and nearly universal start and stop codon sequences. These signals are highly conserved, are relatively easy to model, and have been the focus of a number of algorithms trying to locate all the protein-coding genes in a genome using only the sequence of one or more genomes. -- “Gene Prediction: compare and CONTRAST”, Paul Flicek. Genome Biology 2007, 8:233.
One reason why this problem is so fundamental for the field, is that once one knows where the protein-coding genes are, one can begin to decode the form and function of these proteins. Once one knows the functions of these proteins, one can begin to decode the mechanisms that regulate the synthesis of these proteins (i.e. when and in what quantities these proteins are created by cells). If one can obtain a firm understanding of each of these components of the system, one gains an unprecedented level of understanding and insight into all kinds of biological processes: from understanding bacterial infection to understanding the intricacies of all sorts of cancers (and hopefully through this understanding better treatments).
In this assignment you will be writing a Python program that analyzes a DNA sequence and outputs snippets of DNA that are likely to be protein-coding genes. You will then use your program to find genes in a sequence of DNA from the bacterium Salmonella Enterica. We suspect that this particular DNA sequence is related to Salmonella’s role in the pathogenesis of various diseases such as Typhoid fever. Finally, you will use the genetic search engine protein-BLAST to confirm whether or not the genes predicted by your program are in fact genes, and if so what their functional role might be. This assignment is essentially the Biological equivalent of a mystery novel, and your primary tools in this case will be computational ones! More concretely, given an un-annotated text file of seemingly random symbols, you will write a Python program that will shed light on the nature of these symbols and gain insight into Typhoid fever (including aspects of how it is caused as well as the evolutionary history of the bacteria that causes the disease).
The story of Typhoid Mary is a fascinating one. It is fascinating on both scientific, legal, and philosophical levels. Rather than trying to reproduce it here, we invite you to check out the Wikipedia page on Typhoid Mary. Additionally, you may also want to listen to the fantastic RadioLab podcast called “Patient Zero”. The podcast discusses a number of topics, but the first segment of the podcast is about Typhoid Mary (although, you really should listen to the whole thing; you will not be disappointed).
Image source: http://en.wikipedia.org/wiki/Typhoid_Mary
In this assignment we will be building a gene finding program that can accurately determine regions of the Salmonella bacterium’s DNA that code for proteins. For a quick (and relatively entertaining) overview of the approach you will be using, check out this MIT OpenCourseWare video.
## Getting Set
The first step to getting started on this assignment is get a copy of the starter files on your computer. The process is the same as the one you went through to create your reading journal. For your reference, here are the steps again:
1. Click on the invitation link at https://classroom.github.com/a/i6rW0liB
2. Click the green button “Accept this assignment”.
3. Follow the remaining instructions until you get to your repository page. It will looks something like https://github.com/sd18spring/GeneFinder-myname, except with your GitHub user id instead of myname.
4. Clone the repository to your computer by typing the following into your terminal program. Replace myname with your GitHub user id. (note: these commands will clone your GeneFinder repository in your home directory, please modify the first line to cd to a different directory if you’d rather clone somewhere else).
$cd ~$ git clone https://github.com/sd18spring/GeneFinder-myname.git GeneFinder
$cd GeneFinder$ ls *
The last command will show you all of the files in the GeneFinder repository. The next section explains the purpose of each of these files.
## Week 1
Due Fri, Feb 2
### Getting the Lay of the Land
The last step of the previous section had you listing the contents of the gene_finder subdirectory of your SoftDes repo. Here is a description each of the files:
• gene_finder.py: this is where you will put your code for this assignment.
• amino_acids.py: some predefined variables that will help you write code to translate from DNA sequences to amino acid sequences.
• load.py: some utility functions for parsing and then loading the files in the data directory.
• data/X73525.fa: a FASTA file containing part of the genetic code of the Salmonella bacterium. As discussed in lecture, this part of the genetic code is responsible from some aspects of Salmonella pathogenesis.
• data/3300000497..., data/nitrogenase... :genetic for the metagenome Going Beyond extension
The first thing to do is to use the Atom text editor to open up gene_finder.py. This file has been populated with function declarations, docstrings, and unit tests for all the functions you will need to complete the assignment. Start reading through the functions declared in the file, if you do so before we have had a chance to talk about the biological mechanisms of protein synthesis you may need to consult some other sources (the Wikipedia article for Open-reading Frame is a good place to start).
Now that you have a good sense of the functions you will be filling out, take a look at this function diagram.
This diagram shows all of the functions in the program, and uses a directed arrow to indicate that the function on the “from” side of the arrow calls the function on the to side of the arrow. At this point, some of these arrows might make complete sense (you know exactly why and how the functions would interact). Others will be less obvious. That’s okay! You don’t have to necessarily understand every aspect of the design before you start on the assignment. The idea is that the motivation for the design will become apparent as you implement parts of it.
### Implementation Strategy
We could start by trying to implement any particular box in this diagram. However, we are going to be doing our implementation in a bottom-up ordering. That is, we are going to be implementing the functions that are called by other functions before we implement the calling functions. The motivation for this is that once you have had the experience of implementing the called function (on the “to” side of the arrow), it should be more clear how it can be utilized in the calling function (on the “from” side of the arrow).
### Basic manipulation of DNA snippets and Open Reading Frames (ORFs)
For the first part of the assignment you will be creating some utility functions that will help you build your gene finder. Open up gene_finder.py and fill in your own implementations of the functions described below.
### Unit Testing Instruction
For each function we have given you some unit tests (using doctest). You will want to add additional unit tests (again using doctest). For each unit test you add, write a sentence(ish) explaining your rationale for including the unit test. If you think the unit tests that we have given you are sufficient, please explain why this is the case. This additional text should be included in the docstring of the function immediately before the relevant unit test.
Also, if you want to test a specific function (in this example we will test get_complement) rather than running all of the unit tests you can modify the line at the end of the program from:
doctest.testmod()
to:
doctest.run_docstring_examples(get_complement, globals())
If you want to see verbose output of your doctests, set the verbose flag to True:
doctest.run_docstring_examples(get_complement, globals(), verbose=True)
For this part of the assignment you will write code that takes a DNA sequence and returns a list of all open reading frames in that sequence. Recall that an open reading frame is a sequence of DNA that starts with the start codon (ATG) and extends up to (but not including) the first in frame stop codon (TAG, TAA, or TGA). Open up gene_finder.py and fill in your own implementation of the functions described below:
• get_complement: this function should take a nucleotide as input and return the complementary nucleotide.
To help you get started here are some unit tests (make sure you have read the Unit Testing Instructions):
>>> get_complement("A")
'T'
>>> get_complement("C")
'G'
• get_reverse_complement: this function should return the reverse complementary DNA sequence for the input DNA sequence.
To help you get started here are some unit tests (make sure you have read the Unit Testing Instructions):
>>> get_reverse_complement("ATGCCCGCTTT")
'AAAGCGGGCAT'
>>> get_reverse_complement("CCGCGTTCA")
'TGAACGCGG'
• rest_of_ORF: Takes an input sequence of DNA that is assumed to begin with a start codon, and returns the snippet of DNA from the beginning of the string up to, but not including, the first in frame stop codon. If there is no in frame stop codon, the whole string is returned.
Some unit tests (make sure you have read the Unit Testing Instructions):
>>> rest_of_ORF("ATGTGAA")
'ATG'
>>> rest_of_ORF("ATGAGATAGG")
'ATGAGA'
• find_all_ORFs_oneframe: this function should find all open reading frames in a given sequence of DNA and return them as a list of strings. You should only check for ORFs that start on multiples of 3 from the start of the string. Your function should not return ORFs that are nested within another ORF. In order to accomplish this, once you find an ORF and add it to your list, you should skip ahead in the DNA sequence to the end of that ORF. You will find a while loop to be useful for this purpose. Make sure to utilize your rest_of_ORF function when coding this part.
A unit test (make sure you have read the Unit Testing Instructions):
>>> find_all_ORFs_oneframe("ATGCATGAATGTAGATAGATGTGCCC")
['ATGCATGAATGTAGA', 'ATGTGCCC']
• find_all_ORFs: this function should find all open reading frames in any of the 3 possible frames in a given sequence of DNA and return them as a list of strings. Note that this means that you need to check for ORFs in all three possible frames (i.e. with 0, 1, and 2 offset from the beginning of the sequence). For example, you would want to consider the following codon groupings when looking for all ORFs (groups of +++ or — indicate that the nucleotides above are considered as a single codon).
ATGTGAAGATTA
+++---+++---
-+++---+++--
--+++---+++-
As in above, don’t include ORFs that are nested within other ORFs. Your function should heavily utilize find_all_ORFs_oneframe.
A unit test (make sure you have read the Unit Testing Instructions):
>>> find_all_ORFs("ATGCATGAATGTAG")
['ATGCATGAATGTAG', 'ATGAATGTAG', 'ATG']
• find_all_ORFs_both_strands: this should do exactly the same thing as find_all_ORFs except it should find ORFs on both the original DNA sequence and its reverse complement.
A unit test (have you read the Unit Testing Instructions?) ;-)
>>> find_all_ORFs_both_strands("ATGCGAATGTAGCATCAAA")
['ATGCGAATG', 'ATGCTACATTCGCAT']
### Going Beyond
List comprehensions! Many of these functions can be written more succinctly using list comprehensions (see Section 5.1.3 here). Try to use list comprehensions to rewrite some of your code. Were any of the functions particularly hard (or impossible) to rewrite using list comprehensions? If so, how come?
### Turning in Week 1 Assignment
You be turning in your week 1 assignment by pushing your code to Github (this will use the same process you are using for turning in the reading journals). If you are struggling with the mini-project, we strongly advise you to meet with one of the NINJAs (either for some guidance, or just to look over your program). This week 1 work will be graded using the following rubric:
• No work has been turned in: 0%
• Only a minimal attempt has been made to complete the first part of the mini-project: 50%
• All code is written and works, or a sincere effort has been made to complete the work: 100%
The intermediate checkpoint will be worth 20% of the final grade for this assignment.
## Week 2
Due Fri, Feb 9
This week you will be implementing the rest of the functions necessary to create your gene finder. Once you have done that, you will be using your code to analyze a real DNA sequence suspected to play a role in Typhoid fever.
• longest_ORF: Finds the longest open reading frame on either strand of the DNA. Make sure you leverage code from previous parts of the assignment.
A unit test (make sure you have read the Unit Testing Instructions):
>>> longest_ORF("ATGCGAATGTAGCATCAAA")
'ATGCTACATTCGCAT'
• longest_ORF_noncoding: this function takes as input a DNA sequence and an integer indicating how many random trials should be performed. For each random trial, the DNA sequence should be shuffled and the longest ORF should be computed. The output of the function should be the length of the longest ORF that was found across all random trials (that is the output of longest_ORF_noncoding is an integer). In order to test this code you may find it useful to use the provided Salmonella DNA sequence. For example, if you find a longest ORF of 700, 600, and 300 on your three random trials, this function should output 700.
Note 1: In order to randomly shuffle a string you should use the provided shuffle_string function. If you wanted to implement this function yourself, you could take the following approach: First convert the string to a list using the list function. Once you have a list, you can shuffle the list using the built-in python function random.shuffle. To reassemble the shuffled list back to a string you can use string join function.
Note 2: We are not going to create unit tests for this function. Why not? Can you think of a different method of unit testing that would be appropriate for this function? Are there any other methods you might use to build confidence that your implementation is correct? (These are not rhetorical questions.)
• coding_strand_to_AA: this function converts from a string containing a DNA sequence to a sequence of amino acids. The function should read triplets of DNA nucleotides (codons), look up the appropriate amino acid (either using the provided variables in amino_acids.py or by encoding this information yourself), concatenate the amino acids into a string, and then return the amino acid sequence from the function.
You can convert a three nucleotide string (also called a triplet codon) into the appropriate amino acid in the following manner.
amino_acid = aa_table['CGA']
amino_acid will now be the string ‘R’ (which stands for Arginine). Note that aa_table is actually a dictionary which we haven’t learned about yet, so consider this a sneak peek of a powerful Python feature you will learn soon.
If you wanted to implement your own lookup, you could use the lists aa and codons to complete the mapping. codons is a list of lists where codons[i] contains a list of codons that code for the amino acid stored in aa[i].
Some unit tests (make sure you have read the Unit Testing Instructions):
>>> coding_strand_to_AA("ATGCGA")
'MR'
>>> coding_strand_to_AA("ATGCCCGCTTT")
'MPA
• gene_finder: this function takes as input a sequence of DNA. First, use your longest_ORF_noncoding on the input DNA sequence to compute a conservative threshold for distinguishing between genes and non-genes by running longest_ORF_noncoding for 1500 trials. For instance, the first line of your gene_finder function might be:
threshold = longest_ORF_noncoding(dna, 1500)
Next, find all open reading frames on both strands, and then return a list containing the amino acid sequence encoded by any open reading frames that are longer than the threshold computed above using longest_ORF_noncoding.
To tie it all together you will actually be applying the gene_finder program that you wrote to some real DNA! It is this type of computational sleuthing that has helped unlock many secrets. The first step is to get some DNA to analyze. Included in the data folder is a FASTA file containing a sequence of DNA from Salmonella Enterica believed to be related its pathogenesis. To load the sequence as a FASTA file, use the provided load_seq function.
>>> from load import load_seq
Use your gene_finder function on the Salmonella DNA sequence to get a list of candidate genes. We will be interpreting the results of your analysis during a scaffolded in-class activity.
Also, if you are interested in comparing the results of your gene finder to a state-of-the art one, you can try out one called Glimmer here.
### Turning in Week 2 Assignment
In order to turn in your assignment make sure that your work is pushed to your GitHub repository.
For the main assignment, all your code will be in gene_finder.py. If you choose to do the Going Beyond portion, it is up to you how you structure your code for that portion.
## Going Beyond
### Suggestion 1: Analyzing a meta-genome.
For this assignment you will be analyzing a meta-genome. In metagenomics, communities of microbes are analyzed using samples directly collected from the environment (as opposed to using lab cultures). The benefit of this approach is that it gives better insight into the diversity of microbes in the wild and how this diversity contributes to the functioning of the community. The downside is that it is more difficult computationally to analyze the genetic material collected from these communities. In this portion of the assignment, you will be writing a program to determine which microbe represented in a mystery meta-genome is responsible for Nitrogen fixation.
The first step is to load the data. There are two functions in the load.py file that will help you get started. The first loads a sequence of DNA that is known to code for Nitrogenase (an enzyme crucial in the Nitrogen fixation process).
>>> from load import load_nitrogenase_seq
>>> print(nitrogenase)
'ATGGGAAAACTCCGGCAGATCGCTTTCTACGGCAAGGGCGGGATCGGCAAGTCGACGACCTCGCAGAACACCCTCGCGGCACTGGTCGAGATGGGTCAGAAGATCCTCATCGTCGGCTGCGATCCCAAGGCCGACTCGACCCGCCTGATCCTGAACACCAAGCTGCAGGACACCGTGCTTCACCTCGCCGCCGAAGCGGGCTCCGTCGAGGATCTCGAACTCGAGGATGTGGTCAAGATCGGCTACAAGGGCATCAAATGCACCGAAGCCGGCGGGCCGGAGCCGGGCGTGGGCTGCGCGGGCCGCGGCGTCATCACCGCCATCAACTTCCTGGAAGAGAACGGCGCCTATGACGACGTCGACTACGTCTCCTACGACGTGCTGGGCGACGTGGTCTGCGGCGGCTTCGCCATGCCGATCCGCGAGAACAAGGCGCAGGAAATCTACATCGTCATGTCGGGCGAGATGATGGCGCTCTATGCGGCCAACAACATCGCCAAGGGCATCCTGAAATACGCGAACTCGGGCGGCGTGCGCCTCGGCGGCCTGATCTGCAACGAGCGCAAGACCGACCGCGAGCTGGAACTGGCCGAGGCCCTCGCCGCGCGTCTGGGCTGCAAGATGATCCACTTCGTTCCGCGCGACAATATCGTGCAGCACGCCGAGCTCCGCCGCGAGACGGTCATCCAGTATGCGCCCGAGAGCAAGCAGGCGCAGGAATATCGCGAACTGGCCCGCAAGATCCACGAGAACTCGGGCAAGGGCGTGATCCCGACCCCGATCACCATGGAAGAGCTGGAAGAGATGCTGATGGATTTCGGCATCATGCAGTCCGAGGAAGACCGGCTCGCCGCCATCGCCGCCGCCGAGGCCTGA'
The second step is to load the meta-genome. Again, there is a function in the load.py file loads the meta-genome for you.
>>> from load import load_metagenome
>>> print(metagenome[0])
('Incfw_1000001',
'AACAGCGGGGAATCGTCGACGCAATGCGCGGCATACAGCGTGCCGGCGAGCCCGGCCGACAGAAGACCGGCGAGCGCCCCGGCGAGCGCCGGGCGCGACGGCGCGCCGCGGCGCAGGCCCATCAGCGCGGCACCGAGGAACGGTAGCGACAGCACCGGGATCGAGCCGAGACACAGCAGCGAGTTGTGACCGAGCAGCCGCGTCATCGCCGAGGTTCGATGCGGCAGCATCGCTTCGGCGCCGATCGCGAGGCCGAGGATCGCCAGCGGCGCCAGCAGCAGCAGGCGCCAGCCTTTCGCCGTCGCCTCCGGGCGCGACAGATGCAGCGCGACGATGATCGC')
The variable metagenome contains a list of tuples. Each tuple consists of the name of a DNA snippet and a DNA sequence.
Next, write a program to determine which of these DNA snippets (in metagenome) are responsible for Nitrogen fixation. If all Nitrogenase genes looked the same, this would be relatively straightforward. You would simply loop through all the snippets until one matched the Nitrogenase sequence exactly. However, there are a variety of different Nitrogenase genes. Fortunately, there are fairly large portions of the given Nitrogenase sequence that are conserved (meaning they are the same in almost all genes that code for Nitrogenase). The approach we are going to use to determine whether or not a given snippet is likely to code for a form of Nitrogenase is called longest common substring. The idea is that if we relax our criteria for matching the Nitrogenase sequence exactly, we will still be likely to find the conserved portion of the sequence in our metagenome.
Your program should loop through all of the snippets in the metagenome and compute the longest common substring between the snippet and the Nitrogenase sequence. The snippets that have the longest common substrings with the Nitrogenase reference sequence are likely to code for Nitrogenase. Which are these? What are the longest common substrings?
There are several ways to solve the longest common substring problem. The most straightforward is to use a nested for loop over all possible start positions in both the Nitrogenase and the DNA snippet. For each possible combination of start positions, you then loop through both strings until you find corresponding characters that don’t match. Keep track of the longest substring found so far, and after you have checked all possibilities, return this longest match.
#### Tips for speeding up your program
1. Use PyPy to execute your program (a modified Python interpreter that excels when executing Python programs that depend heavily on loops). I got a 30 fold speedup when using the simple approach to longest common substring described above.
Linux: To install this run sudo apt-get install pypy.
macOS: Install home brew, and then run brew install pypy.
Windows: As of November 2017, only the Python 2.7 compatible version of PyPy is available for Windows. Fortunately, your Gene Finder program will almost certainly work with Windows! (It will work unless you’ve gone way beyond even list comprehensions, to used certain features of Python that aren’t covered in this course. If you’ve done this, you will know.) Download Python 2.7 compatible PyPy from the PyPy download page.
Windows (untested): The PyPy nightly build page has a link to pypy-c-jit-latest-win32.zip. The teaching team hasn’t gotten around to testing this; please let us know if this works (or doesn’t).
2. Implement a smarter algorithm for longest common substring (the dynamic programming solution is the next logical one to try).
3. Use the Python multiprocessing library to use all the processor cores on your laptop to examine several snippets at once. Also see An introduction to parallel programming using Python’s multiprocessing module.
### Suggestion 2: Building a better gene finder
Read more about other approaches to gene-finding in prokaryotes. If you are really gung-ho, pick one, and implement it!
### Suggestion 3: Visualize the data
1. Visualize the data:
For example, generate a picture that shows where the genes are in the DNA strand. Some libraries to look into are:
• Matplotlib. [The image below uses Matplotlib, with a color palette from Seaborn.] Matploglib is a good library to learn if you want to use Python for statistical data visualization; add Seaborn if the default Matplotlib settings offend your visual aesthetics.
• Pycairo for general 2D graphics.
• Kivy for interactive user interfaces.
• pygame for writing games, especially platformer games. We’ll be using pygame for the third mini-project.
You can also explore:
• Draw a histogram that compares the lengths of the genes found to the lengths of the noncoding ORFs from the shuffled sequences.
• Graph the similarity of different snippets to the nitrogenase sequence in Week 2 Going Beyond
• Draw the sequence above as a circle, rather than a line.
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2019-05-25 03:06:17
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https://www.worldsharings.com/2016/04/fuji-xerox-docuprint-cp205w-printer.html
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Fuji Xerox DocuPrint CP205w Printer Driver Download. The printer Fuji Xerox premiums the DocuPrint CP205 print speeds at as so much as 12ppm in colour and 15ppm in monochrome. There is no in-developed duplexing efficiency, so small corporations keen on duplexing would need to do quite a few guide paper shuffling with the DocuPrint CP205. Fuji Xerox premiums the black toner cartridges for the DocuPrint CP205 at as so much as 2000 pages, and the Cyan, Magenta and Yellow Toner cartridges at as much as 1400 pages each. In the get together you would have been going to design a laser printer alongside typical traces, there could also be an exceptional hazard that you just simply'd present you some thing that considered practically exactly like, if no longer equal to the Fuji Xerox DocuPrint CP205. When more often than not the most placing factor a number of product that you're going to see to claim is that it's black (or white) in a market that is dominated by way of beige or soft gray printer objects, you recognize that you're stretching. The Fuji Xerox DocuPrint CP205 measures in at 394x304x234mm and weighs in at 11kg, making it a squat however regular little bit of printing package deal. There is no choice for a larger yield cartridge substitute. I located the black cartridges on-line for an ordinary of AU$89, and the colour cartridges for AU$ninety nine every. A fast little little little bit of division suggests an usual printing internet page expense of round four.45c for a monochrome page and 21.2c per colour page, which is ready what you would anticipate to pay for a printer on this style of cost and abilities style.
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2021-04-10 23:11:04
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https://zbmath.org/?q=an%3A0667.14005
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## Comparison of the regulators of Beilinson and of Borel.(English)Zbl 0667.14005
Beilinson’s conjectures on special values of L-functions, Meet. Oberwolfach/FRG 1986, Perspect. Math. 4, 169-192 (1988).
[For the entire collection see Zbl 0635.00005.]
Using the evaluation map e: $$Spec({\mathbb{C}})\times B.GL_ N({\mathbb{C}})\to B.GL_ N/{\mathbb{C}}$$ and the n-th Chern class $$c_ n$$ in Deligne-Beilinson cohomology $$H^*_{{\mathcal D}}$$ one obtains a map from $$H_{2n-1}(GL_ N({\mathbb{C}}),{\mathbb{Z}})$$ to $${\mathbb{R}}(n-1):=(2\pi i)^{n-1}\cdot {\mathbb{R}}$$, which, composed with the Hurewicz map, gives a map $$K_{2n- 1}({\mathbb{C}})=\pi_{2n-1}(B.GL({\mathbb{C}})^+)\to {\mathbb{R}}(n-1)=H^ 1_{{\mathcal D}}(Spec({\mathbb{C}}),{\mathbb{R}}(n))$$. For a finite number field k this leads to Beilinson’s regulator map $$r_{{\mathcal D}}:\quad H^ 1_{{\mathcal M}}(X,{\mathbb{Q}}(n))\to H^ 1_{{\mathcal D}}(X_{/{\mathbb{R}}},{\mathbb{R}}(n)),$$ where $$X=Spec(k)$$, $$H^ 1_{{\mathcal M}}$$ denotes Beilinson’s motivic cohomology, i.e. the $$k^ n$$-eigensubspace of $$K_{2n-1}(k)\otimes {\mathbb{Q}}$$ under the Adams operations $$\psi^ k$$ and $$H^ 1_{{\mathcal D}}(X_{/{\mathbb{R}}},{\mathbb{R}}(n))$$ is the Gal($${\mathbb{C}}/{\mathbb{R}})$$-invariant part of $$\oplus_{\sigma:k\to {\mathbb{C}}}H^ 1_{{\mathcal D}}(Spec({\mathbb{C}}),{\mathbb{R}}(n)).$$
On the other hand, A. Borel [Ann. Sc. Norm. Super. Pisa, Cl. Sci., IV. Sér. 4, 613-636 (1977; Zbl 0382.57027)] constructed canonical “regulator elements” $$b_{2n-1}$$ in the continuous cohomology groups $$H^{2n-1}_{cont}(GL({\mathbb{C}}),{\mathbb{R}}(n-1))$$. These are the images of $$(2\pi i)^ n$$ times the generators $$u_{2n-1}$$ of $$H^*_{Betti}(U_ N,{\mathbb{Q}})$$ under the isomorphism $$H^*_{Betti}(U_ N,{\mathbb{R}})\otimes {\mathbb{C}}\simeq H^*_{cont}(GL_ N({\mathbb{C}}),{\mathbb{R}})\otimes {\mathbb{C}}$$. Forgetting the topology and using the Hurewicz map one obtains Borel’s regulator map $$r_{Borel}:\quad K_{2n-1}(k)\otimes {\mathbb{Q}}\to H^ 1_{{\mathcal D}}(X_{/{\mathbb{R}}},{\mathbb{R}}(n)).$$ It is the purpose of the paper under review to give a fairly detailed proof of the equivalence (up to a non- zero rational number) of both regulator maps, thus clarifying Beilinson’s sketch of a proof [A. A. Bejlinson, J. Sov. Math. 30, 2036-2070 (1985); translation from Itogi Nauki Tekh., Ser. Sovrem. Probl. Mat. 24, 181-238 (1984; Zbl 0588.14013); appendix to § 2]. An essential step in the proof is the result that the image $$v_{2n-1}$$ of the n-th Chern class $$c_ n\in H^{2n}_{DR}(B.GL_{N/{\mathbb{C}}})$$ under the homomorphism $$H^{2n}_{DR}(B.GL_{N/{\mathbb{C}}})\to H^{2n-1}(g)$$ is equal to a non-zero rational multiple of the image of $$(2\pi i)^ n.u_{2n-1}\in H^{2n-1}_{Betti}(U_ N,{\mathbb{C}})$$ under the comparison isomorphism $$H^*_{DR}(U_ N)\otimes {\mathbb{C}}\simeq H^*_{Betti}(U_ N)\otimes {\mathbb{C}}$$. Here g is the Lie algebra of $$GL_ N({\mathbb{C}})$$. This homomorphism depends on the fact that $$H^{2n}$$ of a suitable part of the Weil algebra equals the g-invariant subspace of the n-fold symmetric power of the dual of g. The construction of the Weil algebra rests on the property of the normalization functor inducing an equivalence of categories between the category of so-called reduced small co-simplicial algebras in a linear tensor category $${\mathcal A}$$ and the category of reduced small differential graded algebras in $${\mathcal A}$$. These notions were introduced by Beilinson in his paper cited above.
A second main ingredient in the proof of equivalence is Beilinson’s interpretation of the van Est isomorphism in continuous cohomology, necessary in Borel’s formulation of the regulator map, as a restriction map to the cohomology of an infinitesimal version of the classifying space, namely the largest small simplicial subscheme.
Reviewer: W.W.J.Hulsbergen
### MSC:
14C35 Applications of methods of algebraic $$K$$-theory in algebraic geometry 17B56 Cohomology of Lie (super)algebras 14A20 Generalizations (algebraic spaces, stacks)
### Citations:
Zbl 0635.00005; Zbl 0382.57027; Zbl 0588.14013
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2022-05-27 16:19:04
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http://mathematica.stackexchange.com/tags/modular-arithmetic/hot
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# Tag Info
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All of the polynomial functions, have an option Modulus which allows you to specify an integer field, like $\mathbb{Z}_5$. In particular, Factor works on your example polynomial Factor[x^2+4, Modulus -> 5] (* (1 + x) (4 + x) *) Additionally, IrreduciblePolynomialQ works to determine irreducibility of $x^2+2$, as follows IrreduciblePolynomialQ[x^2 + ...
15
The problem we encounter here is an instance of rather unexpected limitations of equation solving functionality (i.e. Modulus option in Reduce), e.g. this question : Strange behaviour of Reduce for Mod[x,1] provides another example which has been fixed in the newest version (9.0) of Mathematica. Since Modulus unexpectedly doesn't work here we can take ...
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IntegerDigits works Try powers = IntegerDigits[204, 2] {1, 1, 0, 0, 1, 1, 0, 0} Now, if you want that formatted as a sum of powers of two, you have to hold it. For example Total@MapIndexed[#1 Defer[2]^(First@#2 - 1) &, Reverse@powers] 2^2 + 2^3 + 2^6 + 2^7 EDIT Nicer code, given that your numbers go up to 255 pow2[num_]:=Inner[#1 ...
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This is bit faster: toPrime = 500; sums = Accumulate@FoldList[Times, 1, Range[2, Prime@toPrime - 1]]; primes = Prime[Range[toPrime]]; Mod[sums[[primes - 1]], primes] Precompute factorial sums and primes. Mod is fast on lists.
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Working with LinearSolve we encounter some inconsistency of the related option Modulus -> z if z is not prime. Nonetheless we could do this Mod[ LinearSolve[ {{1, 1, 1}, {4, 2, 1}, {9, 3, 1}}, {31, 3, 11}], 54] {18, 26, 41} Unfortunately we can get only one solution unlike when working with Solve. These posts describe another problems or bugs ...
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Yes, you can use the built in function Outer. It does exactly the kind of thing you are talking about it. Try Outer[Mod, list1, list2] Outer is a generalization of the outer product in Linear Algebra. Its first argument is a function, and the rest of its arguments are lists. Basically, it applies the function in the first argument to every element in the ...
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It's meant to be done divide-and-conquer style. Here is one way to go about that. listMod[n_, {val_}] := {Mod[n, val]} listMod[n_, vals : {_, __}] := With[{len = Length[vals], rem = Mod[n, Times @@ vals]}, Join[listMod[rem, Take[vals, Floor[len/2]]], listMod[rem, Drop[vals, Floor[len/2]]]] ] Your example: n = 31415926535; primeslist = ...
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Solve with Modulus We can use Solve with domain specification like i.e. Integers, or with e.g. integers modulo 5, then instead of specifying the domain one uses Modulus : Solve[x^2 + 4 == 0, x, Modulus -> 5] {{x -> 1}, {x -> 4}} Times @@ ( x - Last @@@ %) Expand[ %, Modulus -> 5] (-4 + x) (-1 + x) 4 + x^2 For an integer $n$, ...
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addmod = Mod[Plus[##],2]& ## is a Sequence of all the arguments given to addmod.
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I do a lot of cybersecurity competitions where we crack crypto, so I'm used to grappling with Mathematica for ring algebra. The sole thing Mathematica honestly isn't great for is cryptography. For this stuff, I generally just use SageMath Cloud, because it has all of the above algorithms built into DiscreteLog. You just throw your values at the function and ...
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If you want to solve an equation over integer rings $\mathbb{Z}_n$ you should specify them with Modulus e.g. Column[Solve[x^3 == 0, x, Modulus -> #] & /@ Range[2, 9]] Edit Since there was no further example of any expression to simplify over a finite ring let's define e.g. a polynomial which cannot be factorized over rationals (as Mathematica ...
7
For moduli that are square-free one can use Chinese remaindering on the coefficient lists to get a result valid for the moduli product. cfs[p1_, p2_, x_, p_] := Reverse[CoefficientList[PolynomialGCD[p1, p2, Modulus -> p], x, 1 + Min[Exponent[p1, x], Exponent[p2, x]]]] FromDigits[ ChineseRemainder[ Transpose[{cfs[f[x], g[x], x, 7], cfs[f[x], ...
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You have several options, either directly implementing incr incr[digs_, base_] := Module[{carry = 1, ndigs = digs, k = 1, nd}, While[k <= Length[digs], {carry, nd} = QuotientRemainder[Part[ndigs, k] + carry, Part[base, k]]; Part[ndigs, k] = nd; If[carry == 0, Break[]]; k++; ]; ndigs ] Or implementing FromMultpleBase and ...
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As already mentioned in the question, we can use Outer, for example (with a $3 \times 2$ matrix) Outer[f, {x1, x2, x3}, {y1, y2}] {{f[x1, y1], f[x1, y2]}, {f[x2, y1], f[x2, y2]}, {f[x3, y1], f[x3, y2]}} so we just need a function f for which f[x, y] returns ChineseRemainder[{x, y}, {7, 30}]. That function could be defined simply as f[x_, y_] := ...
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The difference between $2^n$ and $n^2$ is that $2^n$ is not a function $\bmod 10$ -- that is, $2^{n+10}$ is not congruent to $2^n\bmod 10$. Further $2^n$ is only eventually periodic $\bmod 10^k$, $k \geq 2$. For instance $2^1$ is not congruent to any other $2^n \bmod 100$. On the other hand, polynomial functions are all functions $\bmod\, m$ : f[n+m] is ...
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This isn't directly an answer, and I'll delete it if it is off target. But you might want to use some non-System context functionality for taking polynomial-mod-2 products. Specifically this works with integer lists of coefficients. I'll show an example below. In[1110]:= SeedRandom[1111]; vals = RandomInteger[2^8 - 1, 2] intlists = ...
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Already answered in the comments by DumpsterDoofus and Daniel Lichtblau, to summarize: Machine floating point numbers such as 0.2 are not always exactly representable in binary (no terminating expansion in base 2). Thus floating point arithmetic is susceptible to roundoff error and other accuracy problems. For example, the following are not exactly equal to ...
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As it turns out, there's an (undocumented) function eminently suitable for the task: poly = -1 + x + x^2 - x^4 + x^6 + x^9 - x^10; PolynomialMod[AlgebraPolynomialPowerMod`PolynomialPowerMod[poly, -1, x, x^11 - 1], 32] 5 + 9 x + 6 x^2 + 16 x^3 + 4 x^4 + 15 x^5 + 16 x^6 + 22 x^7 + 20 x^8 + 18 x^9 + 30 x^10 Check the result: ...
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Use a Gröbner basis. The idea is to set up an equation for this multiplicative inverse, in a ring where both $x^{11}-1$ and $32$ are zero (that is, $\mathbf Z[x]/(32,x^{11}-1)$). Then unravel that equation using GroebnerBasis to get the variable representing this reciprocal to f in terms of x: f = -1 + x + x^2 - x^4 + x^6 + x^9 - x^10; defpoly = x^11 - 1; ...
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For the example problem I get about a factor of 4 speedup over PowerMod by memoizing Mont. This of course means that Mont should not contain any global variables so I rewrote the code slightly: MontExp[b_, e_, n_] := Module[ {RLength, R, RM1, RInverse, NPrime, M, Result}, RLength = BitLength[n]; R = 2^RLength; RM1 = R - 1; RInverse = PowerMod[R, -1, ...
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Let's see some beautiful answers pop up. For now, a not too sleek one to break the ice fix[l_, base_] := Module[{take = 0}, Rest@FoldList[ QuotientRemainder[#2[[1]] + take, #2[[2]]] /. {q_, r_} :> (take = q; r) &, 0, Transpose@{l, base}]] inc[{f_, rest___}, base_] := fix[{f + 1, rest}, base] So NestList[inc[#, {10, 5, 3}] &, ...
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There is an option Modulus in certain algebraic functions (Solve, LinearSolve, Det,Factor etc.) to specify that integers are to be treated modulo an integer n. Consider e.g. m0 = {{4, 6, 6}, {6, 3, 2}, {1, 4, 4}}; b0 = {4, 2, 1}; then LinearSolve[ m0, b0, Modulus -> 2] {1, 0, 0} You can work with LinearSolve specifying only the first variable, ...
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Based on Rojo's answer: add[base_][l_, x_] := FoldList[QuotientRemainder @@ ({1, 0} # + #2) &, x, {l, base}\[Transpose]][[2 ;;, 2]] NestList[add[{10, 5, 3}][#, 1] &, {8, 3, 1}, 15] {{8, 3, 1}, {9, 3, 1}, {0, 4, 1}, {1, 4, 1}, {2, 4, 1}, {3, 4, 1}, {4, 4, 1}, {5, 4, 1}, {6, 4, 1}} Alternate formulation: base /: base[l_, blst_] + x_Integer ...
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check[rut_] := Module[{d = 11 - Mod[Total[IntegerDigits[rut]*{3, 2, 7, 6, 5, 4, 3, 2}], 11]}, Which[d == 11, 0, d == 10, "K", True, d]] Takes rut as integer (all digits less check digit), returns check code. Does not validate length of input, so add that if needed. Update: Since you've added check examples, and since they contain both 7 and 8 body ...
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How about Assuming[101 < t < 109, FullSimplify[PiecewiseExpand[Mod[t, 10]]]] rhermans gives a simpler formulation: PiecewiseExpand[Mod[t, 10], 101 < t < 109]
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To fix the memory problems you could rewrite it in a procedural style. It's probably more than a tweak, a bit ugly, and a bit slower. But you can go forever without having to worry about memory. ClearAll@fail; fail = Compile[{{m, _Integer}, {p, _Integer, 1}}, MemberQ[Mod[6 m - 3, #] & /@ p, 2] || MemberQ[Mod[6 m - 3, #] & /@ p, 4]]; ...
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You have a series of conditions you want to be true, so use Select, A1 = {2}; A2 = {3, 5, 6}; A3 = {7, 8, 10}; Select[Range[100], MemberQ[A1, Mod[#, 3]] || MemberQ[A2, Mod[#, 7]] || MemberQ[A3, Mod[#, 11]] &] (* {2, 3, 5, 6, 7, 8, 10, 11, 12, 13, 14, 17, 18, 19, 20, 21, \ 23, 24, 26, 27, 29, 30, 31, 32, 33, 34, 35, 38, 40, 41, 43, 44, 45, \ 47, ...
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This seems to work Mod[Rationalize@1.2, Rationalize@0.2] == 0 I also tried with SetAccuracy, but it didn't always work. "If Your Only Tool Is a Hammer Then Every Problem Looks Like a Nail" What I mean is that I'm using here a function that is probably quite involved (Rationalize) for a problem that doesn't look complex (although it is complex when you ...
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From the comment of Daniel Lichtblau: The Mathematica function you are looking for is MultiplicativeOrder. For example, MultiplicativeOrder[555, 77, {4}] gives 8, because $555^8 \equiv 4 (\mathrm{mod}77)$.
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One way to approach this is to use the Null space of the matrix a, which is a basis for all elements x such that a.x=0. For the OPs problem, this can be done by first finding the null space: a = {{2, 1, 6}, {1, 3, 1}}; n = First[NullSpace[a, Modulus -> 7]] {5, 5, 1} Now, observe that a.x=0 is true exactly when a.(c*x)=0, so it is possible to list all ...
Only top voted, non community-wiki answers of a minimum length are eligible
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2016-05-31 21:50:10
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http://neoxfiles.com/absolute-error/absolute-error-wikipedia.php
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Home > Absolute Error > Absolute Error Wikipedia
# Absolute Error Wikipedia
When this occurs, the term relative change (with respect to the reference value) is used and otherwise the term relative difference is preferred. The system returned: (22) Invalid argument The remote host or network may be down. We wish to compare these costs.[3] With respect to car L, the absolute difference is $10,000 =$50,000 - $40,000. The pink line is one of infinitely many solutions within the green area. this contact form Robert F. We want to find a function f such that f ( x i ) ≈ y i . {\displaystyle f(x_{i})\approx y_{i}.} To attain this goal, we suppose that the function f p.53. The following is an enumeration of some least absolute deviations solving methods. ## Meaning Of Absolute Error When the variable in question is a percentage itself, it is better to talk about its change by using percentage points, to avoid confusion between relative difference and absolute difference. In order to maintain an optimized inventory and effective supply chain, accurate demand forecasts are imperative. By using this site, you agree to the Terms of Use and Privacy Policy. At X confidence, E m = erf − 1 ( X ) 2 n {\displaystyle E_{m}={\frac {{\text{erf}}^{-1}(X)}{\sqrt {2n}}}} (See Inverse error function) At 99% confidence, E m ≈ 1.29 n {\displaystyle This calculation ∑ ( | A − F | ) ∑ A {\displaystyle \sum {(|A-F|)} \over \sum {A}} , where A {\displaystyle A} is the actual value and F {\displaystyle F} The numerators of these equations are rounded to two decimal places. Absolute Error Example When there is no reference value, the sign of Δ has little meaning in the comparison of the two values since it doesn't matter which of the two values is written SIAM Journal on Scientific Computing. 1 (2): 290–301. Define Absolute Error Like confidence intervals, the margin of error can be defined for any desired confidence level, but usually a level of 90%, 95% or 99% is chosen (typically 95%). Note that alternative formulations may include relative frequencies as weight factors. https://en.wikipedia.org/wiki/Mean_absolute_percentage_error In statistics, the mean absolute error (MAE) is a quantity used to measure how close forecasts or predictions are to the eventual outcomes. ISBN 0-87589-546-8 Wonnacott, T.H. Absolute Error Formula Symmetry: The mean absolute scaled error penalizes positive and negative forecast errors equally, and penalizes errors in large forecasts and small forecasts equally. Your cache administrator is webmaster. Moreover, MAPE puts a heavier penalty on negative errors, A t < F t {\displaystyle A_{t} ## Define Absolute Error Please try the request again. In media reports of poll results, the term usually refers to the maximum margin of error for any percentage from that poll. Meaning Of Absolute Error This means that your percent error would be about 17%. Integral Absolute Error Wiki K. The margin of error for a particular sampling method is essentially the same regardless of whether the population of interest is the size of a school, city, state, or country, as weblink Predictable behavior as y t → 0 {\displaystyle y_{t}\rightarrow 0} : Percentage forecast accuracy measures such as the Mean absolute percentage error (MAPE) rely on division of y t {\displaystyle y_{t}} Political Animal, Washington Monthly, August 19, 2004. pp.63–67. Normalized Absolute Error Wiki Text is available under the Creative Commons Attribution-ShareAlike License; additional terms may apply. Computational Statistics & Data Analysis. 39 (1): 35–55. San Francisco: Jossey Bass. http://neoxfiles.com/absolute-error/absolute-error-formula-wikipedia.php The mean absolute error is on same scale of data being measured. Wikipedia® is a registered trademark of the Wikimedia Foundation, Inc., a non-profit organization. Mean Relative Error Please help to improve this article by introducing more precise citations. (April 2011) (Learn how and when to remove this template message) See also Least absolute deviations Mean absolute percentage error The margin of error for a particular individual percentage will usually be smaller than the maximum margin of error quoted for the survey. ## Other properties There exist other unique properties of the least absolute deviations line. That is, car M costs$10,000 more than car L. New York, N.Y: Marcel Dekker. Iteratively re-weighted least squares[7] Wesolowsky’s direct descent method[8] Li-Arce’s maximum likelihood approach[9] Check all combinations of point-to-point lines for minimum sum of errors Simplex-based methods are the “preferred” way to solve Absolute Error Definition Several common choices for the function f(x, y) would be: max (|x|,|y|), max (x, y), min (|x|, |y|), min (x, y), (x + y)/2, and (|x| + |y|)/2.
North Carolina State University. 2008-08-20. This article needs additional citations for verification. The true standard error of the statistic is the square root of the true sampling variance of the statistic. his comment is here Relative difference is often used as a quantitative indicator of quality assurance and quality control for repeated measurements where the outcomes are expected to be the same.
Along with the confidence level, the sample design for a survey, and in particular its sample size, determines the magnitude of the margin of error. doi:10.1137/0901019. Text is available under the Creative Commons Attribution-ShareAlike License; additional terms may apply. By using this site, you agree to the Terms of Use and Privacy Policy.
The method minimizes the sum of absolute errors (SAE) (the sum of the absolute values of the vertical "residuals" between points generated by the function and corresponding points in the data). Another way to define the relative difference of two numbers is to take their absolute difference divided by some functional value of the two numbers, for example, the absolute value of Retrieved 2016-05-15. ^ a b Hyndman, Rob et al, Forecasting with Exponential Smoothing: The State Space Approach, Berlin: Springer-Verlag, 2008. This negative result provides additional information about the experimental result.
This is especially problematic for data sets whose scales do not have a meaningful 0, such as temperature in Celsius or Fahrenheit, and for intermittent demand data sets, where y t For example, experimentally calculating the speed of light and coming up with a negative percent error says that the experimental value is a velocity that is less than the speed of These two may not be directly related, although in general, for large distributions that look like normal curves, there is a direct relationship. Mahwah, NJ: Lawrence Erlbaum Associates. ^ Drum, Kevin.
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2018-05-27 17:37:25
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http://blog.maryquilts.com/2008/03/31/winter-is-not-over-yet/
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## Winter is not over yet
They’re calling for 6-8 inches of snow today – believe it or not – I’m still enjoying it.
It may not look like it – but I’m making progress. This small 8×10 den is my biggest challenge with the move. We’re having the movers do our packing but other than a couple boxes in the longarm room and a couple boxes in my bedroom – this is the only room I feel a need to pack up myself. Since many of the files will stay in storage boxes, it makes sense to sort them now.
The painters come to the townhouse tomorrow at 7AM so I’m going to walk over now to put post-it notes on all the walls so they won’t get *confused* at which colors go where. Hopefully, I can get a haircut on the way over – no appointment but they can usually fit me in.
I love walking in the snow!
Email me
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### 7 Responses to “Winter is not over yet”
• LindaL:
I’m glad someone is still enjoying the snow I woke up to about 3-3 1/2″ on the ground this morning here in NE Nebraska and it’s kept snowing. Sometimes quite heavily. I’m glad you’re making progress on the packing up. We’ve lived in the same house for over 30 yrs. and moving would be a nightmare. House would be bad enough, but DH has two double garages full of stuff. The man doesn’t understand the concept of throwing things away when you clean, he just rearranges it all!!!
Linda
• Mama Koch:
It’s 81 degrees here today.
• zizzybob:
I left New York 39 years ago and beleive me I do not miss the snow. I am getting a chill just looking at the picture you posted. I doubt I will ever live any further north than Orlando.
• D2Quilter:
Well, we’ve had 60 degree weather in UT, but woke up to 3 inches of snow this morning!! I sure love how clean everything looks w/a fresh blanket of snow. Glad you’re enjoying yours!
• Juliann in WA:
I think if I lived somewhere that we knew it was going to snow a lot and had the right kind of clothes and I didn’t have to drive in it, I would probably like to walk in the snow but for now, I am just plain tired of the cold and the mess.
• Deb:
How much snow will it take before you’re tired of it??? Maybe you’re just one of the ones who will always like it.
Glayou got the “cooties” out of the new house!
• meggie:
Mary, You have no idea how I wish I could walk in some snow! But then, I might complain if I had to do it!
I have voted, & I always read your blog, as I dont subscribe to bloglines. I love to vist, though it is very time consuming!
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Email me
Blog Rings
Stash Quilts
Quilt Mavericks
Liberated Quilters
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2014-10-30 17:31:47
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http://valasztas2016.siofok.hu/vanilla-orchid-ycqy/distance-from-point-to-plane-825601
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Cylindrical to Cartesian coordinates find the distance from the point to the line, This means, you can calculate the shortest distance between the point and a point of the plane. Specify the point. MHF Helper. Minimum Distance between a Point and a Plane Written by Paul Bourke March 1996 Let P a = (x a, y a, z a) be the point in question. Peter. Let us use this formula to calculate the distance between the plane and a point in the following examples. The plane satisfies the equation: All points X on the plane satisfy the equation: It means that the vector from P to X is perpendicular to vector . Spherical to Cartesian coordinates. That means in your case the distance in question is nothing but the absolute value of the z-coordinate. First we need to find distance d, that is a perpendicular distance that the plane needs to be translated along its normal to have the plane pass through the origin. The above Python implementation of finding the distance between a point in a plane and a straight line is all I share with you. two points do not define a plane. Here we're trying to find the distance d between a point P and the given plane. How to calculate the distance from a point to a plane. Volume of a tetrahedron and a parallelepiped. If I have the plane 1x minus 2y plus 3z is equal to 5. So how do we find the shortest distance from a point (x1, y1, z1) to the xz-plane? Consider the lower diagram in figure 2. d = |kN| where k is some scalar. Distance between a point and a line. Separate A, B, and C in the equation determined in step 1. Shortest distance between a point and a plane. The distance d(P 0, P) from an arbitrary 3D point to the plane P given by , can be computed by using the dot product to get the projection of the vector onto n as shown in the diagram: which results in the formula: When |n| = 1, this formula simplifies to: showing that d is the distance from the origin 0 = (0,0,0) to the plane P . The Problem. Measure the distance between the point and the plane. If you put it on lengt 1, the calculation becomes easier. I have another algorithm that finds the distance from the origin of the plane, but I''d also like to be able to find the distance to a plane (3 verticies) anywhere in 3D space. C ා basic knowledge series – 1 data type . Distances between a plane and a point are measured perpendicularly. And this is a pretty intuitive formula here. And how to calculate that distance? Next, determine the coordinates of the point. the distance from the nearest point on the plane to the point is. Finding the distance between a point and a plane means to find the shortest distance between the point and the plane. We'll do the same type of thing here. And that is embodied in the equation of a plane that I gave above! It is a good idea to find a line vertical to the plane. Example. The perpendicular A4K4 is the distance from the point to the plane, because it is projected into a segment of natural size. Distance of a Point to a Plane. Thanks Open Live Script. Then length of the perpendicular or distance of P from that plane is: a 2 + b 2 + c 2 ∣ a x 1 + b y 1 + c z 1 + d ∣ Using communication lines, we build a perpendicular to the plane of the quadrilateral EBCD. Distance of a point from a plane - formula Let P (x 1 , y 1 , z 1 ) be any point and a x + b y + c z + d = 0 be any plane. Pretty straightforward question I guess; How do I find the distance from a point in 3D space to a plane? We first need to normalize the line vector (let us call it ).Then we find a vector that points from a point on the line to the point and we can simply use .Finally we take the cross product between this vector and the normalized line vector to get the shortest vector that points from the line to the point. So that's some plane. Answer to: Find the distance from the point (2, 0, -3) to the plane 3x - 4y + 5z = 1. Vi need to find the distance from the point to the plane. Thank You. and Cartesian to Spherical coordinates. If you put it on lengt 1, the calculation becomes easier. Tags: distance, python, straight line. Distance from a point to a plane in space; Distance between two straight lines in space; Distance between two points in space; Solved problems of distance between a straight line and a plane … I hope I can give you a reference and I hope you can support developeppaer more. Currently, I am projecting the point onto the 'infinite' plane that is defined by the normal of the 3 points and testing whether the projected point is within the bounds of the finite plane. Cartesian to Cylindrical coordinates. Distance of a Point from a Plane with the help of Cartesian Form. Please explain how to find between xy and yz plane. analytic-geometry. Therefore, the distance from these points to the plane will be $$\| w_1 - v_1\| = |\beta_1|\|(1, 1, 1)\| = \sqrt{3}$$ and $$\| w_2 - v_2\| = |\beta_2|\|(1, 1, 1)\| = 3\sqrt{3}$$ so the distance is $\sqrt{3}.$ I realise that this doesn't use the hint, but I feel its more direct and straightforward. distance from a point to plane Math and Physics Programming. Such a line is given by calculating the normal vector of the plane. And let me pick some point that's not on the plane. The minimal distance is therefore zero. If a point lies on the plane, then the distance to the plane is 0. Recommended Today. so the distance from the plane to the point normal to the plane is just the projection of the vector normal to the plane . Shortest Distance to a Plane. Dans l'espace euclidien, la distance d'un point à un plan est la plus courte distance séparant ce point et un point du plan. Related topics. Let's assume we're looking for the shortest distance from that point to the xz-plane because there are actually infinite distances from a single point to an entire plane. I am doing cal 3 h.w the text book only show area from two points..."the distance formula in three dimension".. i do know how to do the two points, but this one point question is confusing. We remove the coordinate для for the plane Π1 from the plane Π2. Given a point a line and want to find their distance. Approach: The perpendicular distance (i.e shortest distance) from a given point to a Plane is the perpendicular distance from that point to the given plane.Let the co-ordinate of the given point be (x1, y1, z1) and equation of the plane be given by the equation a * x + b * y + c * z + d = 0, where a, b and c are real constants. Because all we're doing, if I give you-- let me give you an example. Given: a point (x1, y1, z1) a direction vector (a1, b1, c1) a plane ax + by + cz + d = 0 How can I find the distance D from the point to the plane along that vector? Proj(Pvector) = ((Pvector dot N)/|N|^2) Nvector. Plane equation given three points. If the plane is not parallel to the coordinate planes you have to use a formula or you calculate the minimum of all possible distances, using calculus. The problem is to find the shortest distance from the origin (the point [0,0,0]) to the plane x 1 + 2 x 2 + 4 x 3 = 7. Points and Planes. because (0,0,0) is a point on the plane . Thus, if we take the normal vector say ň to the given plane, a line parallel to this vector that meets the point P gives the shortest distance of that point from the plane. Calculate the distance from the point P = (3, 1, 2) and the planes . Ok, how about the distance from a point to a plane? First, determine the equation of the plane. This tells us the distance between any point and a plane. Distance between a Point and a Plane in 3-D Description Measure the distance between a point and a plane in three-dimensional space. 2 Comments. H. HallsofIvy. This example shows how to formulate a linear least squares problem using the problem-based approach. Such a line is given by calculating the normal vector of the plane. Thanks Let's say I have the plane. Specify the plane. Take the 1 and 6 options for which you need to determine: The distance from the point D to the plane defined by the triangle ΔABC. The distance from a point, P, to a plane, π, is the smallest distance from the point to one of the infinite points on the plane. On the plane П1 we take the coordinate Z from the plane П4. A 3-dimensional plane can be represented using an equation in the form AX + BY + CZ + D. Next, gather the constants from the equation in stead 1. Distance from point to plane. IF it is not, I calculate the closest point on each each and select the minimum. Well since the xz-plane extends forever in all directions with y=0, we actually don't need to worry about the x values or the z values! Cause if you build a line using your point and the direction given by a normal vector of length one, it is easy to calculate the distance. Also works for array of points. Finally, you might recognize that the above dot product is simply computed using the function dot, but even more simply written as a matrix multiply, if you have more than one point for which you need to compute this distance. The shortest distance of a point from a plane is said to be along the line perpendicular to the plane or in other words, is the perpendicular distance of the point from the plane. It is a good idea to find a line vertical to the plane. Reactions: HallsofIvy. Find the distance of the point (2, 1, 0) from the plane 2x + y + 2z + 5 = 0. asked Jan 6 in Three-dimensional geometry by Sarita01 ( 53.4k points) three dimensional geometry Determine the distance from a point to a plane. share | cite | improve this question | follow | edited Sep 25 '16 at 0:17. the vector (7,6,8) which represents the point given starts on the plane . Then let PM be the perpendicular from P to that plane. Shortest distance between two lines. Learn more about distance, point, plane, closest distance, doit4me In this paper we consider two similar problems for determining the distance from a point to a plane. If it is within the bounds of the plane, I just use the distance as determined by the equation to plane. Distance from a point to a plane, and the projected point coordinates on the plane. There are an unlimited number of planes that contain the two points {(1,2,2) & (0,0,1)} There is a plane therefore that not only contains those two points but also contains the point P=(-19,-15,1). Spherical to Cylindrical coordinates. Point are measured perpendicularly vi need to find a line and want to find distance! Need to find a line and want to find the shortest distance between the plane Π1 from the point a. 'S not on the plane, and C in the following examples given by calculating the vector! Z from the plane lies on the plane projected point coordinates on the plane a,,. A segment of natural size all we 're trying to find the distance in question is nothing but absolute... Given plane plane Π1 from the plane, because it is a point to a plane and a with... I gave above similar problems for determining the distance between the point and a plane Pvector =. Hope you can support developeppaer more separate a, B, and the given plane their distance the?! Good idea to find a line and want to find a line is given by the... Data type dot N ) /|N|^2 ) Nvector of Cartesian Form as determined by equation. And want to find the distance from a point to a plane bounds of the quadrilateral EBCD measure the between!, how about the distance between distance from point to plane point given starts on the plane is 0 the coordinate for! Idea to find their distance coordinate для for the plane is 0 distance question... Is a good idea to find a line vertical to the plane to! Cartesian coordinates distance of a plane with the help of Cartesian Form Sep 25 '16 at 0:17 how do find! 2Y plus distance from point to plane is equal to 5 plane Π1 from the plane to the point normal to point. You can support developeppaer more l'espace euclidien, la distance d'un point à un est... Is within the bounds of the plane to the plane 1x minus 2y plus 3z equal... So how do I find the shortest distance between a plane perpendicular to the and! The same type of thing here use this formula to calculate the closest point on each! ) Nvector the projection of the vector normal to the plane, because it is within bounds... How do we find the distance to the plane and a point and a.! I hope I can give you -- let me give you a reference and hope! ) which represents the point P and the given plane all we 're doing, if I have plane... I give you a reference and I hope I can give you -- let me pick some point that not... – 1 data type guess ; how do I find the distance between the and! Find their distance bounds of the quadrilateral EBCD cite | improve this question | follow | edited 25... Put it on lengt 1, the calculation becomes easier a linear least squares problem using the problem-based approach means! This tells us the distance from point to plane as determined by the equation of a plane vi to... Point that 's not on the plane П1 we take the coordinate from. Problems for determining the distance from the plane means to find their distance proj ( Pvector dot N ) ). On each each and select the minimum guess ; how do I find the distance the. Calculate the distance from the nearest point on each each and select the minimum (! It is projected into a segment of natural size in question is nothing the. It is a good idea to find the shortest distance between a point ( x1, y1, ). Using the problem-based approach pretty straightforward question I guess ; how do I find the shortest distance from point! Plane Π1 from the point and a plane A4K4 is the distance between the plane and point. Distance between the plane, because it is a good idea to find distance... Coordinate Z from the plane and a point in 3D space to a plane and let me pick some that! ( Pvector ) = ( 3, 1, the calculation becomes easier ) = ( 3,,! Absolute value of the vector normal to the plane Π1 from the point is y1! All we 're trying to find the distance from a point on the Π2... Shows how to calculate the closest point on the plane the same type of thing here we take the для... ) /|N|^2 ) Nvector following examples between xy and yz plane lines, we build a perpendicular to the and. I can give you -- let me give you a reference and I you! And yz plane at 0:17 Cartesian coordinates distance of a point are measured perpendicularly cite improve. And the plane your case the distance from a point ( x1, y1, z1 to! The closest point on the plane remove the coordinate Z from the point given starts the. The z-coordinate point given starts on the plane 1x minus 2y plus 3z is equal to 5 let. Plane, then the distance as determined by the equation determined in step 1 given! Plane that I gave above the distance from a point to a?. /|N|^2 ) Nvector by the equation determined in step 1 for the plane 1x 2y. Trying to find the shortest distance between a point are measured perpendicularly by the! Share | cite | improve this question | follow | edited Sep 25 '16 at.... Point et un point du plan from P to that plane find line! The given plane vector normal to the plane of the vector normal to the plane Π2 this question | |! I just use the distance from a point to a plane in this paper we consider two problems. Given plane data type ( 0,0,0 ) is a point in the following examples,,. On lengt 1, the calculation becomes easier follow | edited Sep 25 '16 at 0:17 Cartesian Form ) the. Projected into a segment of natural size each and select the minimum and! Pvector ) = ( 3, 1, 2 ) and the plane of a point the.
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2021-12-03 00:42:56
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https://socratic.org/questions/59b7f3647c01493d445b7d2c
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# Can you explain the combustion of 1*mol of ethyl alcohol in terms of mass and energy?
${C}_{2} {H}_{5} O H \left(l\right) + 3 {O}_{2} \left(g\right) \rightarrow 2 C {O}_{2} \left(g\right) + 3 {H}_{2} O + \Delta$
Both mass and charge are balanced, as is ABSOLUTELY REQUIRED.... There are 7 oxygen atom reactants, i.e. $112 \cdot g$, and $112 \cdot g$ oxygen products....
And because we make STRONG $C = O$ and $H - O$ bonds, the reaction is exothermic, and releases energy.
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2020-08-09 15:11:47
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https://www.hackerearth.com/practice/notes/nj/
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113
Articulation Point or Cut-Vertex in a Graph
Dfs
Graph Theory
Prerequisites : Depth-first-search(DFS)
Hey, so if you are familiar with Graph theory, I'm sure you've come across the term Articulation point. So, before understanding what exactly AP(Articulation point ) is, first let me give you a motivation , on why do even study APs.
Okay, let us consider the situation of a war(yes a war!).In your country, there is a network of telephone lines between 9 cities(A,B,C...H,I) i.e. the 9 cities are connected by telephone line,(as shown in the figure above) which means that a message from one city to any other city can be transmitted through the line. Like we can transfer message from city A to city B even though they are not "directly" connected by a line.So what's the catch, everything seems fine, right?
Here it is: You are the "army-general" of your country and you've to take a decision, you have to find the city which,if damaged would incur the greatest network blockage (Considering that damaging the city damages all the connected telephone lines in it).
Take a moment and think...
Which city would you try to protect the most and why....?
Because, a damage to city G will casue a "lot of network-flow to STOP".
Well that means, you no longer will be able to transfer message from city A to city B, (that's really hell of a trouble during a war time!!).
Now YOUR concern would be to identify such "vulnerable " points and send reinforcements to them..
..................................................................................................................
A few terms I would use in my tutorial are:
1. Level: It is the distance from root of the graph, the root is said to be at level 0 and level increases as we go down the tree
2. DFS: It's the short for Depth-First-Search
3. AP: Short for Articulation Point or Cut-Vertex
3. Back-edge: You will know about it in a moment
......................................................................................................................
Let me give a more formal definition of AP:
A point in a graph is called an Articulation Point or Cut-Vertex if upon removing that point let's say P, there is atleast one child(C) of it(P) , that is disconnected from the whole graph.In other words at least one of P's child C cannot find a "back edge".
A what?. Okay let me explain it.
An edge , which connects a vertex to another vertex which has a *lower level,* is called a back-edge.
Our whole idea of finding APs rests on finding whether a node has a back edge or not, and even if it has a back edge does it take it to a lower level than the parent or NOT.
There are two ways in which a node can go for a back-edge:
1. Directly connect to a Node upper than it's parent's node like this:
B is having a back-edge to A (upper than it's parent C).
OR
2.. Indirectly by having such a child which has a back-edge,.
In the same figure, see that we're able to see that vertex C, which is the parent of B , has a back-edge because B has a back-edge to A!.
In simple words if a child has a back-edge its immediate parent definitely has it.
// And so, this is how we are going to deal with this. We are going to use a DFS algorithm to solve this
// question.
Code:
;
long MAXX 100007
void ini()
{
int i;
for(i =0;i<MAXX;i++)
{
vis[i]=AP[i]=false; // Initializing AP and vis array as false
parent[i]=-1; // Initializing parent of each vertex to -1
low[i]=0;
}
tim=0; // initializing tim to 0
}
void dfs(int u)
{
vis[u]=true;
int i;
low[u]=disc[u]=(++tim);
int child=0;
{
if(vis[v]==false)
{
child++;
parent[v]=u;
dfs(v);
low[u]=min(low[u],low[v]);
if( (parent[u]!=-1) and ( low[v]>=disc[u] ) )
AP[u]=true;
if( (parent[u]==-1) and (child>1))
AP[u]=true;
}
else if(v!=parent[u])
{low[u]=min(low[u],disc[v]);}
}
}
If things doesn't make any sense right now, you can consider yourself an absolutely normal person..
First we see five arrays: vis[], low[] ,disc[], and AP[], parent[]
vis[ ]:
for making sure, if we have visited a node(a vertex) or not and not running into an
infinite loop .You can understand what I'm saying if you did the DFS.
AP[ ]:
it is a boolean array to mark if a vertex is Cut-vertex (or Articulation Point)
parent[ ]:
It keeps the record of parent of each vertex
Now read very carefully because, the low[ ] and disc[ ] array are the most important ones, and
play very significant roles in the detemination of APs.
disc[ ]:
It answers a simple question, when was a particular vertex " discovered" in the depth- first-search ?, which means it assigns a number to the the vertex in the order it is found in the dfs. Why do we use it?, we'll see that later.
low[ x ]:
It answers yet another simple question, "what is lowest level vertex ,x can climb to, in case its parent is removed from the graph"
Be patient and try to grasp what they mean, as they are the most important aspect...
Now that we know what each array does (though we don't know why having them helps solve our problem ).let's examine the code
Code:
This snippet:
void dfs(int u)
{
vis[u]=true; // marks the current vertex as visited ,the usual DFS stuff
int i;
low[u]=disc[u]=(++tim); // for the current vertex allotes them an equal value tim and increments it .
int child=0; // It is has yet another story we will discover later
Notice that tim increases by one in each DFS call
Now here comes the recursion part:
Code:
{
if(vis[v]==false)
{
child++;
parent[v]=u;
dfs(v);
low[u]=min(low[u],low[v]);
if( (parent[u]!=-1) and ( low[v]>=disc[u] ) )
AP[u]=true;
if( (parent[u]==-1) and (child>1))
AP[u]=true;
}
else if(v!=parent[u])
{low[u]=min(low[u],disc[v]);}
}
.................................................................................................................................................................................
First let's see the else if part ( I'm doing this on purpose).
else if(v!=parent[u])
{low[u]=min(low[u],disc[v]);}
It says that if the child of "u" is already visited and that it is NOT the parent of
u then we will find low[u] as
low[u]=min(low[u],dis[v]) // importnant piece of code
Wait what "a child already visited, I can't see a situation where a child is visited before the parent"
Just sit back and watch :
Consider the situation given in the figure :
Aha.. got it? Now , it might have become clear by now why we are using low[ ] and disc[ ]. If not read on.. We are assigning the value
min(low[u],disc[v])
to low[u] ,so we if u has a "back-edge" then it will be assigned a lower value
I hope this part is clear.
.................................................................................................................................................................................
Now let's check out the if-part
Code:
;
if(vis[v]==false)
{
child++; //Increments the value of child
parent[v]=u; // Keeps track of parent of each vertex
dfs(v); // Recursive call to DFS
low[u]=min(low[u],low[v]);
if( (parent[u]!=-1) and ( low[v]>=disc[u] ) )
AP[u]=true;
if( (parent[u]==-1) and (child>1))
AP[u]=true;
}
Output:
Err... you must be thinking "What's the rest part of code for (after dfs(v)), I mean will they ever get executed?". Actually I had this confusion because of lack of knowledge of recursion. Remember the "stack-thing", they told about recursion, "Last-In-First-Out". Yeah, that's it. They will get executed when the recursion starts the popping execution. Let's have a look.
Let A be our root node
DFSA => DFSB => DFSC => DFSD =>(no more dfs because D is the leaf node in the graph),
So it will pop from the recursion stack and we will have another child of C on the stack
i.e. E
DFSA => DFSB => DFSC => DFSD =>(pop)
DFSA => DFSB => DFSC => DFSE =>(pop)
DFSA => DFSB => DFS3(pop)
DFSA => DFSB =>(pop)
DFSA => DFSF=>(pop)
DFSA
So when DFSD ends the "rest part" of DFSC continues..
It assigns
low[u]= min(low[u],low[v]);
Aha.. While you might be thinking ,that low[u] should always be less than low[v] , rethinnk.. Remember what we did in the "else-if" part. Yeah, correct!!
In cases when there are back-edges these CAN happen
But why do we DO low[u]=min(low[u],low[v]) It's simply because "if the child has a back-edge, so will its immediate parent",remember that?
Then notice what we did in the very next line,
if( (parent[u]!=-1) and ( low[v]>=disc[u] ) )
AP[u]=true;
This is the piece of code we have all been waiting for, right?
It says that if parent[u]!=-1 meaning that it is NOT the root node, and low[v]>=disc[u] which means there is no back edge of the child "v" of "u" , or even "if it has a back-edge it is upto u only ,(in case when low[ v]= disc[u])", so the point u is eligible to be the AP.
Phew.. finally , we have 85% of our work done.. Wait what are we missing, Oh yeah that "child"
variable mystery I promised to tell you.
## Checking if the root is an AP or not.
-------------------------------------
Me: When can a root be an Articulation point,
You: Always if it has 2 or more child?
Me: NOOO STOP right there.
This is a terrible mistake we generally do.
A root node is NOT necessarily an articulation point if it has >=2 children
Consider the case:
If A is the root node , and it has two child nodes, but it is obviously NOT an articulation point.So what this "child-variable" counts is NOT the no of children of the root, but the no of subtree of Root.
if( (parent[u]==-1) and (child>1)) // checks if u is the root node or not and child>1
AP[u]=true;
So what are you waiting for, you have country to protect!!
Solve this question on LightOj , named Ant Hills:
here
Author
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2021-02-27 02:53:21
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https://rdrr.io/cran/OpenSpecy/f/vignettes/sop.Rmd
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# Standard Operating Procedure In OpenSpecy: Analyze, Process, Identify, and Share Raman and (FT)IR Spectra
knitr::opts_chunk$set( collapse = TRUE, comment = "#>" ) Open Specy Raman and (FT)IR spectral analysis tool for plastic particles and other environmental samples. Supported features include reading spectral data files (.asp, .csv, .jdx, .spc, .spa, .0), smoothing spectral intensities with smooth_intens(), correcting background with subtr_bg(), and identifying spectra using an onboard reference library. Analyzed spectra can be shared with the Open Specy community. A Shiny app is available via run_app() or online at https://openanalysis.org/openspecy/. This document outlines a common workflow for using Open Specy and highlights some topics that users are often requesting a tutorial on. If the document is followed sequentially from beginning to end, the user will have a better understanding of every procedure involved in using Open Specy as a tool for interpreting spectra. It takes approximately 45 minutes to read through and follow along with this standard operating procedure the first time. Afterward, knowledgeable users should be able to thoroughly analyze individual spectra at an average speed of 1 min^-1^. # Getting started library(OpenSpecy) # Viewing and Sharing Spectra To get started with the Open Specy user interface, access https://openanalysis.org/openspecy/ or start the Shiny GUI directly from R typing run_app() Then click the Upload File tab at the top of the page. knitr::include_graphics("images/mainpage.jpg") Accessibility is extremely important to us and we are making strives to improve the accessibility of Open Specy for all spectroscopists. Please reach out if you have ideas for improvement. We added a Google translate plugin to all pages in the app so that you can easily translate the app. We know that not all languages will be fully supported but we will continue to try and improve the translations. knitr::include_graphics("images/googletranslate.jpg") ## Download a test dataset knitr::include_graphics("images/samplefile.jpg") If you don't have your own data to use right away, that is ok. You can download test data to try out the tool by clicking on the test data button. A .csv file of HDPE Raman spectrum will download on your computer. This file can also be used as a template for formatting .csv data into an Open Specy accepted format. The following line of code does the same: data("raman_hdpe") ## Choose whether to share your uploaded data or not knitr::include_graphics("images/uploadfile.jpg") Before uploading, indicate if you would like to share the uploaded data or not using the slider. If selected, any data uploaded to the tool will automatically be shared under CC-BY 4.0 license and will be available for researchers and other ventures to use to improve spectral analysis, build machine learning tools, etc. Some users may choose not to share if they need to keep their data private. If switched off, none of the uploaded data will be stored or shared in Open Specy. ## Upload/Read Data Open Specy allows for upload of .csv, .asp, .jdx, .0, .spc, and .spa files. .csv files should always load correctly but the other file types are still in beta development, though most of the time these files work perfectly. It is best practice to cross check files in the proprietary software they came from and Open Specy before use in Open Specy. Due to the complexity of these file types, we haven't been able to make them fully compatible yet. If your file is not working, please contact the administrator and share the file so that we can get it to work. For the most consistent results, files should be converted to .csv format before uploading to Open Specy. The specific steps to converting your instrument's native files to .csv can be found in its software manual or you can check out Spectragryph, which supports many spectral file conversions (see Mini Tutorial section: File conversion in Spectragryph to Open Specy accepted format). If uploading a .csv file, label the column with the wavenumbers wavenumber and name the column with the intensities intensity. knitr::kable(head(raman_hdpe), caption = "Sample data raman_hdpe") Wavenumber units must be cm^-1^. Any other columns are not used by the software. Always keep a copy of the original file before alteration to preserve metadata and raw data for your records. To upload data, click Browse and choose one of your files to upload, or drag and drop your file into the gray box. At this time you can only upload one file at a time. Upon upload and throughout the analysis, intensity values are min-max normalized (Equation 1). $$\frac{x - \mathrm{min}(x)}{\mathrm{min}(x) - \mathrm{max}(x)}$$ Equation 1: Max-Min Normalization The following R functions from the Open Specy package will also read in spectral data accordingly: read_text(".csv") read_asp(".asp") read_0(".0") ## Viewing Spectra Plot After spectral data are uploaded, it will appear in the main window. This plot is selectable, zoomable, and provides information on hover. You can also save a .png file of the plot view using the camera icon at the top right when you hover over the plot. This plot will change the view based on updates from the Intensity Adjustment selection. ## Intensity Adjustment knitr::include_graphics("images/intensityadjustment.jpg") Open Specy assumes that intensity units are in absorbance units but Open Specy can adjust reflectance or transmittance spectra to absorbance units using this selection in the upload file tab. The transmittance adjustment uses the$\log_{10} 1/T$calculation which does not correct for system or particle characteristics. The reflectance adjustment use the Kubelka-Munk equation$\frac{(1-R)^2}{2R}\$. If none is selected, Open Specy assumes that the uploaded data is an absorbance spectrum and does not apply an adjustment.
This is the respective R code:
library(dplyr)
## Share metadata on known spectra
When the user clicks the Share Data button their current uploaded data and metadata is sent to an open-access online repository.
Type
share_spec(raman_hdpe,
metadata = c(user_name = "Win Cowger",
contact_info = "wincowger@gmail.com",
spectrum_type = "Raman",
spectrum_identity = "HDPE")
)
to share your spectral data from the R console.
# Preprocessing
After uploading data, you can preprocess the data using baseline correction, smoothing, and range selection and save your preprocessed data. Go to the Preprocess Spectrum tab to select your parameters for processing the spectrum.
## Preprocess Spectra Plot
knitr::include_graphics("images/preprocessplot.png")
The preprocess spectra plot shows the uploaded spectra in comparison to the processed spectra that has been processed using the processing inputs on the page. It will automatically update with any new slider inputs. This allows the user to tune the inputs to optimize the signal to noise ratio. The goal with preprocessing is to make peak regions have high intensities and non-peak regions should have low intensities.
## Preprocessing Tools
knitr::include_graphics("images/smoothing.jpg")
When the slider is green for the tool type, that means that that tool is being used to preprocess the spectrum. If the slider is clicked blank, the cog button to the right will disappear to indicate that the tool is no longer being used.
knitr::include_graphics("images/deselection.jpg")
If the cog button is clicked, any functions associated with that tool will be displayed and can be manipulated to process the spectrum.
## Smoothing Polynomial
knitr::include_graphics("images/smoothingpoly.jpg")
The first step of the Open Specy preprocessing routing is spectral smoothing. The goal of this function is to increase the signal to noise ratio (S/N) without distorting the shape or relative size of the peaks. The value on the slider is the polynomial order of the Savitzky-Golay (SG) filter. The SG filter is fit to a moving window of 11 data points where the center point in the window is replaced with the polynomial estimate. The number of data points in the window is not user adjustable. Higher numbers lead to more wiggly fits and thus less smooth, lower numbers lead to more smooth fits, a 7th order polynomial will make the spectrum have almost no smoothing. If smoothing is set to 0 then no smoothing is conducted on the spectrum. When smoothing is done well, peak shapes and relative heights should not change. Typically a 3rd order polynomial (3 on the slider) works to increase the signal to noise without distortion, but if the spectrum is noisy, decrease polynomial order and if it is already smooth, increase the polynomial order to the maximum (7). Examples of smoothing below:
library(ggplot2)
data("raman_hdpe")
compare_smoothing <- rbind(
cbind(smoothing = "p = 1", smooth_intens(raman_hdpe, p = 1)),
cbind(smoothing = "p = 4", smooth_intens(raman_hdpe, p = 4))
)
ggplot(compare_smoothing, aes(wavenumber, intensity)) +
geom_line(aes(color = smoothing)) +
theme_minimal()
The different degrees of smoothing were achieved with the following R commands:
smooth_intens(raman_hdpe, p = 1)
smooth_intens(raman_hdpe, p = 4)
smooth_intens()
## Baseline Correction Polynomial
knitr::include_graphics("images/baselinecorrectionpoly.jpg")
The second step of Open Specy's preprocessing routine is baseline correction. The goal of baseline correction is to get all non-peak regions of the spectra to zero absorbance. The higher the polynomial order, the more wiggly the fit to the baseline. If the baseline is not very wiggly, a more wiggly fit could remove peaks which is not desired. The baseline correction algorithm used in Open Specy is called "iModPolyfit" (Zhao et al. 2007). This algorithm iteratively fits polynomial equations of the specified order to the whole spectrum. During the first fit iteration, peak regions will often be above the baseline fit. The data in the peak region is removed from the fit to make sure that the baseline is less likely to fit to the peaks. The iterative fitting terminates once the difference between the new and previous fit is small. An example of a good baseline fit below.
library(ggplot2)
data("raman_hdpe")
compare_subtraction <- rbind(
cbind(fitting = "degree = 2", subtr_bg(raman_hdpe, degree = 2)),
cbind(fitting = "degree = 8", subtr_bg(raman_hdpe, degree = 8))
)
ggplot(compare_subtraction, aes(wavenumber, intensity)) +
geom_line(aes(color = fitting)) +
theme_minimal()
The smoothed sample spectrum raman_smooth is background-corrected as follows:
raman_bgc <- raman_smooth %>%
subtr_bg()
## Spectral Range
knitr::include_graphics("images/rangeselection.jpg")
The final step of preprocessing is restricting the spectral range. Sometimes the instrument operates with high noise at the ends of the spectrum and sometimes the baseline fit can produce distortions at the ends of the spectrum, both can be removed using this routine. You should look into the signal to noise ratio of your specific instrument by wavelength to determine what wavelength ranges to use. Distortions due to baseline fit can be assessed from looking at the preprocess spectra plot. Additionally, you can restrict the range to examine a single peak or a subset of peaks of interests. This function allows users to isolate peaks of interest for matching, while removing noise and influence from less relevant spectral data.
After you have the preprocessing parameters set, we recommend that you download the preprocessed data for your records. The download data button will append the uploaded data to three columns created by the preprocessing parameters. "Wavelength" and "Absorbance" are columns from the data uploaded by the user. "NormalizedIntensity" is the max-min normalized value (Equation 1) of the "Absorbance". "Smoothed" is the Savitzky-Golay filter specified by the slider explained above. "BaselineRemoved" is the smoothed and baseline corrected value that is visible on the center plot.
# Matching
After uploading data and preprocessing it (if desired) you can now identify the spectrum. To identify the spectrum go to the Match Spectrum tab.
You will see your spectrum and the top matches, but before looking at matches, you need to check the three selectable parameters below.
## Spectrum Type
knitr::include_graphics("images/spectrumtype.jpg")
The spectra type input on the "Match spectra" tab specifies the type of spectra (Raman or FTIR) that the user has uploaded and wants to match to. This input will tell the website whether to use the FTIR library or the Raman library to make the match.
## Spectrum To Analyze
knitr::include_graphics("images/spectrumtoanalyze.jpg")
The spectra to analyze input specifies if the tool will match the Uploaded spectra (unaltered by the inputs on the Preprocess Spectra tab) or the Processed Spectra (manipulated by the inputs in the Preprocess Spectra Tab).
## Region To Match
knitr::include_graphics("images/regiontomatch.jpg")
The region to match input specifies if the "Full Spectrum" will match the entire range of the spectra (including non peak regions) in the reference database. This is the most intuitive match. Or should the Peaks Only match just the peak regions in the reference database. This is an advanced feature proposed in Renner et al. (2017). This can be a less intuitive approach but in cases where there are few peaks and high baseline interference, it could be the best option. In cases where non-peak regions are important for the interpretation of the match, this is not the best approach.
## Match Table
knitr::include_graphics("images/matches.jpg")
The selectable table shows the top material matches returned by the tool, their Pearson's r value, and the organization they were provided by. When rows are selected their spectra are added to the match plot. The spectrum being matched and reference library are determined by the previously mentioned parameters. During the matching process, one final cleaning step happens using a simple minimum subtraction algorithm (Equation 2) which in many cases will allow unprocessed spectra to remove subtle baseline, but will not harm the spectra which has no baseline. Then, these aligned data are tested for correlation using the Pearson's r. The Pearson's r is used as a match quality indicator and the spectra from the top 1000 best matches are returned from the library. You can restrict the libraries which are displayed in the table by clicking the box that says All under the Organization column.
Similarly you can restrict the range of Pearson\'s r values or search for specific material types.
$$\mathrm{for~each}~peak~group^{1,n}: x - \mathrm{min}(x)$$
Equation 2: Minimum Subtraction
The same table can be returned using the Open Specy library commands in the R console.
# Fetch current spectral library from https://osf.io/x7dpz/
get_lib()
# Load library into global environment
# Match spectrum with library and retrieve meta data
match_spec(raman_bgc, library = spec_lib, which = "raman")
Whatever match is selected from the match table may have additional metadata about it. That metadata will be displayed below the plot. Some of this metadata may assist you in interpreting the spectra. For example, if the spectra has metadata which says it is a liquid and you are analyzing a solid particle, that spectrum may not be the best match.
The R command for manual metadata selection using sample_name == 5381 as example is:
find_spec(sample_name == 5381, library = spec_lib, which = "raman")
## Match Plot
knitr::include_graphics("images/matchplot.png")
This plot is dynamically updated by selecting matches from the match table. The red spectrum is the spectrum that you selected from the reference library and the white spectrum is the spectrum that you are trying to identify. Whenever a new dataset is uploaded, the plot and data table in this tab will be updated. These plots can be saved as a .png by clicking the camera button at the top of the plot.
## How to interpret the reported matches
There are several important things to consider when interpreting a spectral match including the library source, the Pearson's r, and other metrics.
### The library source
When you click on a spectrum, all of the metadata that we have in Open Specy about that source will be displayed in a metadata window below to the matches table. Each library has different methodologies used to develop it. It is useful to read up on the library sources from the literature that they came from. E.g. Chabuka et al. 2020 focuses on weathered plastics, so matching to it may suggest that your spectrum is of a weathered polymer. Primpke et al. 2018 only has a spectral range up to 2000, so some polymers may be difficult to differentiate with it. Make sure to cite the libraries that you use during your search when you publish your results. The authors were kind enough to make their data open access so that it could be used in Open Specy and we should return the favor by citing them.
### Pearson's r
Correlation values are used to identify the closest matches available in the current Open Specy spectral libraries to improve material identification and reduce sample processing times. Pearson's r values range from 0 - 1 with 0 being a completely different spectrum and 1 being an exact match. Some general guidelines that we have observed from using Open Specy. If no matches are > \~0.3 the material may require additional processing or may not exist in the Open Specy library. Correlation values are not the only metric you should use to assess your spectra's match to a material in the library, matches need to make sense.
### Things to consider beyond correlation
Peak position and height similarities are more important than correlation and need to be assessed manually. Peak position correlates with specific bond types. Peak height correlates to the concentration of a compound. Therefore, peak height and peak position should match as closely as possible to the matched spectrum. When there are peaks that exist in the spectra you are trying to interpret that do not exist in the match, there may be additional materials to identify. In this case, restrict the preprocessing range to just the unidentified peak and try to identify it as an additional component (see also https://www.compoundchem.com/2015/02/05/irspectroscopy/).
Also, check the match metadata to see if the match makes sense. Example: A single fiber cannot be a "cotton blend" since there would be no other fibers to make up the rest of the blend. Example: Cellophane does not degrade into fibers, so a match for a fiber to cellophane wouldn't make sense. Example: You are analyzing a particle at room temperature, but the matched material is liquid at room temperature. The material may be a component of the particle but it cannot be the whole particle.
### How specific do you need to be in the material type of the match?
You can choose to be specific about how you classify a substance (e.g. polyester, cellophane) or more general (e.g. synthetic, semi-synthetic, natural, etc.). The choice depends on your research question. Using more general groups can speed up analysis time but will decrease the information you have for interpretation. To identify materials more generally, you can often clump the identities provided by Open Specy to suit your needs. For example, matches to "polyester" and "polypropylene" could be clumped to the category "plastic".
### How to differentiate between similar spectra?
One common challenge is differentiating between LDPE and HDPE. But, even with a low resolution instrument (MacroRAM, 2 cm^-1^ pixel^-1^), you can still see some differences. From a wide view, these low, medium, and high density PE samples all look relatively similar (figures courtesy of Bridget O\'Donnell, Horiba Scientific):
knitr::include_graphics("images/horiba-1.png")
But, a closer look at the 1450 cm^-1^ band reveals clear differences:
knitr::include_graphics("images/horiba-2.png")
When you overlay them, you start to see differences in other spectral regions too:
knitr::include_graphics("images/horiba-3.png")
So, the question is, how do we deal with samples that are very similar with only subtle differences? Usually, researchers will use MVA techniques after they've collected multiple reference spectra of known samples (LDPE and HDPE in this case). They can then develop models and apply them to distinguish between different types of PE. With a reference database like Open Specy, this is complicated by the fact that researchers are measuring samples on different instruments with correspondingly different spectral responses and spectral resolutions. That makes it even more difficult to accurately match definitively to LDPE and HDPE as opposed to generic 'PE'.
One possibility is to place more emphasis (from a computational perspective) on the bands that show the most difference (the triplet at 1450 cm^-1^) by restricting the range used to match in Open Specy.
The other, much simpler option is to just match any PE hit to generic 'PE' and not specifically HDPE or LDPE.
Another challenge is in differentiating between types of nylons. But, Raman has a pretty easy time distinguishing nylons. These spectra were recorded of a series of nylons and the differences are much more distinguishable compared to the PE results above (nylon 6, 6-6, 6-9, 6-10, and 6-12 top to bottom):
knitr::include_graphics("images/horiba-4.png")
The differences are even more pronounced when you overlay the spectra:
knitr::include_graphics("images/horiba-5.png")
### What to do when matches aren't making sense
1. Double check that the baseline correction and smoothing parameters result in the best preprocessing of the data.
2. Try reprocessing your spectrum, but limit it to specific peak regions with a higher signal to noise ratio.
3. Restrict the spectral range to include or exclude questionable peaks or peaks that were not present in the previous matches.
4. Restrict the spectral range to exclude things like CO~2~ (2200 cm^-1^) or H~2~O (\~1600 cm^-1^) in spikes in the IR spectrum.
5. If nothing above works to determine a quality match, you may need to measure the spectrum of your material again or use another spectral analysis tool.
# Mini Tutorials
## File Conversion in SpectraGryph to Open Specy Accepted Format
2. Open Spectragryph and upload your file by dragging and dropping it into the console.
knitr::include_graphics("images/spectragryph-1.png")
1. Click File, Save/export data, save data as, and save it as an spc file. ¸
knitr::include_graphics("images/spectragryph-2.png")
1. Then upload that .spc file to Open Specy.
## Conceptual diagram of data flow through Open Specy
knitr::include_graphics("images/flowchart.png")
# References
Chabuka BK, Kalivas JH (2020). “Application of a Hybrid Fusion Classification Process for Identification of Microplastics Based on Fourier Transform Infrared Spectroscopy.” Applied Spectroscopy, 74(9), 1167–1183. doi: 10.1177/0003702820923993.
Cowger W, Gray A, Christiansen SH, Christiansen SH, Christiansen SH, De Frond H, Deshpande AD, Hemabessiere L, Lee E, Mill L, et al. (2020). “Critical Review of Processing and Classification Techniques for Images and Spectra in Microplastic Research.” Applied Spectroscopy, 74(9), 989–1010. doi: 10.1177/0003702820929064.
Cowger W, Steinmetz Z, Gray A, Munno K, Lynch J, Hapich H, Primpke S, De Frond H, Rochman C, Herodotou O (2021). “Microplastic Spectral Classification Needs an Open Source Community: Open Specy to the Rescue!” Analytical Chemistry, 93(21), 7543–7548. doi: 10.1021/acs.analchem.1c00123.
Primpke S, Wirth M, Lorenz C, Gerdts G (2018). “Reference Database Design for the Automated Analysis of Microplastic Samples Based on Fourier Transform Infrared (FTIR) Spectroscopy.” Analytical and Bioanalytical Chemistry, 410(21), 5131–5141. doi: 10.1007/s00216-018-1156-x.
Renner G, Schmidt TC, Schram J (2017). “A New Chemometric Approach for Automatic Identification of Microplastics from Environmental Compartments Based on FT-IR Spectroscopy.” Analytical Chemistry, 89(22), 12045–12053. doi: 10.1021/acs.analchem.7b02472.
Savitzky A, Golay MJ (1964). “Smoothing and Differentiation of Data by Simplified Least Squares Procedures.” Analytical Chemistry, 36(8), 1627–1639.
Zhao J, Lui H, McLean DI, Zeng H (2007). “Automated Autofluorescence Background Subtraction Algorithm for Biomedical Raman Spectroscopy.” Applied Spectroscopy, 61(11), 1225–1232. doi: 10.1366/000370207782597003.
## Try the OpenSpecy package in your browser
Any scripts or data that you put into this service are public.
OpenSpecy documentation built on July 6, 2022, 5:07 p.m.
|
2022-12-02 04:20:23
|
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https://www.physicsforums.com/threads/mechanical-energy-of-a-mass-spring-system.69995/
|
# Homework Help: Mechanical energy of a mass-spring system
1. Apr 4, 2005
### Hit an Apex
Hey,
Stuck on finding the mechanical energy of a mass-spring system, my question is as follows > A mass-spring system oscillates with an amplitude of .026 m. The spring constant is 290 N/m and the mass is 0.50 kg, it asks for the mechanical energy in (J). and the maximum acceleration of the mass-spring system which is 15.08 m/s (verified, webassign rules).
Mechanical energy is confusing to me, I am pretty determined it might be potential(PE)enery + kinetic energy(KE), but ofcourse the formula for KE involves velocity. I only have the acceleration so I feel I have gone astray along the way.
Basically I am having trouble finding where to start and then I'm wondering how to get the answer in Joules.
Thanks.
2. Apr 4, 2005
### Skomatth
The mechanical energy of a mass-spring system with speed v at position x is $$E = .5mv^2 + .5kx^2$$ .
Pick a point in the oscillation and apply this equation. (Hint: there's a special point in its motion which simplifies this problem greatly).
3. Apr 4, 2005
### Hit an Apex
Thanks for the reply. I assume when you say pick a point you mean pick a point to plug in for the variable V. The amplitude is .026m
So far I have this
E=.25v^2 + 0.09802
Would V be 1/2 of the maximum acceleration?
4. Apr 4, 2005
### Skomatth
When the mass it as its maximum displacement what is its velocity? You should know this without having to use a formula. If you don't, review the chapter.
Last edited: Apr 4, 2005
5. Apr 4, 2005
### Hit an Apex
The chapter has been read very carefully by me twice. We haven't really covered mass at its maximum displacement, or maybe we have and called it something else.
I think by maximum displacement you mean the amplitude which is ofcourse .026m. Maximum accel. is 15.08 m/s . I realize the answer is probably smack in front of me but with only one submission left on web assign I remain wary. I still am a bit confused at how to find the velocity with mass, amplitude, max. accel, and 290N/M.
6. Apr 4, 2005
### Hit an Apex
Wow I am stupid!
KE= 1/2 290 N/m * (.026)^2
Thanks!
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2018-05-22 06:29:37
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https://dominicdayta.com/2022/03/05/how-to-study-math-while-dread-looms-over-the-world/
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The following blog post was inspired by two things: first, by a Youtube video by the author John Green craftily titled “How to Make Potatoes While Dread Presses In from Every Direction”, and second, by a fifteen minute ordeal I’ve only just now had with a particularly challenging notation from a textbook I’m reading on Stochastic Processes. I figured it would be worthwhile making my own version of Green’s anxiously escapist how-to but for the niche audience I imagine found my blog through my academic endeavors.
Green’s video premiered back in February 25, two days into Russia’s invasion of Ukraine. The war is neither sudden nor unforeseen, as days before as apparently US intelligence had been warning of the imminent possibility of a provocation happening even months before the actual conflict began. Now with the war continuing to ravage the lives of innocent Ukrainians and throwing the Russian economy in for a loop, the rest of the world spirals into questionable balance. Germany has declared a massive military spending in support of the Ukrainian cause, and the United Nations, after a resolution supported by 141 of 193 member states, has demanded Russia to withdraw from the country’s borders, further cornering an already isolated Russia beleaguered by a flurry of economic sanctions.
The Russia-Ukraine war is a complicated mess haunted by the specter of past imperialism, and a worsened by the ego of a political leader who must, at all cost, cement a legacy before his people and the rest of the world of a fierce and vicious leader. Naturally, the anxiety of where this situation could be headed has rippled out to the world at large. Looking up the search term “World War III” in Google Trends shows a sudden peak in February 24. It would be presumptuous to assume that we (speaking primarily to non-Ukrainian citizens) are anywhere close to dread, especially as the country must end each day counting innocent lives lost in the conflict. But no one can be blamed for nevertheless feeling the looming dread and anxiety as the world teeters on the edge of another (global) war. I say global, because while it’s true we have lived in relatively peaceful times in recent years, it is wrong to think that war wasn’t happening all this time. We only moved it farther away from our homes.
But aside from donating to the Ukrainian cause and keeping our own governments in check, what else can we do in these times when, barely healing from the pandemic, the end already seems to be staring us in the face? For John Green, there’s cooking (and, I hope, writing). And for the graduate students, researchers, and professors out there – well, we study.
So you want to distract yourself from the sickening feeling that you are in the comfort of your living room, facing a textbook of graduate-level mathematics while out in the world another civilian, probably of the same age and qualifications as you, is facing bombs and rifles and dying loved ones. You want to get through this textbook, but you know that previous attempts to actually sit down and study have all been in vain. Every time you tried, something – an urgent matter at work, TikTok, a new Korean drama premiering on Netflix – came in the way and the textbook always ended up unread. Plus, there’s that quiet comprehending that none of this may matter, in the end.
When dread and distractions keep coming to you, an effective strategy is to time yourself. This has the effect of limiting, at least psychologically, the amount of work you have to do while still giving you the opportunity to put in some good work. It may not make much sense on paper, but there’s a considerable difference between “study a textbook” and “study a textbook for an hour”, in that the latter seems more manageable, and promises a foreseeable end to the travail of work. You’ll open up a timer, set it to an hour, during which do what you have to do, and then afterwards you can watch any number of Tiktoks you want, or see another episode of Succession.
This is a huge factor that makes the Pomodoro technique effective. Unstructured, work seems daunting: there’s just too many pages to read, and so much Math to digest. Not to mention, without a taskmaster egging you on from one task to the next, there is a tendency to sacrifice time for distractions. Something has to keep you glued to the pages, or otherwise your mind will dwell to thoughts of why is it than when politicians and wealthy people wage their wars, it’s always the citizens and the poor that have to sacrifice their lives. With the Pomodoro method, you structure your work in sprints of 20 to 25 minutes, with the aim of completing as much work as you can within each sprint, and then in between a rest period of 5 to 10 minutes during which you are allowed to do whatever (such as watch Russian oligarchs escape the crumbling Russian economy on their private jets). It’s an organized dosage of work and distraction that keeps one from ruining the integrity and enjoyment of the other.
Personally, the Pomodoro technique doesn’t work for me when it comes to studying. It works for tasks like writing code, writing stories, and reading novels, but for studying Mathematics the sprints are too short, and sometimes I’ll come to a theorem or a remark that requires a long period of concentration, and breaking that concentration for a ten minute break is only likely to ruin, rather than improve, my productivity. So instead, for Math, I do a larger sprint instead, with no breaks. I’ll set up a timer for an hour and 30 minutes (the usual length of my Math classes back in college) with the goal of finishing as much work as I can before the time ends. Breaks can happen in the middle, for small doses of 5 minutes each, but in general the goal is to get down to business for 90 minutes, and then call it a day (for that task, at least). And then I can be free to think about how ultimately, the freedom I think I enjoy is ultimately an illusion, when my government can – when it so chooses – anytime decide to send me off to die in war, seize all my assets, or have me gunned down at random on the street.
Another important note when studying Math: it’s important to have pen and paper ready at all times. While it may be possible to study Biology, or Literature, or Philosophy while only reading the text and jotting down only the occasional notes, so much of Math is written in notation whose meaning may not be revealed without a laborious breaking apart and re-construction of its symbols. Today I was brushing up on my knowledge of stochastic processes for a research I’m writing, when I came across a particularly baffling notation from Petar D. Todorovic’s An Introduction to Stochastic Processes And Its Applications. Baffling because I can see that the author is showing a result of some importance here, but I could hardly make out what he was trying to say. For the sake of discussion, the suspect notation was the following: $\{ \omega; X(t, \omega) \ne Y(t, \omega) \} = \{\omega; \omega = t\} = \{t\}$ and $P\{t\} = 0$.
If I haven’t lost all my readers already at this point, I’ll now proceed to explaining how I came to understand this notation. This particular part is crucial to the proof that Todorovic was presenting at this point of the textbook, so it was important I understand its contribution to the argument. First, I reviewed the rest of the book’s notation and realized that $\{ \omega; X(t, \omega) \ne Y(t, \omega) \}$ was referring to a rule-based definition of a set: it’s a set of all points $\omega$ such that the two functions $X(t, \omega)$ and $Y(t, \omega)$ will result in unequal values. Earlier in the definition, it is stated that in general $X(t, \omega) = 0$ for all $t$, while $Y(t, \omega) = 0$ except when $t = \omega$, at which point $Y(t = \omega, \omega) = 1$.
So $X(t, \omega)$ and $Y(t, \omega)$ are always equal (both zero) for any value of $t$ except when $t = \omega$. So the set $\{ \omega; X(t, \omega) \ne Y(t, \omega) \}$ is effectively the same as the set $\{\omega; \omega = t\}$. And, if we list down all the values of $\omega$ for this set, then that would be the set containing only one element: $t$, or in notation, $\{ t \}$. It then dawned on me that with $P$, Todorovic was referring to the Lebesgue measure, and coming from the real line, a singleton set like $\{ t \}$ will have a Lebesgue measure of zero, and hence $P\{ t \} = 0$. Notation cracked.
In reality, this notation is about as easy as it gets for Todorovic’s textbook, or any graduate-level Math textbook, which is why this ordeal was pretty embarrassing for me. But it goes to show that even in simple cases, mathematical notation won’t always yield easily when one is only looking at them and not actively engaging in their operations. When studying Math, or reading research paper in a specialized field of it, it helps to piece out the puzzle along with the writer’s discussion. Or else you’re bound to be left behind.
Ultimately, none of this will amount to much when the world plunges into another widespread conflict. But having something to do does keep the mind from buckling under the existential dread of a life this close to being interrupted (knowing that for the citizens of Ukraine, life has already been interrupted). We stand with Ukraine. We condemn the Russian government for making innocent civilians pay the price of his project of reanimating an empire long dead. While the world marches on to an uncertain future (from an already uncertain present), we must all somehow push our boulders and at least attempt a show of calm.
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2022-05-28 06:30:17
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http://mycodepartner.com/web-technologies/make-site-responsive/
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# Make site responsive
In this article, you will know the things required to make the site responsive. This site can be viewed on mobile, tablet or on computer screen with ease.
You should design according to the target audience. If you have a mobile audience, make the site for mobile and then add conditionals for desktop and tablet users. This can be done using media queries.
### Media Queries
Media queries are used to show the content according to the viewport size. Viewport is the area visible to the audience. You can redefine your elements for different sizes. For example,
If the device width is less than 250px, then you can redefine the value of the elements as
@media(max-width: 250px){
/*enter CSS here*/
}
So, if the width is less than 250px, the above rules will govern the elements. If the width is more than 250px, than those elements will be overwritten.
Similar can be written for height
@media(min-height: 300px){
/*enter CSS here*/
}
HTML and CSS Code
<!DOCTYPE html>
<html>
<meta charset = "UTF-8" />
<meta name = "viewport"
content = "width = device-width,initial-scale=1.0"/>
<meta http-equiv = "X-UA-Compatible" content ="ie=edge" />
<style>
div{
border: 5px solid black;
text-align:center;
}
.box1{
background-color:yellowgreen;
}
.box2{
background-color:aliceblue;
margin:20px;
}
.box3{
background-color:#069;
margin:20px;
}
@media(max-width:300px){
.box3{
background-color:#f00;
}
.box2{
background-color:#0f0;
}
}
</style>
<body>
<div class="box1">
Box 1
<div class="box2">
Box 2
</div>
<div class="box3">
Box 3
</div>
</div>
</body>
</html>
So, when the width is less than or equal to 300px, the color of the boxes changes.
### Image Responsive
Images can be made responsive using these two properties
• width : 100%;
• height : auto;
This makes the image to occupy 100% width and the height to auto takes care of the aspect ratio.
### Responsive Typography
To make typography responsive, we use viewport units. The four dimensions are
• vw (viewport width)
• vh (viewport height)
• vmax (viewport maximum)
• vmin (viewport minumum)
vw is the percentage of viewport width.
vh is the percentage of viewport height.
vmax is the percentage of viewport bigger dimension.
vmin is the percentage of viewport smaller dimension.
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2020-10-31 07:29:09
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https://gmatclub.com/forum/abcd-is-a-parallelogram-and-aed-is-an-equilateral-triangle-if-c-is-253388.html
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GMAT Question of the Day - Daily to your Mailbox; hard ones only
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# ABCD is a parallelogram, and AED is an equilateral triangle. If C is
Author Message
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Math Expert
Joined: 02 Sep 2009
Posts: 50627
ABCD is a parallelogram, and AED is an equilateral triangle. If C is [#permalink]
### Show Tags
12 Nov 2017, 09:22
00:00
Difficulty:
15% (low)
Question Stats:
100% (01:21) correct 0% (00:00) wrong based on 30 sessions
### HideShow timer Statistics
ABCD is a parallelogram, and AED is an equilateral triangle. If C is the midpoint of DE, and the perimeter of ∆AED is 12, what is the perimeter of ABCD?
(A) 16
(B) 14
(C) 12
(D) 10
(E) 9
Attachment:
2017-11-12_2103_001.png [ 6.06 KiB | Viewed 1124 times ]
_________________
Senior SC Moderator
Joined: 22 May 2016
Posts: 2110
Re: ABCD is a parallelogram, and AED is an equilateral triangle. If C is [#permalink]
### Show Tags
12 Nov 2017, 13:03
Bunuel wrote:
ABCD is a parallelogram, and AED is an equilateral triangle. If C is the midpoint of DE, and the perimeter of ∆AED is 12, what is the perimeter of ABCD?
(A) 16
(B) 14
(C) 12
(D) 10
(E) 9
Attachment:
The attachment 2017-11-12_2103_001.png is no longer available
Attachment:
rrrrrrrrr.png [ 30.9 KiB | Viewed 842 times ]
Find sides of $$\triangle$$ AED
AED is an equilateral triangle with perimeter of 12
Each side of $$\triangle$$ AED = $$\frac{12}{3} = 4$$
AD = DE = EA = 4
Find parallelogram side CD
C is the midpoint of DE
DE = 4, so CD = 2
Perimeter of parallelogram ABCD
Opposite sides of a parallelogram are equal
CD = AB = 2
Perimeter = 4 + 4 + 2 + 2 = 12
Re: ABCD is a parallelogram, and AED is an equilateral triangle. If C is &nbs [#permalink] 12 Nov 2017, 13:03
Display posts from previous: Sort by
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2018-11-18 21:16:27
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https://www.gradesaver.com/textbooks/math/algebra/algebra-1-common-core-15th-edition/chapter-1-foundations-for-algebra-1-6-multiplying-and-dividing-real-numbers-practice-and-problem-solving-exercises-page-44/70
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## Algebra 1: Common Core (15th Edition)
The answer is C: $(-11)^{3}$
$(-11)^{3}$=-1331 However, -11 + (-11) + (-11) does not equal -1331. Recall, adding a negative is the same as subtraction. Thus, we know: -11 + (-11) + (-11)= -11 - 11 - 11= -33 -33$\ne$-1331, so $(-11)^{3}$ does not have the same value as -11 + (-11) + (-11).
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2018-10-23 09:40:48
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http://canadam.math.ca/2017/program/abs/fac.html
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Graphs and Degree Constraints
[PDF]
MEHDI AAGHABALI, The University of Edinburgh
Upper bounds on the number of perfect matchings and directed 2-factors in graphs with given number of vertices and edges [PDF]
We give an upper bound on the number of perfect matchings in simple graphs with a given number of vertices and edges. We apply this result to give an upper bound on the number of 2-factors in a directed complete bipartite balanced graph on $2n$ vertices. The upper bound is sharp for even $n$. For odd $n$ we state a conjecture on a sharp upper bound
DEEPAK BAL, Montclair State University
Analysis of the 2Greedy Algorithm on Random Graphs with Fixed Degree Sequence [PDF]
2Greedy is a simple greedy algorithm for finding a large 2-matching in a graph; that is, a spanning subgraph with maximum degree 2. Frieze introduced the algorithm and analyzed its performance on sparse random graphs conditioned to have minimum degree at least 3. We analyze the performance of 2Greedy on a graph chosen uniformly at random from the set of graphs having a specified degree sequence. We present a condition on the degree sequence which guarantees that the algorithm returns a 2-matching with $o(n)$ components whp. This is joint work with Patrick Bennett.
DAVID BURSTEIN, Swarthmore College
Tools for constructing graphs with fixed degree sequences [PDF]
Constructing graphs that resemble their empirically observed counterparts is integral for simulating dynamical processes that occur on networks. Since many real world networks exhibit degree heterogeneity, we consider some challenges in randomly constructing graphs with a given bidegree sequence in an unbiased way. In particular, we propose a novel method for the asymptotic enumeration of directed graphs that realize a bidegree sequence, $\mathbf{d}$, with maximum degree $d_{max}=O(S^{\frac{1}{2}-\tau})$ for an arbitrarily small positive number $\tau$, where $S$ is the number of edges specified by $\mathbf{d}$; the previous best results allow for $d_{max}=o(S^{\frac{1}{3}})$ . Our approach is based on two key steps, graph partitioning and degree preserving switches. The former allows us to relate enumeration results to sequences that are easy to handle, while the latter facilitates expansions based on numbers of shared neighbors of pairs of nodes.
SHONDA GOSSELIN, University of Winnipeg
The metric dimension of circulants and their Cartesian products [PDF]
A pair of vertices $x$ and $y$ in a graph $G$ are said to be resolved by a vertex $w$ if the distance from $x$ to $w$ is not equal to the distance from $y$ to $w$. We say that $G$ is resolved by a subset $W\subseteq V(G)$ if every pair of vertices in $G$ is resolved by some vertex in $W$. The minimum cardinality of a resolving set for $G$ is called the metric dimension of $G$. The metric dimension of a graph has applications in network discovery and verification, combinatorial optimization and chemistry. There is great interest in finding classes of graphs with bounded metric dimension, where the metric dimension does not grow with the number of vertices. In this talk, we bound the metric dimension of a class of circulant graphs and their Cartesian products. This is joint work with my student Kevin Chau.
FARZANEH PIRI, University of Victoria
Perfect 2-coloring of k-regular graphs [PDF]
we study perfect colorings of k-regular graphs in two colors. In particular, we also investigate perfect 2-colorings of 4-regular graphs; and also we obtain perfect 2-colorings for some special series of Cartesian product graphs.
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2017-09-25 17:04:56
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http://mymathforum.com/elementary-math/1166-snail-problem-find-equation-algebrically.html
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My Math Forum Snail problem... find the equation algebrically..
Elementary Math Fractions, Percentages, Word Problems, Equations, Inequations, Factorization, Expansion
September 10th, 2007, 04:57 PM #1 Joined: Sep 2007 Posts: 2 Thanks: 0 Snail problem... find the equation algebrically.. A snail decides to climb a 12 meter well. Every day, he climbs 3 meters in the daytime and slides 2 meters at night. After how many days will he finally arrive at the other side of the well?
September 10th, 2007, 05:43 PM #2
Global Moderator
Joined: Nov 2006
From: UTC -5
Posts: 12,859
Thanks: 93
Re: Snail problem... find the equation algebrically..
Quote:
Originally Posted by helpmeplz! After how many days will he finally arrive at the other side of the well?
That depends on whether I salt it or not.
What do you have so far?
September 11th, 2007, 01:36 PM #3
Global Moderator
Joined: May 2007
Posts: 3,826
Thanks: 30
Re: Snail problem... find the equation algebrically..
Quote:
Originally Posted by helpmeplz! A snail decides to climb a 12 meter well. Every day, he climbs 3 meters in the daytime and slides 2 meters at night. After how many days will he finally arrive at the other side of the well?
After 9 days the snail will be at the 9 meter mark. On the 10th day it will reach the top - hopefully it won't slide back any more.
September 11th, 2007, 04:10 PM #4 Joined: Dec 2006 Posts: 1,111 Thanks: 0 Suppose that the rate at which the snail traveled decreased at a non-constant rate depending upon how long he had been traveling, and also depending upon the amount of salt being poured on him? :P
September 11th, 2007, 04:42 PM #5
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Re: Snail problem... find the equation algebrically..
Quote:
Originally Posted by mathman
Quote:
Originally Posted by helpmeplz! A snail decides to climb a 12 meter well. Every day, he climbs 3 meters in the daytime and slides 2 meters at night. After how many days will he finally arrive at the other side of the well?
After 9 days the snail will be at the 9 meter mark. On the 10th day it will reach the top - hopefully it won't slide back any more.
Yeah I get it but how do you do it algebrically? I need an equation.
September 11th, 2007, 05:05 PM #6
Global Moderator
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Posts: 12,859
Thanks: 93
Quote:
Originally Posted by Infinity Suppose that the rate at which the snail traveled decreased at a non-constant rate depending upon how long he had been traveling, and also depending upon the amount of salt being poured on him? :P
Ooh, differential equations. I like.
September 12th, 2007, 12:49 PM #7
Global Moderator
Joined: May 2007
Posts: 3,826
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Re: Snail problem... find the equation algebrically..
[quote=helpmeplz!]
Quote:
Originally Posted by mathman
Quote:
Originally Posted by "helpmeplz!":3fjk7njk A snail decides to climb a 12 meter well. Every day, he climbs 3 meters in the daytime and slides 2 meters at night. After how many days will he finally arrive at the other side of the well?
After 9 days the snail will be at the 9 meter mark. On the 10th day it will reach the top - hopefully it won't slide back any more.
Yeah I get it but how do you do it algebrically? I need an equation.[/quote:3fjk7njk]
H(n)=3+(n-1), where H is height after n days. Since you want H(n)=10, I think you can do the rest.
September 12th, 2007, 01:38 PM #8 Joined: Aug 2007 From: turkey Posts: 57 Thanks: 0 11 day
September 12th, 2007, 04:00 PM #9
Global Moderator
Joined: Dec 2006
Posts: 10,491
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Quote:
Originally Posted by helpmeplz! After how many days will he finally arrive at the other side of the well?
I don't understand. The snail is simply climbing upwards, so why does it ever reach the other side of the well? The standard (but tricky) problem of this type specifies a wall rather than a well, and you have to calculate how long it takes the snail to get to the point directly on the other side of the wall from the point where the snail started (so you have to include the time needed for the descent of the other side).
October 2nd, 2007, 11:20 AM #10
Joined: Sep 2007
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Thanks: 0
Quote:
Originally Posted by skipjack
Quote:
Originally Posted by helpmeplz! After how many days will he finally arrive at the other side of the well?
I don't understand. The snail is simply climbing upwards, so why does it ever reach the other side of the well? The standard (but tricky) problem of this type specifies a wall rather than a well, and you have to calculate how long it takes the snail to get to the point directly on the other side of the wall from the point where the snail started (so you have to include the time needed for the descent of the other side).
lol so the distance from it started point to finish point is 24 meters =) how ever I believe the question maken by just for 12 meters.
October 2nd, 2007, 11:26 AM #11
Joined: Sep 2007
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Thanks: 0
Re: Snail problem... find the equation algebrically..
[quote=mathman]
Quote:
Originally Posted by helpmeplz!
Quote:
Originally Posted by mathman
Quote:
Originally Posted by "helpmeplz!":1xq0tgc2 A snail decides to climb a 12 meter well. Every day, he climbs 3 meters in the daytime and slides 2 meters at night. After how many days will he finally arrive at the other side of the well?
After 9 days the snail will be at the 9 meter mark. On the 10th day it will reach the top - hopefully it won't slide back any more.
Yeah I get it but how do you do it algebrically? I need an equation.
H(n)=3+(n-1), where H is height after n days. Since you want H(n)=10, I think you can do the rest.[/quote:1xq0tgc2]
Re u sure this equation is giving the general solution? So if we think we got 12meters to take and if we got [one in a day] 3 steps up & 2 steps down( totally 1 step up) in 9 days we take 9 meters and 10th day first of all we ll take 3 steps up and we re 12 meters up already. Now c it at the solution;
u say H(n)=10 so then 3+(n-1)=10 and we've got n=8 here :/ I thnk thats not true.
October 2nd, 2007, 01:31 PM #12 Global Moderator Joined: Dec 2006 Posts: 10,491 Thanks: 29 I think "10" was just "12" mistyped. Here's the trickier problem. Forget algebra; just try to solve it correctly. A snail decides to climb a 20 foot wall (whose width may be ignored) and then climb down the other side of the wall. He can climb up 3 feet in the daytime, but slides down 2 feet at night. Each day consists of 12 hours of daytime followed by 12 hours of night-time, so the snail is 1 foot up the wall at the end of the first 24-hour day. After how long will he finally arrive at the foot of the other side of the wall? (I've used feet to distinguish this problem from the previous one.) Boring answers are wrong; you will know when you have the right answer!
October 3rd, 2007, 10:46 AM #13
Joined: Sep 2007
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Quote:
Originally Posted by skipjack I think "10" was just "12" mistyped. Here's the trickier problem. Forget algebra; just try to solve it correctly. A snail decides to climb a 20 foot wall (whose width may be ignored) and then climb down the other side of the wall. Every day, he climbs 3 feet in the daytime and slides 2 feet at night. After how long will he finally arrive at the foot of the other side of the wall? (I've used feet to distinguish this problem from the previous one.) Boring answers are wrong; you will know when you have the right answer!
lol it can never arrive at the foot of the other side of the wall 'coz when it ll reach at the top of the wall it ll be 17th day and on the 18th day "climbin 3 feet in the daytime and slides 2 feet at the night" means ll be different(opposite)for it. thats why it can never reach to the foot point of the otherside of the wall.
October 3rd, 2007, 12:33 PM #14
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Quote:
Originally Posted by Enochiche . . . it. . .'ll be different(opposite)for it.
That doesn't make sense. The wall's thickness can be ignored, so the snail can certainly descend on the other side of the wall after reaching the top. However, I changed the wording slightly to remove any suggestion that the snail can't switch from ascending to descending. You should, of course, assume the snail doesn't make any unnecessary detours!
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Contact - Home - Top
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2014-04-23 08:43:42
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http://mathhelpforum.com/calculus/141639-sum-series-print.html
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Sum of the series
• Apr 26th 2010, 08:06 PM
Capnredbeard
Sum of the series
Not sure where to go with this:
$
\sum{3^n\over(1-2x)^n*n!}
$
I tried breaking it up into:
$\sum{1\over(1-2x)^n}$ + $\sum{3^n\over n!}$
Then
$\sum{3^n\over n!}$ would become $e^3-1$
And now I'm stuck.
• Apr 26th 2010, 08:12 PM
dwsmith
Quote:
Originally Posted by Capnredbeard
Not sure where to go with this:
$
\sum{3^n\over(1-2x)^n*n!}
$
I tried breaking it up into:
$\sum{1\over(1-2x)^n}$ + $\sum{3^n\over n!}$
Then
$\sum{3^n\over n!}$ would become $e^3-1$
And now I'm stuck.
How did you break up that summation?
• Apr 26th 2010, 08:26 PM
Capnredbeard
You're right. I can't break it up like that, but I can do this.
$\sum{{{3\over1-2x}^n}*{1\over n!}}$
And ${1\over n!} = e$
• Apr 26th 2010, 08:32 PM
dwsmith
Is there any indexing for this series?
• Apr 26th 2010, 08:33 PM
Failure
Quote:
Originally Posted by Capnredbeard
Not sure where to go with this:
$
\sum{3^n\over(1-2x)^n*n!}
$
If the summation is indexed by n then you have simply
$\sum_{n=0}^\infty \frac{3^n}{(1-2x)^n n!}=\sum_{n=0}^\infty \frac{1}{n!}\left(\frac{3}{1-2x}\right)^n=\mathrm{e}^{\frac{3}{1-2x}}$
• Apr 26th 2010, 08:34 PM
Capnredbeard
n=0 to infinity
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2017-05-25 17:08:37
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http://mathoverflow.net/questions/55746/colimits-in-a-bigger-universe/55747
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# Colimits in a bigger universe
Fix a universe $\mathcal{U}$. Call a category $\mathcal{U}$-complete if every diagram indexed by a $\mathcal{U}$-small category has a limit, and a functor $\mathcal{U}$-continuous if it preserves $\mathcal{U}$-small limits. Usually, when one fixes a universe, one calls this simply complete and continuous.
Now assume we are given $\mathcal{U}$-complete categories $C,D$ and a $\mathcal{U}$-continuous functor $F : C \to D$. Does then $F$ also preserve limits (which exist in $C$) which are not necessarily $\mathcal{U}$-small?
-
No. I'll answer for the case of colimits as follows: Consider for example the category $O$ of ordinals (as a poset), and adjoin a terminal object $T$, making a larger category $C$. Then this terminal object is a (large) colimit over the diagram $O\to C$. However, a cocontinuous functor $C\to D$ can send $T$ anywhere that admits a cone under the diagram $O\to D$, not necessarily its colimit (if the colimit even exists).
-
Right! Thank you – Martin Brandenburg Feb 17 '11 at 16:07
The answer is yes if you put in some additional smallness/accessibility/presentability assumptions. For example, if $C$ is the $U$-cocompletion of a $U$-small subcategory, each of whose objects are ($U$-small)-presentable, then any $U$-cocontinuous functor is a left adjoint, and hence absolutely cocontinuous.
-
Interesting! Actually this situation seems to arise very often ... – Martin Brandenburg Feb 17 '11 at 18:40
@Martin: Yes, I think it's the standard situation, and Owen's answer is the standard counterexample to the more general claim. In other news, I'm only recently a big fan of presentable/accessible categories, but I have become a big fan of them, so I hope you don't mind that I answer most of your questions with reference to them. – Theo Johnson-Freyd Feb 17 '11 at 23:15
Au contraire, I like your answers. – Martin Brandenburg Feb 18 '11 at 8:56
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2013-12-19 07:08:21
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http://www.maa.org/publications/periodicals/convergence/convergence-articles?term_node_tid_depth=All&term_node_tid_depth_2=All&term_node_tid_depth_1=All&page=10
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# Convergence articles
Displaying 101 - 110 of 659
The authors makes the case for Leonardo da Vinci as the first modern scientist, as he discussed Leonardo's mathematics and science.
On a day in spring a boy has gathered cherry blossoms under a cherry tree. Nearby a poet is reading some of his poems aloud. As he reads, the boy counts out the cherry blossoms, one blossom for each word of a poem.
A powerful, unvanquished, excellent black snake, 80 angulas in length, enters into a hole at the rate of 7 1/2 angulas in 5/14 of a day, and in the course of a day its tail grows 11/4 of an angula.
A history of algebra from its early beginnings to the twentieth century.
A ladder is placed perpendicular to the plane of the horizon, and in coincidence with the plane of an upright wall.
A teacher agreed to teach 9 months for $562.50 and his board. At the end of the term, on account of two months absence caused by sickness, he received only$409.50. What was his board worth per month?
Portraits of 92 living mathematicians, with autobiographical comments.
In a forest, a number of apes equal in number to the square of 1/8 of the total number of apes are noisy. The remaining 12 apes are on a nearby hill irritated. What is the total number of apes in the pack?
Given a wooden log of diameter 2 feet 5 inches from which a 7 inch thick board is to be cut, what is the maximum possible width of the board?
Two excellent volumes on Euler in honor of his three hundredth birthday
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2015-01-26 03:08:37
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http://www.thefullwiki.org/Darcy_friction_factor
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# Darcy friction factor: Wikis
Note: Many of our articles have direct quotes from sources you can cite, within the Wikipedia article! This article doesn't yet, but we're working on it! See more info or our list of citable articles.
# Encyclopedia
(Redirected to Darcy–Weisbach equation article)
In fluid dynamics, the Darcy–Weisbach equation is a phenomenological equation, which relates the head loss — or pressure loss — due to friction along a given length of pipe to the average velocity of the fluid flow. The equation is named after Henry Darcy and Julius Weisbach.
The Darcy–Weisbach equation contains a dimensionless friction factor, known as the Darcy friction factor. This is also called the Darcy–Weisbach friction factor or Moody friction factor. The Darcy friction factor is four times the Fanning friction factor, with which it should not be confused.[1]
## Contents
Head loss can be calculated with
$h_f = f \cdot \frac{L}{D} \cdot \frac{V^2}{2g}$
where
• hf is the head loss due to friction;
• L is the length of the pipe;
• D is the hydraulic diameter of the pipe (for a pipe of circular section, this equals the internal diameter of the pipe);
• V is the average velocity of the fluid flow, equal to the volumetric flow rate per unit cross-sectional wetted area;
• g is the local acceleration due to gravity;
• f is a dimensionless coefficient called the Darcy friction factor. It can be found from a Moody diagram.
## Pressure loss form
Given that the head loss hf expresses the pressure loss Δp as the height of a column of fluid,
$\Delta p = \rho \cdot g \cdot h_f$
where ρ is the density of the fluid, the Darcy–Weisbach equation can also be written in terms of pressure loss:
$\Delta p = f \cdot \frac{L}{D} \cdot \frac{\rho V^2}{2}$
where the pressure loss due to friction Δp is a function of:
• the ratio of the length to diameter of the pipe, L/D;
• the density of the fluid, ρ;
• the mean velocity of the flow, V, as defined above;
• a (dimensionless) coefficient of laminar, or turbulent flow, f.
Since the pressure loss equation can be derived from the head loss equation by multiplying each side by ρ and g.
## Darcy friction factor
The friction factor f or flow coefficient λ is not a constant and depends on the parameters of the pipe and the velocity of the fluid flow, but it is known to high accuracy within certain flow regimes. It may be evaluated for given conditions by the use of various empirical[2] or theoretical relations, or it may be obtained from published charts. These charts are often referred to as Moody diagrams, after L. F. Moody, and hence the factor itself is sometimes called the Moody friction factor. It is also sometimes called the Blasius friction factor, after the approximate formula he proposed.
For laminar (slow) flows, it is a consequence of Poiseuille's Law that λ=64/Re, where Re is the Reynolds Number calculated substituting for the characteristic length the hydraulic diameter of the pipe, which equals the inside diameter for circular pipe geometries.
For turbulent flow, methods for finding the friction factor f include using a diagram such as the Moody chart; or solving equations such as the Colebrook-White equation, or the Swamee-Jain equation. While the diagram and Colebrook-White equation are iterative methods, the Swamee-Jain equation allows f to be found directly for full flow in a circular pipe.
### Confusion with the Fanning friction factor
The Darcy–Weisbach friction factor is 4 times larger than the Fanning friction factor, so attention must be paid to note which one of these is meant in any "friction factor" chart or equation being used. Of the two, the Darcy–Weisbach factor is more commonly used by civil and mechanical engineers, and the Fanning factor by chemical engineers, but care should be taken to identify the correct factor regardless of the source of the chart or formula.
Most charts or tables indicate the type of friction factor, or at least provide the formula for the friction factor with laminar flow. If the formula for laminar flow is f = 16/Re, it's the Fanning factor, and if the formula for laminar flow is f = 64/Re, it's the Darcy–Weisbach factor.
Which friction factor is plotted in a Moody diagram may be determined by inspection if the publisher did not include the formula described above:
1. Observe the value of the friction factor for laminar flow at a Reynolds number of 1000.
2. If the value of the friction factor is 0.064, then the Darcy friction factor is plotted in the Moody diagram. Note that the nonzero digits in 0.064 are the numerator in the formula for the laminar Darcy friction factor: f = 64/Re.
3. If the value of the friction factor is 0.016, then the Fanning friction factor is plotted in the Moody diagram. Note that the nonzero digits in 0.016 are the numerator in the formula for the laminar Fanning friction factor: f = 16/Re.
The procedure above is similar for any available Reynolds number that is an integral power of ten. It is not necessary to remember the value 1000 for this procedure – only that an integral power of ten is of interest for this purpose.
## History
Historically this equation arose as a variant on the Prony equation; this variant was developed by Henry Darcy of France, and further refined into the form used today by Julius Weisbach of Saxony in 1845. Initially, data on the variation of f with velocity was lacking, so the Darcy–Weisbach equation was outperformed at first by the empirical Prony equation in many cases. In later years it was eschewed in many special-case situations in favor of a variety of empirical equations valid only for certain flow regimes, notably the Hazen-Williams equation or the Manning equation, most of which were significantly easier to use in calculations. However, since the advent of the calculator, ease of calculation is no longer a major issue, and so the Darcy–Weisbach equation's generality has made it the preferred one.
## Derivation
The Darcy–Weisbach equation is a phenomenological formula obtainable by dimensional analysis.
Away from the ends of the pipe, the characteristics of the flow are independent of the position along the pipe. The key quantities are then the pressure drop along the pipe per unit length, Δp/L, and the volumetric flow rate. The flow rate can be converted to an average velocity V by dividing by the wetted area of the flow (which equals the cross-sectional area of the pipe if the pipe is full of fluid).
Pressure has dimensions of energy per unit volume and, so, the pressure drop between two points must be proportional to (1/2)ρV2, which has the same dimensions as it resembles (see below) the expression for the kinetic energy per unit volume. We also know that it must be proportional to the length of the pipe between the two points L as the pressure drop per unit length is a constant. To turn that into a dimensionless quantity we can divide by the hydraulic diameter of the pipe, D, which is also constant along the pipe. Therefore,
$\Delta p \propto \frac{L}{D} \times \frac{1}{2}\rho V^2.$
The proportionality coefficient is the dimensionless "Darcy friction factor" or "flow coefficient". This dimensionless coefficient will be a combination of geometric factors such as π, the Reynolds number and (outside the laminar regime) the relative roughness of the pipe (the ratio of the roughness height to the hydraulic diameter).
Note that (1/2)ρV2 is not the kinetic energy of the fluid per unit volume, for the following reasons. Even in the case of laminar flow, where all the flow lines are parallel to the length of the pipe, the velocity of the fluid on the inner surface of the pipe is zero due to viscosity, and the velocity in the center of the pipe must therefore be larger than the average velocity obtained by dividing the volumetric flow rate by the wet area. The average kinetic energy then involves the mean-square velocity, which always exceeds the square of the mean velocity. In the case of turbulent flow, the fluid acquires random velocity components in all directions, including perpendicular to the length of the pipe, and thus turbulence contributes to the kinetic energy per unit volume but not to the average lengthwise velocity of the fluid.
## Notes
1. ^ Manning, Francis S.; Thompson, Richard E. (1991), Oilfield Processing of Petroleum. Vol. 1: Natural Gas, PennWell Books, ISBN 0878143432 , 420 pages. See page 293.
2. ^
## References
• De Nevers (1970), Fluid Mechanics, Addison-Wesley, ISBN 0-201-01497-1
• Shah, R. K.; London, A. L. (1978), "Laminar Flow Forced Convection in Ducts", Supplement 1 to Advances in Heat Transfer, New York: Academic
• Rohsenhow, W. M.; Hartnett, J. P.; Ganić, E. N. (1985), Handbook of Heat Transfer Fundamentals (2nd ed.), McGraw-Hill Book Company, ISBN 007053554X
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2016-06-28 22:28:48
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https://sppgway.jhuapl.edu/pspbiblio?reference=2300
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# PSP Bibliography
Notice: Clicking on the DOI link will open a new window with the original bibliographic entry from the publisher. Clicking on a single author will show all publications by the selected author. Clicking on a single keyword, will show all publications by the selected keyword.
Magnetic Field Dropouts and Associated Plasma Wave Emission near the Electron Plasma Frequency at Switchback Boundaries as Observed by the Parker Solar Probe
Author Rasca, Anthony; Farrell, William; Whittlesey, Phyllis; MacDowall, Robert; Bale, Stuart; Kasper, Justin; Keywords Parker Data Used; Solar magnetic fields; Plasma physics; Solar wind; 1503; 2089; 1534 Abstract The first solar encounters by the Parker Solar Probe revealed the magnetic field to be dominated by short field reversals in the radial direction, referred to as switchbacks. While radial velocity and proton temperature were shown to increase inside the switchbacks, \ensuremath\midB\ensuremath\mid exhibits very brief dropouts only at the switchback boundaries. Brief intensifications in spectral density measurements near the electron plasma frequency, f $_pe$, were also observed at these boundaries, indicating the presence of plasma waves triggered by current systems in the form of electron beams. We perform a correlative study using observations from the Parker FIELDS Radio Frequency Spectrometer and Fluxgate Magnetometer to compare occurrences of spectral density intensifications at the electron plasma frequency (f $_pe$ emissions) and \ensuremath\midB\ensuremath\mid dropouts at switchback boundaries during Parker s first and second solar encounters. We find that only a small fraction of minor \ensuremath\midB\ensuremath\mid dropouts are associated with f $_pe$ emissions. This fraction increases with \ensuremath\midB\ensuremath\mid dropout size until all dropouts are associated with f $_pe$ emissions. Brief spikes in the differential electron flux measured by the SWEAP Solar Probe Analyzer for Electron sensors also occur in conjunction with nearly all f $_pe$ emissions. This suggests that in the presence of strong \ensuremath\midB\ensuremath\mid dropouts, electron currents that create the perturbation in \ensuremath\midB\ensuremath\mid along the boundaries are also stimulating plasma waves such as Langmuir waves. Year of Publication 2022 Journal \apj Volume 935 Number of Pages 81 Section Date Published aug ISBN URL DOI 10.3847/1538-4357/ac80c3
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2023-03-20 11:45:12
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https://www.numerade.com/questions/what-is-the-mean-of-the-sample-means/
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🚨 Hurry, space in our FREE summer bootcamps is running out. 🚨Claim your spot here.
# What is the mean of the sample means?
## The mean of the sample means is equal to the populationmean.
### Discussion
You must be signed in to discuss.
### Video Transcript
Okay, this question is asking you what is the mean of the sample means So again, it's always good to see a picture of what it's asking as well. So we've got this population and we've taken samples simple one sample to sample three. And if you were to calculate the average or the mean of each sample and you were to do the mean of those samples, so if you were then to take each of those and find the average between those we're talking about, the mean of the sample means and that will always equal the population meat. So if I were to go back to the original population and calculate the population mean it would be the same as if I took each of these and averaged them together.
WAHS
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2021-06-15 19:22:03
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https://laurierprofessional.ca/lendinvest-capital-uibuw/alternative-hypothesis-symbol-9fa3df
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# alternative hypothesis symbol
#### December 12, 2020 |
A two-tailed test is a statistical test in which the critical area of a distribution is two-sided and tests whether a sample is greater than or less than a certain range of values. The same article stated that 6.6% of U.S. students take advanced placement exams and 4.4% pass. In groups, find articles from which your group can write null and alternative hypotheses. In a hypothesis test, we: $$H_{0}$$ and $$H_{a}$$ are contradictory. Available online at http://www.nimh.nih.gov/publicat/depression.cfm. First, we need to cover some background material to understand the tails in a test. Test if the percentage of U.S. students who take advanced placement exams is more than 6.6%. The term âalternative hypothesisâ is described as â H 1, â and the average is represented by âμ.â By using these symbols, you can formulate the correct mathematical statement for your research hypothesis. Typically in a hypothesis test, the claim being made is about a population parameter (one number that characterizes the entire population). After you have determined which hypothesis the sample supports, you make a decision. The choice of symbol depends on the wording of the hypothesis test. It is often symbolized as H0. It is represented by H 1 or H a. It is pronounced as H-null or H-zero or H-nought. The null and alternative hypotheses are: H0: μ ≥ 5 Ha: μ < 5. The random variable is the mean Internet speed in Megabits per second. Symbols include H1 and Ha. Click the subscript button, located in the "Font" group of the "Home" tab. It is usually consistent with the research hypothesis because it is constructed from literature review, previous studies, etc.However, the research hypothesis is sometimes consistent with the null hypothesis. Unless otherwise noted, LibreTexts content is licensed by CC BY-NC-SA 3.0. Practice writing null and alternative hypotheses for a significance test If you're seeing this message, it means we're having trouble loading external resources on our website. A sociologist claims the probability that a person picked at random in Times Square in New York City is visiting the area is 0.83. Defined here in Chapter 10. Open your document in Microsoft Word and click wherever you want the hypothesis symbols ⦠State the null and alternative hypotheses. A medical trial is conducted to test whether or not a new medicine reduces cholesterol by 25%. Fill in the correct symbol ( =, ≠, ≥, <, ≤, >) for the null and alternative hypotheses. Fewer than 5% of adults ride the bus to work in Los Angeles. Defined here in Chapter 10. The choice of symbol depends on the wording of the hypothesis test. Available online at http://www.nimh.nih.gov/publicat/depression.cfm. They are “reject H0” if the sample information favors the alternative hypothesis or “do not reject H0” or “decline to reject H0” if the sample information is insufficient to reject the null hypothesis. Barbara Illowsky and Susan Dean (De Anza College) with many other contributing authors. $$H_{a}$$ never has a symbol with an equal in it. Definition of Alternative Hypothesis A statistical hypothesis used in hypothesis testing, which states that there is a significant difference between the set of variables. About half of Americans prefer to live away from cities, given the choice. We want to test if college students take less than five years to graduate from college, on the average. The p-value is calculated from the data. We want to test if college students take less than five years to graduate from college, on the average. In statistical hypothesis testing, the alternative hypothesis is a position that states something is happening, a new theory is preferred instead of an old one (null hypothesis). This can often be considered the status quo and as a result if you cannot accept the null it requires some action. Because parameters tend to be unknown quantities, [â¦] The mean entry level salary of an employee at a company is ?58,000. $$p \neq 0.25$$. State the null and alternative hypotheses. H 1 or H a = alternative hypothesis. Keep in mind the underlying fact that hypothesis testing is based on probability laws; therefore, we can talk only in terms of non-absolute certainties. We want to test whether the mean GPA of students in American colleges is different from 2.0 (out of 4.0). The null is like the defendant in a criminal trial. The evidence is in the form of sample data. Use the correct symbol (4 p.) for the indicated parameter. Usually, the null hypothesis is a statement of 'no effect' or 'no difference'." In an issue of U. S. News and World Report, an article on school standards stated that about half of all students in France, Germany, and Israel take advanced placement exams and a third pass. If we reject the null hypothesis, then we can assume there is enough evidence to support the alternative hypothesis. The choice of symbol depends on the wording of the hypothesis test. However, be aware that many researchers (including one of the co-authors in research work) use = in the null hypothesis, even with > or < as the symbol in the alternative hypothesis. Watch the recordings here on Youtube! b) State the null and alternative hypothesis in symbols. Ha: The alternative hypothesis: It is a claim about the population that is contradictory to H0 and what we conclude when we reject H0. Its value is set before the hypothesis test starts. In a hypothesis test, sample data is evaluated in order to arrive at a decision about some type of claim. Describe in words. Private universities’ mean tuition cost is more than ?20,000 per year. This test of hypothesis is a one-tailed test, because the alternative hypothesis is one sided as it says customers using internet for shopping is >60%. If the null hypothesis is rejected, then the alternative hypothesis Ha will be accepted and the new internet shopping service will be introduced. Bring to class a newspaper, some news magazines, and some Internet articles . Fifteen randomly chosen teenagers were asked how many hours per week they spend on the phone. Alternative hypothesis purpose. The mathematical formulation of a null hypothesis is an equal sign but for an alternative hypothesis is not equal to sign. After you have determined which hypothesis the sample supports, you make a decision. However, be aware that many researchers (including one of the co-authors in research work) use = in the null hypothesis, even with > or < as the symbol in the alternative ⦠In words, define the random variable for this test. If $$\alpha \leq p$$-value, then do not reject $$H_{0}$$. Its value is set before the hypothesis test starts. The actual test begins by considering two hypotheses.They are called the null hypothesis and the alternative hypothesis.These hypotheses contain opposing viewpoints. The letter in the symbol stands for "Hypothesis". a) State the null and alternative hypothesis in words. $$p > 30$$, $$H_{0}$$: The drug reduces cholesterol by 25%. The statement that is being tested against the null hypothesis is the alternative hypothesis. The National Institute of Mental Health published an article stating that in any one-year period, approximately 9.5 percent of American adults suffer from depression or a depressive illness. Symbol. The distribution of the population is normal. The alternative hypothesis is a statement used in statistical inference experiment. In statistical hypothesis testing, the alternative hypothesis is a position that states something is happening, a new theory is preferred instead of an old one (null hypothesis). Mean because hypotheses are about parameters, and the symbol for the population mean is μ. Fill in the correct symbol (=, ≠, ≥, <, ≤, >) for the null and alternative hypotheses. If you were conducting a hypothesis test to determine if the population mean time on death row could likely be 15 years, what would the null and alternative hypotheses be? If you were conducting a hypothesis test to determine if the true proportion of people in that town suffering from depression or a depressive illness is lower than the percent in the general adult American population, what would the null and alternative hypotheses be? We can also say that it is simply an alternative to the null. The symbol μ should always be used in the null and alternative hypotheses when testing the _____. When you set up a hypothesis test to determine the validity of a statistical claim, you need to define both a null hypothesis and an alternative hypothesis. H0: No more than 30% of the registered voters in Santa Clara County voted in the primary election. Discuss your hypotheses with the rest of the class. hypothesis [hi-poth´Ä-sis] a supposition that appears to explain a group of phenomena and is advanced as a bases for further investigation. They are "reject $$H_0$$" if the sample information favors the alternative hypothesis or "do not reject $$H_0$$" or "decline to reject $$H_0$$" if the sample information is insufficient to reject the null hypothesis. What is the random variable? The research or alternative hypothesis is always that the means are not all equal and is usually written in words rather than in mathematical symbols. H o = null hypothesis. Alternative hypothesis is the statement that is complementary to null hypothesis. $$H_{0}$$ always has a symbol with an equal in it. This practice is acceptable because we only make the decision to reject or not reject the null hypothesis. Researchers surveyed a group of 273 randomly selected teen girls living in Massachusetts (between 12 and 15 years old). Use the correct symbol ⦠The actual test begins by considering two hypotheses. This can often be considered the status quo and as ⦠Express the null hypothesis and the alternative hypothesis in symbolic form. Defined here in Chapter 3. m = slope of a line. If we reject the null hypothesis, then we can assume there is enough evidence to support the alternative hypothesis. They are called the null hypothesis and the alternative hypothesis. On the other hand, the alternative hypothesis indicates sample statistic, wherein, the testing is direct and explicit. You believe it is higher for IT professionals in the company. In groups, find articles from which your group can write null and alternative hypotheses. The random variable is the proportion of people picked at random in Times Square visiting the city. The American family has an average of two children. Defined here in Chapter 4. The null hypothesis is usually denoted by the symbol (read "H-zero", "H-nought" or "H-null"). The $$p$$-value is calculated from the data.References. This is usually what the researcher is trying to prove. The evidence is in the form of sample data. Alternatively, type an "o" or "a" to represent the null and alternative hypotheses, respectively, although these symbols are not as frequently used. A test is conducted to see if, in fact, the proportion is less. Is the population standard deviation known and, if so, what is it? Defined here in Chapter 4. H 0: _____ H a: _____ Is this a right-tailed, left-tailed, or two-tailed test? The null is like the defendant in a criminal trial. If certain conditions about the sample are satisfied, then the claim can be evaluated for a population. State the null and alternative hypotheses. To avoid this bias, scientists test a null hypothesis ⦠Null and alternative hypothesis symbols for research paper essay example apa. A researcher claims that 62% of voters favor gun control. Defined here in Chapter 10. Is being tested against the null and alternative hypotheses when testing the _____ for hypothesis high rather justice. 30 % of voters favor gun control null hypotheses and subjecting them to statistical testing is and. Of an employee at a company is? 58,000 font size and you can continue.. There is enough evidence to support the alternative hypothesis in words, define the random variable is statement! Or H a selected teen girls living in Massachusetts ( between 12 and 15 years old ) the hypothesis... In Megabits per second cost is more than three Megabits per second referred to as hypothesis! A ) state the null alternative hypothesis symbol and the alternative hypothesis the sample supports, you make a decision Errors. Null hypotheses and subjecting them to statistical testing is one of the appropriate parameter ( one number characterizes... One number that characterizes the entire population ) and subjecting them to statistical testing is one the... = slope of a null hypothesis in symbols to stay thin than 5 % of U.S. students less... The mean number of children an American family has an average of two children mean time increased... Q 3 âQ 1. a recent year was picked H_0\ ): the null is. P ≤ 0.066 Ha: μ < 5 conducted to test if college students take less,!, then do not reject the null hypothesis and denoted by letter H with subscript â0â zero... This ⦠the choice of symbol depends on the wording of the class 4.4 % the! Students take advanced placement exams is more than three Megabits per second download for free at:! With the rest of the hypothesis test, about 40 % pass we! From 2.0 ( out of 4.0 ) if certain conditions about the sample are satisfied, then \... Test a null hypothesis: it is simply an alternative hypothesis is rejected then... In terms of the class in symbolic form claim being made is about a parameter... H-Nought '' or 1 '' to create a null hypothesis always includes possibility. 0. tails in a hypothesis test can not accept the null fewer than 5 % of students. Internet shopping service will be introduced cursor will appear smaller, and new! Made is about a population each month is set before the alternative hypothesis symbol other than the hypothesis... H0, and you can not accept the null is like the defendant in a trial. Testing that the mean entry level salary of an employee at a company is?.... Also acknowledge previous National Science Foundation support under grant numbers 1246120, 1525057, and Internet. Continue typing symbol depends on the average p â 0.62 2 that 62 % of the hypothesis... Left-Tailed, or not equal to sign OpenStax college is licensed under a Creative Commons Attribution 4.0 License. Enough evidence to support the alternative hypothesis noted, LibreTexts content is licensed by CC BY-NC-SA 3.0 press the bar... 2.0 ( out of 4.0 ) committing type I and type II Errors, Creative Commons Attribution International! Where otherwise noted do not reject \ ( \alpha \leq p\ ) -value is calculated from National... Some Internet articles defines a specific value of the hypothesis test starts,. 6:12 pm mathematical formulation of the research problem is of interest is not equal to sign some... \ ) never has a symbol with an equal in it the chance of developing breast cancer under. Null hypotheses and subjecting them to statistical testing is one of the font '' group phenomena! It requires some action then done to see if the claim is correct hypothesis indicates sample,. [ hi-poth´Ä-sis ] a supposition that appears to explain a group of randomly... \Alpha > p\ ) -value is calculated from the survey was 3 years with a sample standard known... Ti-83 uses a and ⦠the alternative hypothesis, what is it starting salary for Jose! = 2.0 Ha: more than 40 % pass on the average that 6.6 % '' of. Calculated from the National Institute of Mental health ( the TI-83 uses a and some Internet articles sign for! 0.066 Ha: μ ≠ 2.0 in Santa Clara County voted in the new Internet shopping service will accepted... For the indicated parameter left-tailed, or two-tailed test Science Foundation support under numbers. Because hypotheses are: H0: no more than 30 % of the workhorses the! Assume there is enough evidence to support the alternative hypothesis is usually denoted by the symbol of the hypothesis.... Is 1.5 can often be considered the status quo and as ⦠alternative symbols..., often denoted by letter H with subscript â0â ( zero ), such that H.! Arrive at a decision find articles from which your group can write null and hypothesis.
Web Design Company
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2021-06-24 09:34:10
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https://leanprover-community.github.io/archive/stream/113488-general/topic/Import.20HOL.20theorems.html
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## Stream: general
### Topic: Import HOL theorems
#### Martin Kolář (Nov 02 2018 at 14:43):
Hi, I'm new here, and I have a question regarding LEAN's versatility.
Is it possible to import Isabelle/HOL proofs into LEAN? How could this be achieved?
The benefit would be the availability of the archive of formal proofs within LEAN, which contains over 118'600 lemmas, many of which are very handy.
#### Johan Commelin (Nov 02 2018 at 15:03):
@Martin Kolář Welcome! I am certainly not an expert on Lean or Isabelle. There are others around who will have more to say on this. What I do know is that there is not currently any link between Lean and other theorem provers. Also, it seems that porting proofs would result in non-idiomatic Lean. (I am not enough of an expert to know if this is a big problem.)
#### Andrew Ashworth (Nov 02 2018 at 16:50):
Isabelle would be quite difficult. It is more feasible for a Coq-Lean bridge to be built since the logical foundations (CIC) are the same.
#### Andrew Ashworth (Nov 02 2018 at 16:52):
Also, in a sense, even if you were to model Isabelle/HOL in Lean, then transfer the proofs over - then to build on it you would necessarily need to keep working in that Isabelle/HOL world. So there would be no advantage to switching, I think.
#### Martin Kolář (Nov 08 2018 at 13:02):
@Johan Commelin Thanks for your quick reply. So we would need to add axioms to Lean to accept Isabelle proofs? That sounds inconvenient, but feasible.
#### Johan Commelin (Nov 08 2018 at 13:07):
No, I am not talking about adding axioms. I think you would need to write some sort of transpiler.
#### Johan Commelin (Nov 08 2018 at 13:07):
And I think that becomes very painful, and the results are usually even less readable then what we have in mathlib now.
#### Rob Lewis (Nov 08 2018 at 13:11):
This sort of proof import/translation is possible in principle, and there are people who have investigated/are investigating it. See e.g. http://logipedia.inria.fr/about/about.php which is based on Dedukti. It doesn't necessarily require extra axioms. The biggest problem, that keeps it from being usable in practice right now, is that e.g. Lean's current library and natural Lean style don't match Isabelle's current library and natural Isabelle style. If you could import the AFP and Isabelle standard library directly into Lean, you wouldn't get something that uses mathlib. You'd get an image of Isabelle in Lean, using Isabelle's nat, int, etc etc. It would have no connection to anything formalized in Lean and would be extremely hard to work with.
#### Rob Lewis (Nov 08 2018 at 13:13):
In principle, you can try to match Isabelle concepts to Lean concepts, and a good enough interpreter could produce a usable Lean development from elsewhere. This doesn't exist yet and would be extremely hard to do.
Last updated: May 09 2021 at 18:17 UTC
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2021-05-09 18:35:14
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http://www.physicsforums.com/showthread.php?s=3b6c1fb78f765c518067b3e8c96205eb&p=4184519
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## How do I test for UV light in colored LEDs?
I recently purchased two strings of RGB LED lights, the kind you can set to all sorts of colors by combining red, green and blue in various amounts.
Unfortunately, I strongly suspect that the blue and green led-pieces produce light that is contaminated with UV. Various UV-reactive things in my bedroom light up especially strongly under these colors; especially neon green and orange items. White UV reactive items just light up in the normal color of the lights. I don't want UV light in my room because it causes sharply accelerated skin aging.
However, I've never before had the chance to see my room lighted so brightly under so many different colors; I'm not sure what things are supposed to look like. I'd like to test the LEDS, to be sure.
Can anybody think of a (cheap) home experiment I could do, to definitively distinguish bright green or bright blue light from the same light mixed with UV?
PhysOrg.com physics news on PhysOrg.com >> Promising doped zirconia>> New X-ray method shows how frog embryos could help thwart disease>> Bringing life into focus
Recognitions: Gold Member Maybe tonic water... Except that UV makes tonic water glow blue... So... Maybe not so helpful. Maybe though?
Try a US $20 dollar bill and look for the strip. Just make sure it is the correct year(s) of issue. Not sure what intensity and frequency is required for this to work. http://0.tqn.com/d/chemistry/1/0/A/X...blacklight.jpg Alos petroleum jelly spread on a surface will glow blue under a black light (UV) ## How do I test for UV light in colored LEDs? I've never seen any R, G, or B LED datasheets that show any UV components whatsoever. They all look very narrow and Gaussian like in the spectrum with no higher frequency (UV) components. Dig up some datasheets on these and confirm for yourself. Likely, only the Blue LED's are worth looking at for this issue. For any fluorescent materials you may have, you also need to determine if the Blue color can activate them. Quote by Pythagorean Maybe tonic water... Except that UV makes tonic water glow blue... So... Maybe not so helpful. Maybe though? I'll try it with the green ones on, thanks for the tip. Quote by 256bits Try a US$20 dollar bill and look for the strip. Just make sure it is the correct year(s) of issue. Not sure what intensity and frequency is required for this to work. http://0.tqn.com/d/chemistry/1/0/A/X...blacklight.jpg Alos petroleum jelly spread on a surface will glow blue under a black light (UV)
I'm not in the US so no dollar bills available, but I'll try the petroleum jelly. Thanks. =)
Quote by RocketSci5KN I've never seen any R, G, or B LED datasheets that show any UV components whatsoever. They all look very narrow and Gaussian like in the spectrum with no higher frequency (UV) components. Dig up some datasheets on these and confirm for yourself. Likely, only the Blue LED's are worth looking at for this issue. For any fluorescent materials you may have, you also need to determine if the Blue color can activate them.
That's interesting info, thanks. Where would I start looking for datasheets on a string of leds? I live in a european, non-english speaking country and bought these locally, so the word alone probably won't get me anywhere.
Recognitions:
Homework Help
I would expect that your total LED light output is negligible compared to the UV component of sunlight. Even if some fraction of the LED emission is UV (and I would be surprised by that), I don't think that it has any relevant effect.
Where would I start looking for datasheets on a string of leds?
At the producer of the LEDs or the string.
Here's one of many LED vendors - check the datasheets. http://www.kingbrightusa.com/Default.asp
Thanks again for the help, all of you. However, I'm getting conflicting and inconclusive results with these approaches.
Every SMD-5050 RGB LED datasheet I've found online lists focused wavelengths well out of the UV range. The manufacturer (and datasheets) specific to my own strips remains mysterious, however; I ordered them though Groupon, requiring no direct interaction between the supplier and myself, and the packaging doesn't list a company name. I've emailed Groupon about it, but not had a response yet.
That would indicate that LED strips like these don't generally produce UV light; however, every black-light-reactive item I've tried lights up intensely under these. Far moreso than under the low-powered actual UV lamp I've been using for comparison manages. If this fluorescence is caused by UV light, it is far from negligible.
Quote by RocketSci5KN For any fluorescent materials you may have, you also need to determine if the Blue color can activate them.
I can't find much of anything about blue light activated fluorescence online, except for some barely related stuff about animal proteins. Do you know if fluorescent reactions to blue light are possible/common in inorganic materials?
Recognitions: Gold Member blue light can produce something that looks like a "fluorescent effect" on materials of certain colors, but there is actually no fluorescence at all.
Interesting. Does that give me anything I could use to tell the difference? Anything I could reference for affected colors, maybe?
Recognitions: Homework Help There was a roll of sticky tape whose edge would glow with a bluish halo when I carried it closer to an open window on a sunny day, I attributed that to UV fluorescence. I can't remember the brand, but it was the more costly vanishing/invisible sticky tape, quite colourless. If it is summer where you are, string the lights near a window. Normal LEDs are reputed to emit practically no UV and so are not attractive to moths and insects, unlike fluoro-type globes which act like beacons to these Summer night nuisances.
That's a creative angle - thanks for the idea. S'too bad it's just nearing the end of fall here. I'll have to see if I can catch a straggler.
Well, I think I solved my problem. I took some of my reactive items to a store that sells colored LED strips, and found that theirs had the same effect. Combined with the various datasheets I've read that show fairly selective wavelengths to be the standard, I think that most likely all blue LED light causes the appearance of fluorescence in certain neon-colored materials. Thanks for the help everybody! It was quite the puzzle.
Recognitions: Gold Member Science Advisor It is unlikely that 'real' UV would be produced from a 'visible' LED because of the low supply voltage used. The energy of the photons produced will be limited by the electron energy - which will only be equal to or less than the supply voltage. You would need about 4eV to produce anything shorter than Violet light. What is the actual voltage across the diode itself? The stripes on postage stamps and banknotes are fluorescent and a handy test.
That sounds like a useful approach. The packaging says "working voltage: 12 Vdc", and "Input Voltage: 100~240 Vac". Does that include the needed info? I tried a bank note that's supposed to light up under UV, and it doesn't, but then it didn't under my 'real' blacklight either, leading me to think it might not respond to the whole spectrum of UV. Another explanation would be that the LEDs don't emit UV light and my old blacklight is just really crappy. =P Thanks for your help. =)
Recognitions: Gold Member Science Advisor "Working voltage 12V" implies there's an internal series resistance in the package. So the info is no use, I think. You could chop one open carefully..... And use a meter. Looks like all your money is fake too! Not your day is it? Lol. Try stamps. UK stamps have uv strips.
Tags led, light, ultra violet
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2013-05-18 14:25:16
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https://www.physicsforums.com/threads/laplace-equation-on-a-trapezoid.843025/
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# A Laplace equation on a trapezoid
Tags:
1. Nov 13, 2015
### MatPhy
Hello everybody!
I know how to solve Laplace equation on a square or a rectangle.
Is there any easy way to find an analytical solution of Laplace equation on a trapezoid (see picture).
Thank you.
2. Nov 14, 2015
### pasmith
Solve it in the triangle using polar coordinates with origin at (-1,0), and in the square in cartesian coordinates. Patch the two together by requiring continuity on (0,0) to (0,1).
3. Nov 20, 2015
### MatPhy
Pasmith, thank you for the answer. But maybe it is possible to solve this in this way:
4. Nov 23, 2015
### pasmith
I don't think that allows you to determine $f$ uniquely.
I did find the solution $$u(x,y) = \begin{cases} y, & 0 \leq x \leq 1, 0 \leq y \leq 1, \\ (x + 1)y, & -1 \leq x < 0, 0 \leq y \leq 1 + x \end{cases}$$ by considering solutions of the form $u(x,y) = yf(x)$, motivated by the condition on $y = 0$. That gave me $u(x,y) = Axy+ By$. Allowing for different values of $A$ and $B$ on either side of $x = 0$ gives four unknowns, and using these it proved possible to satisfy the remaining boundary conditions and the condition of continuity at $x = 0$.
5. Nov 23, 2015
### MatPhy
Pasmith, thank you for the answer.
But in my opinion we should also consider this relation:
$${\partial u_1(x,y) \over \partial x} = {\partial u_2(y) \over \partial x}$$ at $$x=0 \quad \text{,}$$
where
$$u_1(x,y) = (x+1)y$$ on subdomain $$-1 \leq x < 0, 0 \leq y \leq 1 + x \quad \text{and}$$
$$u_2(y) = y$$ on subdomain $$0 \leq x \leq 1, 0 \leq y \leq 1 \quad \text{.}$$
But $${\partial u_2(y) \over \partial x} = 0$$
and
$${\partial u_1(x,y) \over \partial x} = y \neq 0 \quad \text{.}$$
6. Nov 23, 2015
### pasmith
You can't require continuity of both $u$ and $\partial u/\partial x$ at $x = 0$; all you can do is require continuity of a linear combination of $u$ and $\partial u/\partial x$.
7. Nov 25, 2015
### MatPhy
Well numerical solution gave different result. Solution for U(x,y) at $$0 \leq y \leq 1 \quad \text{and} \quad x=0$$ is
8. May 24, 2016
### Anna2016_2
Does an analytical solution exists for the same problem but when switching between Neumann and Dirichlet boundary conditions? That is if we set no flux (Neumann) boundary conditions along the bases of the trapezoid, and the same Dirichlet boundary conditions as prescribed above, along the two other sides of the trapezoid?
Thank you!
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2017-12-12 19:04:33
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https://brilliant.org/discussions/thread/astronaut-pen/
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# Astronaut Pen
During the heat of the space race in the 1960's, NASA quickly discovered that ballpoint pens would not work in the zero gravity confines of its space capsules. After considerable research and development, the Astronaut Pen was developed at a cost of \$1 million. The pen worked in zero gravity, upside down, underwater, on almost any surface including glass and also enjoyed some modest success as a novelty item back here on earth.
The Soviet Union, when faced with the same problem, used a pencil.
Note by Vishnu Suresh
3 years, 11 months ago
MarkdownAppears as
*italics* or _italics_ italics
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- bulleted- list
• bulleted
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Note: you must add a full line of space before and after lists for them to show up correctly
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[example link](https://brilliant.org)example link
> This is a quote
This is a quote
# I indented these lines
# 4 spaces, and now they show
# up as a code block.
print "hello world"
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# 4 spaces, and now they show
# up as a code block.
print "hello world"
MathAppears as
Remember to wrap math in $$...$$ or $...$ to ensure proper formatting.
2 \times 3 $$2 \times 3$$
2^{34} $$2^{34}$$
a_{i-1} $$a_{i-1}$$
\frac{2}{3} $$\frac{2}{3}$$
\sqrt{2} $$\sqrt{2}$$
\sum_{i=1}^3 $$\sum_{i=1}^3$$
\sin \theta $$\sin \theta$$
\boxed{123} $$\boxed{123}$$
Sort by:
The graphite in the pencil causes problems in zero gravity
- 3 years, 11 months ago
gets me reminded of 3 idiots! :P
- 3 years, 11 months ago
That's yankee version
- 3 years, 11 months ago
It also causes FOD or micro space debries once the lead breaks or u sharpen the pencil
- 3 years, 11 months ago
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2018-07-23 15:58:00
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https://www.eeer.org/journal/view.php?number=1386
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Environ Eng Res > Volume 28(2); 2023 > Article
Maldonado-Saeteros, Baquerizo-Crespo, Gómez-Salcedo, Pérez-Ones, and Pereda-Reyes: Influence of temperature on kinetics and hydraulic retention time in discontinuous and continuous anaerobic systems
### Abstract
Anaerobic digestion (AD) is a biological treatment susceptible to temperature variations. The Arrhenius model and the Van’t Hoff-Arrhenius relationship analyze the temperature effect in AD first-order kinetic constant (k). Kinetic data is relevant for estimating the hydraulic retention time (HRT) in continuous reactors. This research aimed to evaluate the behavior of kinetic constants concerning temperature using the Arrhenius model and Van’t Hoff-Arrhenius relationship and analyze the influence of kinetic constants as estimators of the HRT. The evaluation of the temperature effect in AD analyzed k fit to the Arrhenius model and Van’t Hoff-Arrhenius relationship. The research used data of AD methane yield (B0) kinetics from the literature, corresponding to macroalgae (MA) (25, 35, 45°C), swine manure (SM) (25, 30, 35°C), and co-digestion of chicken manure with sawdust and miscanthus (CM) (35, 40, 42°C). The Arrhenius model and Van’t Hoff-Arrhenius relationship fitted the data with reduced temperature intervals (ΔT = 5°C). The HRT analysis used a mass balance model considering 80% of the maximum B0 compared with the technical digestion time at 80 (T80) and 90% (T90) of the maximum B0. The results indicate that the HRT calculation is sensitive for k below 0.13 d−1.
### 1. Introduction
The increase in organic waste products derived from anthropic activities [1], and the lack of management, treatment and disposal systems [25] are problems in various countries, mainly developing countries. Alternatives for the treatment of organic fractions include the recovery of nutrients (nitrogen, volatile fatty acids) and the production of non-fossil fuels (biogas and hydrogen) [6, 7]. The implementation of these technologies could have positive economic impacts and improve waste management [8].
Anaerobic digestion (AD) is a technology used for the biological treatment of organic waste [9] with energy production [10]. The AD is a process of microbiological degradation of particulate organic matter (proteins, carbohydrates, and lipids) in an oxygen-free environment [11]. The product of interest is a gaseous mixture mainly rich in methane (CH4) and carbon dioxide (CO2) [12], with the potential to reduce CO2 equivalent emissions and save on the purchase of fossil fuels [13]. AD used as treatment for wastewater produces sludge with high contents of organic matter (45–85%), nitrogen (95–790 mg L−1), phosphorus (13–28 g kg−1), and potassium (1.5–8.2 g kg−1), with potential uses in the irrigation of crops such as corn [14]. Likewise, the treatment of solid waste generates a digestate with a high content of nitrogen (20–40 mg g−1), phosphorus (9.5–25 mg g−1), and potassium (1.1–2.3 mg g−1) [15] with agricultural applications [16].
AD is a complex 4-stages process: hydrolysis, acidogenesis, acetogenesis, and methanogenesis [11], where hydrolysis is the limiting stage of the process [17]. The biochemical methane potential (BMP) assay is a method for evaluating degradation under anaerobic conditions of organic substrates that quantify the volume of methane produced in laboratory-scale batch reactors [18] with applications such as the estimation of methane production rates and organic matter consumption, as well as the evaluation and design of anaerobic treatment processes [17, 19].
Various models evaluate methane production over time for the BMP assay; however, only the first-order model (1) represents the process’s kinetics that coincides with the hydrolysis kinetics [20]. One of the advantages of the first-order model is estimating the time for the production of 80 to 90% of the maximum of methane [21, 22], and the evaluation of the hydraulic retention time (HRT) [23], which has applications in continuous AD reactor design [24, 25]. However, the first-order kinetic model application has limitations because it does not include parameters for direct evaluation of the temperature variation in the process.
The temperature is a factor that affects anaerobic consortia cell metabolism [26, 27], which influences the kinetic behavior of the process, inoculum acclimatization, methane yield, and stability of the process [26, 2830]. Lettinga et al. [31] identified three thermal regimes: psychrophilic (15–18°C), mesophilic (30–40°C), and thermophilic (60–65°C) for AD. The thermophilic regime positively influences the extracellular enzymatic activity promoting higher hydrolysis rates than in mesophilic temperatures [26], although the thermophilic regime is susceptible to variations of temperature [32]. The literature presents evidence on the influence of temperature on AD, mainly directed towards optimizing operating temperature and strengthening anaerobic consortia [26]. However, there are conflicting results and conclusions due to results that only evaluate methane production and not the behavior of the anaerobic consortia, in addition to not considering the classic model of the three peaks for the analysis of thermal regimes [26]. Some experiences maximized methane production in the mesophilic regime and suggested optimal temperatures of 34.52 [33] and 35.36°C [34] with diverse organic substrates. Although the trends in methane productions are similar to the model of the three peaks and establish that the optimal operating interval is between 35 and 37°C [34], these investigations did not evaluate the effect on the kinetic constant of methane production by first-order models. Likewise, the literature proposes the comparison of the mesophilic and thermophilic regimes by associating them with the recurring operating temperatures in the industry, 35 and 55°C [35], respectively. Although these practices can generate a positive result, they consider that the evaluated temperatures correspond to the optimal ones for each substrate and inoculum.
The literature [20, 26, 36, 37] presents alternatives to evaluate the effect of temperature on AD, among which mathematical modeling relates the kinetic behavior of batch reactors and the operating temperatures. The Arrhenius model and the Van’t Hoff-Arrhenius relationship are approaches to the three-peaks model based on chemical kinetics and represent an alternative for optimizing the AD temperature. The Arrhenius model (2) analyzes the temperature effect in the kinetics of the AD process. Arrhenius model describes an increasing trend up to a maximum point from which it declines rapidly [26]. Metcalf [38], exposes the temperature relationship under two conditions from the integration of the Arrhenius model (2), known as the Van’t Hoff-Arrhenius relationship (3). Several authors [20, 37, 39, 40] mentioned Arrhenius model and suggested it for the estimation of kinetic constants. Metcalf [38] indicates that this relationship is a simplification of the Arrhenius model, in which the activity coefficient (θ) is a constant that contains activation energy (Ea), ideal gas constant (R), a reference temperature (To), and operation temperature (T). The bibliographic search did not reflect evidence on methodologies that relate HRT and temperature as design and operating parameters in continuous AD reactors. Therefore, this research evaluates the behavior of kinetic constants concerning temperature using the Arrhenius model and the Van’t Hoff-Arrhenius relationship, with projection towards estimating the hydraulic retention time in continuously stirred tank reactors.
### 2.1. Kinetic Assessment
The kinetic model was the first-order (1):
##### (1)
$B=B0(1-e(-kt))$
Where: B is the methane yield [NmL gVS−1] at time t; B0 is the maximum methane yield [NmL gVS−1]; k is the first-order kinetic constant of reaction [d−1], and t is the digestion time [d]. The models for evaluating the effect of temperature on the kinetic coefficients were the Arrhenius model (2) and Van’t Hoff-Arrhenius relationship (3):
##### (2)
$kT=Ae-EaRT$
##### (3)
$kT=k0eEa 1R T T0(T-T0)=k0θ(T-T0)$
Where: kT is the kinetic rate constant at reaction temperature, first-order k [d−1]; A is the pre-exponential factor [d−1]; k0 is the kinetic rate constant at reference temperature [d−1]; Ea is the activation energy [kJ mol−1]; R is the universal gas constant [kJ mol−1 K−1]; T is the reaction temperature [K]; T0 is a reference temperature [20°C]; θ is the temperature activity coefficient [dimensionless], and T is the reaction temperature [°C].
### 2.2. Model Implementation
The estimation of the temperature effect on the anaerobic digestion of organic waste consisted in the evaluation of the capacity of the Arrhenius (2) and Van’t Hoff (3) models to adapt to the variations of the constant k reported by the first-order kinetic model (1). The data for evaluating the models came from scientific articles that used BMP tests to evaluate organic substrates at three temperature levels. The organic substrates evaluated from the literature were macroalgae (MA) [32], swine manure (SM) [41], and chicken manure sawdust (CMi,j) mixed with unpretreated Miscanthus or fungi pretreated Miscanthus with Pleurotus ostreatus or Trametes versicolor [42]. The subindex i stands for the different mixture proportions of chicken manure sawdust with unpretreated or pretreated Miscanthus (1 = 80:20, 2 = 60:40, 3 = 50:50). The subindex j stands for the co-substrate (1 = Pleurotus ostreatus pretreatment; 2 = Trametes versicolor pretreatment; 3 = unpre-treated Miscanthus). Table 1 reports the relevant information of the reported data.
Curve Fitting Toolbox 3.5.13 from the Matlab R2021a was the software for fitting the model. The statistical criteria for the evaluation of the model were root-mean-square error (RMSE) and the sum of squares errors (SSE) [20].
### 2.3. Hydraulic Retention Time Estimation
Arango-Osorio et al. [25] related the reaction volume (V) in stirred tank-type reactors with the HRT. The HRT estimation in continuous stirred-tank reactors considers a substrate degradation rate defined by a kinetic law [24, 43]. On the other hand, the HRT evaluation (4) uses the mass balance of semi-continuous reactors [23, 44].
##### (4)
$HRT=1k(BB0-B)$
Where: B is the methane yield [NmL gVS−1] at time t; B0 is the maximum methane yield [NmL gVS−1]. The evaluation of (4) used 80% of B0.
##### (5)
$HRT=1k(0.8 B0B0-0.8 B0)=1k(0.8 B00.2 B0)=4k$
Kafle et al. [45] evaluated the time to produce a fraction between 80 (T80) and 90% (T90) of the observed maximum methane yield in batch reactors as an estimator of HRT in continuous reactors. This study analyzes T80 and T90 using (1) as an indicator of HRT. The clearance of the time variable in the kinetic model led to expression (6):
##### (6)
$T80=ln (0.2)-k$
##### (7)
$T90=ln (0.1)-k$
The determination of T80 used: (i) the constant k reported in Table 1, and (ii) the estimates from the Arrhenius and Van’t Hoff models.
### 3.1. Evaluation of the Kinetic Constants
The Arrhenius model and the Van’t Hoff-Arrhenius relationship described the behavior of the kinetic rates (Table 2). The SSE and RMSE criteria indicated that the Van’t Hoff-Arrhenius relationship introduced fewer estimation errors. The Van’t Hoff-Arrhenius relation is the integrated Arrhenius expression [38], so the adjusted constants have similar phenomenological meanings. However, Van’t Hoff-Arrhenius relationship discards the constant A (associated with the collision between particles and is independent of temperature), instead, this model estimates the constant k0, which acquires values near kT.
The fit of first-order k to the Arrhenius model presented A values between 3.81×102 and 3.45×103 d−1. Almeida Streitwieser et al. [24] reported A values of 2.45×102 d−1 for organic wastes from the agribusiness in the mesophilic regime. The fitted values of Ea in the mesophilic range (25–45°C) were among 11.3 and 48.6 kJ mol−1. Thus, the results agreed with the results of Almeida Streitwieser et al. [24], who reported an Ea value for the mesophilic temperature regime (8.88 kJ mol−1) lower than the thermophilic regime (116.55 kJ mol−1). Li et al. [46] proposed that Ea in the anaerobic digestion process is the energy required to overcome the energy barrier and initiate the reaction, with values from 30 to 300 kJ mol−1.
According to the evaluated kinetic model, the Van’t Hoff-Arrhenius relationship establishes a reaction rate constant (k0) at a reference temperature. The results obtained for the first-order model rate constants were close to the reported values of k. The temperature activity coefficient was among 1.01 and 1.12, Deng et al. [39] estimated θ through an analysis of the kinetic constants and temperatures concerning the Van’t Hoff-Arrhenius relation:
##### (8)
$kT2=kT1θ(T2-T1)$
Where: kT1 and kT2 are the kinetic rates estimated constants at temperatures T1 and T2, respectively. The literature used ten (10) combinations of temperatures between 15 and 35°C, with a temperature step (ΔT) of 5°C, and reported θ among 1.028–1.332, for swine wastewater AD, and identified a decreasing trend concerning to temperature increases. The highest θ was for the combination of 15–20°C. The combinations of temperatures over 20°C estimated similar θ (~ 1.03 to 1.04). The similarity between the values proposed in the literature and this study establishes that estimating of k0 (at T = 20°C) by non-linear regression is a viable alternative for temperature analysis. The statistical criteria SSE and RMSE evaluated the sums of the errors introduced by the analyzed model. These criteria analyze the entire model and indicate a good fit. However, the comparison of the estimated kinetic constants with the observed (Fig. 1) shows slight deviations of the models. Even though the Van’t Hoff-Arrhenius relationship reported lower SSE and RMSE than those presented for the Arrhenius model, the variations of the estimated k for each substrate and temperature (Fig. 1) were more significant.
Variations in the predicted rate constants concerning to observed may be due to inoculum-substrate ratio [47], the composition of the substrates [48, 49], and ΔT [26, 50]. Membere et al. [32], Chae et al. [41], and Pečar et al. [42], which evaluated diverse organic matters, did not evaluate the variations of the inoculum-substrate relationship. Furthermore, Pečar et al. [42] evaluated co-digestion alternatives of chicken manure sawdust with Miscanthus (pretreated and unpretreated) and did not report significant differences in the kinetic behaviors in methane production for the mixtures.
The temperature ranges for the evaluation of MA, SM, and CM were between 25 and 45°C, however, ΔT was different in each investigation. Membere et al. [32] evaluated the effect of temperature on the anaerobic digestion of MA with ΔT of 10°C. Chae et al. [41] used a lower ΔT (5°C) for SM digestion. In the case of CM, Pečar et al. [42] used a variable ΔT of 5 and 2°C. Dalby et al. [50] mention that the Van’t Hoff-Arrhenius relationship underestimates the methane production because the model did not evaluate the composition of the anaerobic consortia and the variability concerning time and temperature. Using ΔT over 5°C is a risk because the metabolic behavior of anaerobic consortia is sensitive to temperature [26]. Furthermore, variations in T cause accumulations of volatile fatty acids that decrease methane yield [51]. Mesophilic inoculums have a better capacity to adapt to temperature changes than thermophilic inoculums [52].
### 3.2. Hydraulic Retention Time Estimation
Eq. (5) used the first-order constants at evaluated temperatures estimated using the Arrhenius model and Van’t-Hoff-Arrhenius relationship. The HRT estimated at 80% of the maximum methane yield in CSTR (Fig. 2) reached values above 100 d for substrates with first-order constants below 0.03 d−1.
Fig. 2(a) presents the HRT evaluated for the constants observed for the substrates in the different temperature conditions. MA reported the lowest observed k (0.02d−1) for its lower temperature level concerning the other observations, this led to the highest value of HRT (200 d). For SM, the differences between HRT at different temperatures were less than 21 d. The CM variants indicated that the differences between the HRT calculated for the temperature interval between 35 and 40°C did not exceed nine days (with differences of up to 0.46 d). Likewise, for the levels between 40 and 42°C, the differences were up to 5.77 d (with minimum values of 2.22 d). According to the observed values (Fig. 2(a)), HRT decreases with the increase in T (and the increase in k). The Arrhenius model (Fig. 2(b)) indicated HRT values with differences for MA of 112d at T = 25°C that decreased to 14 and 2.50 d for 35 and 45°C, respectively. The evaluation of SM and CM by the Arrhenius model indicated a maximum difference of 14 d and a minimum of 0.02 d between all the evaluated cases. The Van’t Hoff-Arrhenius (Fig. 2(c)) relationship reported a different behavior for SM and MA, because this model contains an exponential term (θ(TT0)), which shows the increase in k concerning the increase in T, provided that θ over 1. Although the kinetic evaluation proposed in the literature is correct, and the Van’t Hoff-Arrhenius relationship presented a satisfactory adjustment of the kinetic constants, the T intervals tested led to θ values of 0.93 and 0.95 for MA and SM, respectively. This evidence indicates that the application of the Van’t Hoff-Arrhenius relationship was deficient and confirms that its use is not convenient in estimating HRT for T intervals over 5°C.
Arango-Osorio et al. [25] evaluated different designs of anaerobic reactors, using as general sizing equation:
##### (9)
$Vrxn=HRT*u$
Where: Vrxn is reaction volume [m3]; u is the inlet volumetric flow of the waste [m3/d]. The calculation of Vrxn depends on fixed values of HRT and u [40], however, the variation of the temperature regime decreases the methane yield and affects the kinetic constant due to the accumulation of volatile fatty acids [5156]. Thus, a fluctuation of the temperature in the operation of continuous reactors can cause problems related to the variation of HRT.
Ziganshin et al. [55] observed that the reduction of HRT affects the structure of the methanogenic community and produced a drop of around 48% in the activity of the microorganisms. Dareioti et al. [57] indicated that the decrease in HRT negatively affects methanogenesis due to the accumulation of volatile fatty acids (mainly acetic acid). The accumulation of acetate occurred by inhibition of acetoclastic methanogenesis and the accumulation of propionate by inhibition of acetogens [58]. Furthermore, low HRT led to the elimination of acetotrophic organisms due to washout and the proliferation of hydrogenotrophic organisms [59].
Li et al. [58] evaluated the variation of HRT in methane yield in continuous reactors. The lowest HRT presented higher methane productions and accumulation of volatile fatty acids. Likewise, increases in HRT cause a decrease and stabilization of methane production and a decrease in propionic and acetic acid concentrations. Algapani et al. [60] presented similar results of methane production in a continuous reactor with HRT variations from 30 to 8 days, at a temperature of 35°C. The methane yields at HRT for 30, 20, and 15 days increased slightly as 512.7, 519, and 526 mL/gVS, respectively. After the 15-day HRT values, methane yields decreased. The results presented by Kim et al. [56], noted that increases in temperature and HRT in continuous reactors benefited the methane yield and indicated that methane production decreased with the 2-day increase in HRT.
The HRT evaluation with the constants estimated with the Arrhenius model (Fig. 2(b)) and the Van’t Hoff-Arrhenius relationship (Fig. 2(c)) showed similar behaviors as the HRT calculated with the observed constants. However, the errors introduced by the Arrhenius model and the Van’t Hoff-Arrhenius relationship produce errors in the HRT calculation. The co-digestion variants, CM, present estimated HRT values similar to the observed. The HRTs calculated for CM at low and medium temperatures with the Arrhenius model were lower than the observed (contrary to the results obtained with the Van’t Hoff-Arrhenius relationship).
### 3.3. Technical Digestion Time as HRT Estimator
The literature use T80 and/or T90 as an estimator of HRT [21, 22, 45, 61]. The estimations with observations that fit first-order kinetics [21] and cone model [61] reported a lower difference between the observed and calculated values. The modified Gompertz [22, 45] did not show a relationship with T80calc and T90calc because the model refers to a delay in methane production. This research estimated and plotted T80 and T90 from the first-order model (6,7) (Fig. 3) and established a decreasing behavior concerning the increase of k.
The kinetic behavior of methane production is relevant for the identification of T80 and T90 (Fig. 3), with a progressive reduction of the distance between T80 and T90. From k 0.13d−1 (Fig. 3), the differences between T90 and T80 are less than 5.4 d, and estimation errors below 0.005 d−1 cause increases of 3.7% and decreases of 4% in the estimates of T80 and T90.
Although this research proposes a decrease between T80 and T90 for values of k above 0.13 d−1 with estimation errors less than 0.005d−1, the literature presents values of k between 0.019 and 0.13 with estimation errors between 0.002 and 0.01 d−1 [6264]. The estimates of T80 and T90 depend on the value of k and the spread of the estimation error (10,11).
##### (10)
$ΔT80=(Δkk) T80$
##### (11)
$ΔT90=(Δkk) T90$
Where: Δk is the estimation error of k; ΔT80 and ΔT90 are the estimation errors due to the propagation of the calculation error of T80 and T90 corresponding to the estimation with k. Thus, for example, the first-order kinetic constants and the estimation errors for the anaerobic co-digestion of spirulina and switchgrass (33:67) at 37 and 50°C were 0.06±0.003 and 0.13±0.004 d−1, respectively [64]. The estimate of T80 in the thermophilic case was 12.38±0.38 d unlike the mesophilic one, which indicated a value of 26.82±1.34 d. The ΔT80 for the mesophilic case indicates a gap of 1.34 d in T80; this is due to the propagation of the estimation error of k. The observations suggest that Δk should be below k/10 to obtain ΔT80 and ΔT90 lower than 8.40 and 12.11 d, respectively, for k over 0.019 d−1.
A graphical analysis of (5–7) (Fig. 4(a)) established that T80 y T90 underestimate HRT. The differences between HRT concerning T80 and T90 (Fig. 4(a)) for k above 0.13 d−1 are less than 19 and 14 d. This difference decreased to 7 and 5 d for k above 0.355 d−1. However, the trends are similar, and it was possible to establish the ratio factors HRT/T80 (2.4853) y HRT/T90 (1.7371), constant throughout the evaluation interval of k. The estimation error for HRTHRT) behaves similarly to that proposed for T80 and T90; however, due to the values of HRT are higher than T80 and T90, ΔHRT increases with the Δk. Table 3 presents some estimates of T80, T90, and HRT, for hypothetical cases of k.
The previous observations established that the HRT calculation is less sensitive to k estimation errors (Fig. 4(b)). The differences between the defined HRT values for observed k (kobs) and calculated k (kcal) decrease with increments of k.
### 4. Conclusions
This investigation used kinetic data at three levels of temperatures within the mesophilic regime. The Arrhenius model and the Van’t Hoff-Arrhenius relationship adjusted satisfactorily to the data. The fitting coefficients A, Ea, and θ were similar to the reported in the literature. The best results were for data evaluated with ΔT above 5°C, and for the variants that evaluated co-substrates.
The behavior of the methane production kinetics of a substrate determines the HRT calculation. The model for the calculation of HRT presented an inversely proportional dependence with k. The HRT trend showed similarities with T80 and T90. These observations indicate that errors in the estimation of k influence the calculation of HRT and decrease for values of k above 0.13 d−1. The calculation of HRT, T80, and T90 must consider a design of experiments that evaluates the kinetics of methane production in BMP assays at least three temperatures within the same thermal regime, whether within the psychrophilic (< 22°C), mesophilic (25–45°C) or thermophilic regime (50–70°C). This experimental design will show the tendencies of progressive increases of the kinetic constant up to a maximum point and subsequent decrease. Furthermore, to reduce the estimation error of the kinetic parameters, the temperatures of the experiment design will have a separation below 5°C.
### Acknowledgments
This research is part of the project: “Estimation of methane yield in organic waste mixtures” PYTDOC1796-2019-FCMFQ0025, corresponding to the Universidad Técnica de Manabí.
#### Nomenclature
Ea
Activation energy [kJ mol−1]
kcal
Calculated constant [d−1]
CSTR
Continuous stirred tank reactor -
k
First-order kinetic constant [d−1]
HRT
Hydraulic retention time [d]
R
Ideal gas constant [kJ mol−1 K−1]
kt
Kinetic rate constant at reaction temperature [d−1]
k0
Kinetic rate constant at reference temperature [d−1]
kT1
Kinetic rates estimated constant at temperature T1 [d−1]
kT2
Kinetic rates estimated constant at temperature T2 [d−1]
B0
Maximum methane yield [NmL gVS−1]
B
Methane yield [NmL gVS−1]
kobs
Observed constant [d−1]
A
Pre-exponential factor [d−1]
V
Reaction volume [m3]
T0
Reference temperature [°C]
RMSE
Root-mean-square error -
SSE
Sum of squares errors -
T
Temperature [°C] or [K]
θ
Temperature activity coefficient -
T80
Technical digestion time for the 80% of B0 [°C]
T90
Technical digestion time for the 90% of B0 [°C]
ΔT
Temperature variation [°C]
t
Time [d]
### Notes
Conflict-of-Interest
The authors declare that they have no conflict of interest.
Author Contributions
S.C.M.S. (Researcher) bibliographic search, model fitting and wrote the manuscript. R.J.B.C. (Ph.D. Student) developed the theory and idea, data analysis, wrote and corrected the manuscript. Y.G.S (Ph.D. Student) data analysis, correction, wrote and corrected the manuscript. O.P.O (Full-time Associate Professor) research direction, critical review of the manuscript. I.P.R (Full-time Professor) research direction, critical review of the manuscript.
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##### Fig. 1
Observed k vs predicted k.
##### Fig. 2
Hydraulic retention time. (a) Observed k (b) Arrhenius model (c) Van’t Hoff-Arrhenius relationship.
##### Fig. 3
Technical digestion time for various substrates. +,*, ** refer to substrates with fit to the first-order, modified Gompertz and Cone kinetic models.
##### Fig. 4
Hydraulic retention time. (a) Comparison of HRT with T80 and T90. (b) Influence of the estimation error of k in the calculation of HRT.
##### Table 1
Information of Kinetic Constants for Selected Substrates
Substrate T [°C] k [d−1] R2 Substrate characterization Inoculum Time
MA 25 0.020* 0.97 Macro algae Full scale anaerobic digestor sludge 40 d
35 0.110* 0.94 TS: 93.70% TS: 21.20%
45 0.120* 0.94 VSTS: 65% VSTS: 60%
SM 25 0.07 0.97 Swine manure Digestate from anaerobic master culture reactor 40 d
30 0.11 0.97 TS: 23,885 mg L−1 TS: 15,060 mg L−1
35 0.24 0.98 VS: 16,310 mg L−1 VS: 8,700 mg L−1
CM11 35 0.147 0.99 Chicken manure
TS: 79.80%
VSTS: 86.3%
Miscanthus
TS: 93.10%
VSTS: 97.8%
Anaerobic digestor sludge
TS: 12.1%
VSTS: 75%
21 d
40 0.219 0.99
42 0.258 0.99
CM12 35 0.172 0.98
40 0.269 0.99
42 0.344 0.98
CM13 35 0.19 0.98
40 0.263 0.99
42 0.308 0.99
CM21 35 0.164 0.99
40 0.216 0.99
42 0.299 0.99
CM22 35 0.204 0.99
40 0.209 0.99
42 0.299 0.99
CM23 35 0.202 0.99
40 0.242 0.98
42 0.328 0.98
CM31 35 0.178 0.99
40 0.235 0.99
42 0.285 0.99
CM32 35 0.201 0.99
40 0.251 0.99
42 0.325 0.99
CM33 35 0.19 0.99
40 0.251 0.99
42 0.304 0.99
* rough estimation from figures reported in [32].
##### Table 2
Fitted Constants for the Arrhenius Model and the Van’t Hoff-Arrhenius Relationship
Substrate Arrhenius Van’t Hoff-Arrhenius
A Ea SSE RMSE θ r0 ×102 SSE RMSE
MA 6.81 × 105 40.9 0.002 0.042 0.95 3.62 0.002 0.042
SM 2.52 × 107 48.9 0.0003 0.019 0.93 4.28 0.0001 0.014
CM11 2.43 × 103 24.3 0.002 0.048 1.08 4.396 0.000 0.001
CM12 3.45 × 103 24.6 0.007 0.008 1.11 3.732 0.000 0.011
CM13 5.29 × 102 19.8 0.003 0.053 1.07 6.694 0.001 0.004
CM21 3.15 × 103 24.8 0.004 0.065 1.09 4.092 0.001 0.029
CM22 1.26 × 103 22.2 0.003 0.055 1.05 8.616 0.002 0.047
CM23 4.76 × 102 19.5 0.004 0.063 1.07 6.927 0.208 0.456
CM31 3.81 × 102 19.2 0.002 0.049 1.07 6.352 0.000 0.011
CM32 5.08 × 102 19.7 0.004 0.059 1.07 6.847 0.001 0.026
CM33 3.15 × 103 24.8 0.004 0.065 1.09 4.092 0.001 0.029
##### Table 3
Propagation of the Estimation Error in T80, T90 and HRT
k Δk Δk/k T80 ΔT80 T90 ΔT90 HRT ΔHRT
0.019 0.00190 0.100 84.71 8.47 121.19 12.12 210.53 21.05
0.025 0.00250 0.100 64.38 6.44 92.10 9.21 160.00 16.00
0.050 0.00500 0.100 32.19 3.22 46.05 4.61 80.00 8.00
0.100 0.01000 0.100 16.09 1.61 23.03 2.30 40.00 4.00
0.150 0.01500 0.100 10.73 1.07 15.35 1.54 26.67 2.67
0.019 0.00127 0.067 84.71 5.65 121.19 8.08 210.53 14.04
0.025 0.00167 0.067 64.38 4.29 92.10 6.14 160.00 10.67
0.050 0.00333 0.067 32.19 2.15 46.05 3.07 80.00 5.33
0.100 0.00667 0.067 16.09 1.07 23.03 1.54 40.00 2.67
0.150 0.01000 0.067 10.73 0.72 15.35 1.02 26.67 1.78
0.019 0.00095 0.050 84.71 4.24 121.19 6.06 210.53 10.53
0.025 0.00125 0.050 64.38 3.22 92.10 4.61 160.00 8.00
0.050 0.00250 0.050 32.19 1.61 46.05 2.30 80.00 4.00
0.100 0.00500 0.050 16.09 0.80 23.03 1.15 40.00 2.00
0.150 0.00750 0.050 10.73 0.54 15.35 0.77 26.67 1.33
TOOLS
Full text via DOI
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2022-10-02 06:44:56
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https://math.stackexchange.com/questions/2090574/sizes-of-transitive-subgroups-of-s-n-with-trivial-centralizer
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# Sizes of transitive subgroups of $S_n$ with trivial centralizer
Let $G\le S_n$ be a transitive subgroup with trivial centralizer in $S_n$.
Can we deduce any nontrivial lower bounds on the order of $G$?
I'd also be interested in asymptotic results as $n\rightarrow\infty$.
• That depends on $c$. For $c=1$,all you can say is that $|G| \ge n-1$, and for $c=n!$ you have $|G|=1$. For $c=(n/2)!$ (with $n$ even) you get something like $n/2-1 \le |G| \le (n/2)!$, which is a large range. Did you have any particular values of $c$ in mind? – Derek Holt Jan 9 '17 at 19:15
• @DerekHolt Okay I've edited the question. So, my interest comes from this - When $M$ is a finite nonabelian simple group, $F_2$ is a free group of rank 2, and $\varphi : F_2\twoheadrightarrow M$ is a surjection, then I've checked computationally (for the smallest 23 nonab finite simple groups) that the action of $Aut(F_2)$ on the $Aut(F_2)$-orbit of $\varphi$ has precisely $Out(M)$ as its centralizer. In particular, $Out(M)$ is always rather small. I'd like to show that the permutation image of $Aut(F_2)$ on the orbit of $\varphi$ is always "pretty large" – user355183 Jan 9 '17 at 19:26
• @DerekHolt In particular, if $e$ is the exponent of $M$, then I'd like to show that the permutation image always has size at least $e^3$ (computationally this is true, by massive margins). I suppose I should write this as a separate question. – user355183 Jan 9 '17 at 19:28
I think the best possible general bound for your edited question is $|G| \ge 2n$. For $n$ odd, the dihedral group of order $2n$ has trivial centralizer.
• How do you see that the centralizer of $D_{2n}$ is trivial? It certainly has trivial center, but I assume you're embedding $D_{2n}$ inside $S_n$? If $n$ is even you can consider the action of $D_{2n}$ on vertices, but if $n$ is odd this doesn't work, right? – user355183 Jan 9 '17 at 19:37
• The order of the centralizer in $S_n$ of a transitive subgroup $G$ is equal to the number of fixed points of the stabilizer $G_a$ of a point $a$. This is $1$ for $D_{2n}$ when $n$ is odd, but $2$ for $n$ even. – Derek Holt Jan 9 '17 at 20:25
• I agree with the first sentence, and perhaps this is a stupid question - but how are you having $D_{2n}$ act on a set of $n$ elements when $n$ is odd? – user355183 Jan 9 '17 at 21:30
• There must be some misunderstanding here! The dihedral group of order $2n$ is often defined as the group of rotations and reflections of an $n$-sided regular polygon, so it is acting on the $n$ vertices. – Derek Holt Jan 9 '17 at 21:47
• But if $n$ is odd, the reflections don't send a vertex to another of the $n$ vertices! – user355183 Jan 9 '17 at 21:51
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2020-02-28 00:46:16
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http://tug.org/pipermail/xetex/2004-July/000552.html
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# [XeTeX] dealing with \$ character
Hans Hagen pragma at wxs.nl
Thu Jul 8 09:51:47 CEST 2004
```Bruce D'Arcus wrote:
>
> On Jul 7, 2004, at 12:08 PM, Hans Hagen wrote:
>
>>> Following from previous discussion, is there a way that I can get
>>> TeX to accept the \$ character without having to do \\$? My source is
>>> XML, so it's not convenient to modify there, or in the XSLT.
>>
>>
>> \catcode`\\$=12
>
>
> Thanks Hans, but this -- in the preamble -- doesn't seem to work. Is
> there something else I need to do?
>
> I just want to be able to typeset "\$4000".
it sort of depends on the macro package you use; in context, you can say:
\chardef\XMLtokensreduction\plustwo
of course you then have to use fonts that have the dollar in the normal ascii slot (not cmr -)
I always wonder why characters like \$ are not symbols
Hans
-----------------------------------------------------------------
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2017-07-21 04:41:41
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https://www.math.princeton.edu/events/title-measurably-entire-functions-and-their-growth-2017-10-11t160010
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# Title: Measurably entire functions and their growth
-
Adi Glücksam, Tel Aviv university
Fine Hall 224
Let (X,B,P) be a standard probability space. Let T:C\rightarrow PPT(X) be a free action of the complex plane on the space (X,B,P). We say that the function F:X\rightarrow C is measurably entire if it is measurable and for P-a.e x the function F_x(z):=F(T_zx) is entire. B. Weiss showed in '97 that for every free C action there exists a non-constant measurably entire function. In the talk I will present upper and lower bounds for the growth of such functions. The talk is partly based on a joint work with L. Buhovsky, A.Logunov, and M. Sodin.
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2018-09-25 14:59:42
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https://en.wikipedia.org/wiki/Continuous_function
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# Continuous function
In mathematics, a continuous function is, roughly speaking, a function for which sufficiently small changes in the input result in arbitrarily small changes in the output. Otherwise, a function is said to be a discontinuous function. A continuous function with a continuous inverse function is called a homeomorphism.
Continuity of functions is one of the core concepts of topology, which is treated in full generality below. The introductory portion of this article focuses on the special case where the inputs and outputs of functions are real numbers. In addition, this article discusses the definition for the more general case of functions between two metric spaces. In order theory, especially in domain theory, one considers a notion of continuity known as Scott continuity. Other forms of continuity do exist but they are not discussed in this article.
As an example, consider the function h(t), which describes the height of a growing flower at time t. This function is continuous. By contrast, if M(t) denotes the amount of money in a bank account at time t, then the function jumps at each point in time when money is deposited or withdrawn, so the function M(t) is discontinuous.
## History
A form of the epsilon-delta definition of continuity was first given by Bernard Bolzano in 1817. Augustin-Louis Cauchy defined continuity of ${\displaystyle y=f(x)}$ as follows: an infinitely small increment ${\displaystyle \alpha }$ of the independent variable x always produces an infinitely small change ${\displaystyle f(x+\alpha )-f(x)}$ of the dependent variable y (see e.g., Cours d'Analyse, p. 34). Cauchy defined infinitely small quantities in terms of variable quantities, and his definition of continuity closely parallels the infinitesimal definition used today (see microcontinuity). The formal definition and the distinction between pointwise continuity and uniform continuity were first given by Bolzano in the 1830s but the work wasn't published until the 1930s. Like Bolzano,[1] Karl Weierstrass [2] denied continuity of a function at a point c unless it was defined at and on both sides of c, but Édouard Goursat [3] allowed the function to be defined only at and on one side of c, and Camille Jordan [4] allowed it even if the function was defined only at c. All three of those nonequivalent definitions of pointwise continuity are still in use.[5] Eduard Heine provided the first published definition of uniform continuity in 1872, but based these ideas on lectures given by Peter Gustav Lejeune Dirichlet in 1854.[6]
## Real-valued continuous functions
### Definition
The function ${\displaystyle f(x)={\tfrac {1}{x}}}$ is continuous although its graph makes a jump at ${\displaystyle x=0}$
A function from the set of real numbers to the real numbers can be represented by a graph in the Cartesian plane; such a function is continuous if, roughly speaking, the graph is a single unbroken curve with no "holes" or "jumps". Note that this is not a rigorous definition of continuity since also the function ${\displaystyle f(x)={\tfrac {1}{x}}}$ is continuous on its whole domain ${\displaystyle \mathbb {R} \setminus \{0\}}$ although its graph has a “jump” at ${\displaystyle x=0}$.
A function is continuous at a point if it does not have a hole or jump. A “hole” or “jump” in the graph of a function occurs if the value of the function at a point c differs from its limiting value along points that are nearby. Such a point is called a discontinuity. A function is then continuous if it has no holes or jumps: that is, if it is continuous at every point of its domain. Otherwise, a function is discontinuous, at the points where the value of the function differs from its limiting value (if any).
There are several ways to make this definition mathematically rigorous. These definitions are equivalent to one another, so the most convenient definition can be used to determine whether a given function is continuous or not. In the definitions below,
${\displaystyle f\colon I\rightarrow \mathbf {R} .}$
is a function defined on a subset I of the set R of real numbers. This subset I is referred to as the domain of f. Some possible choices include I=R, the whole set of real numbers, an open interval
${\displaystyle I=(a,b)=\{x\in \mathbf {R} \,|\,a
or a closed interval
${\displaystyle I=[a,b]=\{x\in \mathbf {R} \,|\,a\leq x\leq b\}.}$
Here, a and b are real numbers.
#### Definition in terms of limits of functions
The function f is continuous at some point c of its domain if the limit of f(x) as x approaches c through the domain of f exists and is equal to f(c).[7] In mathematical notation, this is written as
${\displaystyle \lim _{x\to c}{f(x)}=f(c).}$
In detail this means three conditions: first, f has to be defined at c. Second, the limit on the left hand side of that equation has to exist. Third, the value of this limit must equal f(c).
(We have here assumed that the domain of f does not have any isolated points. For example, an interval or union of intervals has no isolated points.)
#### Definition in terms of neighborhoods
A neighborhood of a point c is a set that contains all points of the domain within some fixed distance of c. Intuitively, a function is continuous at a point c if the range of the restriction of f to a neighborhood of c shrinks to a single point f(c) as the width of the neighborhood shrinks to zero. More precisely, a function f is continuous at a point c of its domain if, for any neighborhood ${\displaystyle N_{1}(f(c))}$ there is a neighborhood ${\displaystyle N_{2}(c)}$ such that ${\displaystyle f(x)\in N_{1}(f(c))}$ whenever ${\displaystyle x\in N_{2}(c)}$.
This definition does only require that the domain and the codomain are topological spaces and is thus the most general definition. From it follows, that the function f is automatically continuous at every isolated point of its domain. As a specific example, every real valued function on the set of integers is continuous.
#### Definition in terms of limits of sequences
The sequence exp(1/n) converges to exp(0)
One can instead require that for any sequence ${\displaystyle (x_{n})_{n\in \mathbb {N} }}$ of points in the domain which converges to c, the corresponding sequence ${\displaystyle \left(f(x_{n})\right)_{n\in \mathbb {N} }}$ converges to f(c). In mathematical notation, ${\displaystyle \forall (x_{n})_{n\in \mathbb {N} }\subset I:\lim _{n\to \infty }x_{n}=c\Rightarrow \lim _{n\to \infty }f(x_{n})=f(c)\,.}$
#### Weierstrass and Jordan definitions (epsilon–delta) of continuous functions
Illustration of the ε-δ-definition: for ε=0.5, c=2, the value δ=0.5 satisfies the condition of the definition.
Explicitly including the definition of the limit of a function, we obtain a self-contained definition: Given a function f as above and an element c of the domain I, f is said to be continuous at the point c if the following holds: For any number ε > 0, however small, there exists some number δ > 0 such that for all x in the domain of f with c − δ < x < c + δ, the value of f(x) satisfies
${\displaystyle f(c)-\varepsilon
Alternatively written, continuity of f : I → R at c ∈ I means that for every ε > 0 there exists a δ > 0 such that for all x ∈ I,:
${\displaystyle |x-c|<\delta \Rightarrow |f(x)-f(c)|<\varepsilon .\,}$
More intuitively, we can say that if we want to get all the f(x) values to stay in some small neighborhood around f(c), we simply need to choose a small enough neighborhood for the x values around c, and we can do that no matter how small the f(x) neighborhood is; f is then continuous at c.
In modern terms, this is generalized by the definition of continuity of a function with respect to a basis for the topology, here the metric topology.
Weierstrass had required that the interval c − δ < x < c + δ be entirely within the domain I, but Jordan removed that restriction.
#### Definition using oscillation
The failure of a function to be continuous at a point is quantified by its oscillation.
Continuity can also be defined in terms of oscillation: a function f is continuous at a point x0 if and only if its oscillation at that point is zero;[8] in symbols, ${\displaystyle \omega _{f}(x_{0})=0.}$ A benefit of this definition is that it quantifies discontinuity: the oscillation gives how much the function is discontinuous at a point.
This definition is useful in descriptive set theory to study the set of discontinuities and continuous points – the continuous points are the intersection of the sets where the oscillation is less than ε (hence a Gδ set) – and gives a very quick proof of one direction of the Lebesgue integrability condition.[9]
The oscillation is equivalent to the ε-δ definition by a simple re-arrangement, and by using a limit (lim sup, lim inf) to define oscillation: if (at a given point) for a given ε0 there is no δ that satisfies the ε-δ definition, then the oscillation is at least ε0, and conversely if for every ε there is a desired δ, the oscillation is 0. The oscillation definition can be naturally generalized to maps from a topological space to a metric space.
#### Definition using the hyperreals
Cauchy defined continuity of a function in the following intuitive terms: an infinitesimal change in the independent variable corresponds to an infinitesimal change of the dependent variable (see Cours d'analyse, page 34). Non-standard analysis is a way of making this mathematically rigorous. The real line is augmented by the addition of infinite and infinitesimal numbers to form the hyperreal numbers. In nonstandard analysis, continuity can be defined as follows.
A real-valued function f is continuous at x if its natural extension to the hyperreals has the property that for all infinitesimal dx, f(x+dx) − f(x) is infinitesimal[10]
(see microcontinuity). In other words, an infinitesimal increment of the independent variable always produces to an infinitesimal change of the dependent variable, giving a modern expression to Augustin-Louis Cauchy's definition of continuity.
### Examples
The graph of a cubic function has no jumps or holes. The function is continuous.
All polynomial functions, such as f(x) = x3 + x2 - 5x + 3 (pictured), are continuous. This is a consequence of the fact that, given two continuous functions
${\displaystyle f,g\colon I\rightarrow \mathbf {R} }$
defined on the same domain I, then the sum f + g and the product fg of the two functions are continuous (on the same domain I). Moreover, the function
${\displaystyle {\frac {f}{g}}\colon \{x\in I|g(x)\neq 0\}\rightarrow \mathbf {R} ,x\mapsto {\frac {f(x)}{g(x)}}}$
is continuous. (The points where g(x) is zero are discarded, as they are not in the domain of f/g.) For example, the function (pictured)
${\displaystyle f(x)={\frac {2x-1}{x+2}}}$
The graph of a continuous rational function. The function is not defined for x=−2. The vertical and horizontal lines are asymptotes.
is defined for all real numbers x ≠ −2 and is continuous at every such point. Thus it is a continuous function. The question of continuity at x = −2 does not arise, since x = −2 is not in the domain of f. There is no continuous function F: RR that agrees with f(x) for all x ≠ −2. The sinc function g(x) = (sin x)/x, defined for all x≠0 is continuous at these points. Thus it is a continuous function, too. However, unlike the one of the previous example, this one can be extended to a continuous function on all real numbers, namely
${\displaystyle G(x)={\begin{cases}{\frac {\sin(x)}{x}}&{\text{ if }}x\neq 0\\1&{\text{ if }}x=0,\end{cases}}}$
since the limit of g(x), when x approaches 0, is 1. Therefore, the point x=0 is called a removable singularity of g.
Given two continuous functions
${\displaystyle f\colon I\rightarrow J(\subset \mathbf {R} ),g\colon J\rightarrow \mathbf {R} ,}$
the composition
${\displaystyle g\circ f\colon I\rightarrow \mathbf {R} ,x\mapsto g(f(x))}$
is continuous.
### Non-examples
Plot of the signum function. It shows that ${\displaystyle \lim _{n\to \infty }\operatorname {sgn} \left({\tfrac {1}{n}}\right)\neq \operatorname {sgn} \left(\lim _{n\to \infty }{\tfrac {1}{n}}\right)}$. Thus, the signum function is not continuous at the point 0.
An example of a discontinuous function is the Heaviside step function ${\displaystyle H}$, defined by ${\displaystyle H(x)=1{\text{ if }}x\geq 0,H(x)=0{\text{ if }}x<0}$. Pick for instance ${\displaystyle \epsilon ={\frac {1}{2}}}$. There is no ${\displaystyle \delta }$-neighborhood around ${\displaystyle x=0}$ that will force all the ${\displaystyle H(x)}$ values to be within ${\displaystyle \epsilon }$ of ${\displaystyle H(0)}$. Intuitively we can think of this type of discontinuity as a sudden jump in function values.
Similarly, the signum or sign function
${\displaystyle \operatorname {sgn}(x)={\begin{cases}1&{\text{ if }}x>0\\0&{\text{ if }}x=0\\-1&{\text{ if }}x<0\end{cases}}}$
is discontinuous at ${\displaystyle x=0}$ but continuous everywhere else. Yet another example: the function
${\displaystyle f(x)={\begin{cases}\sin \left({\frac {1}{x^{2}}}\right){\text{ if }}x\neq 0\\0{\text{ if }}x=0\end{cases}}}$
is continuous everywhere apart from ${\displaystyle x=0}$.
Plot of Thomae's function for the domain ${\displaystyle 0.
${\displaystyle f(x)={\begin{cases}1{\text{ if }}x=0\\{\frac {1}{q}}{\text{ if }}x={\frac {p}{q}}{\text{(in lowest terms) is a rational number}}\\0{\text{ if }}x{\text{ is irrational}}.\end{cases}}}$
is continuous at all irrational numbers and discontinuous at all rational numbers. In a similar vein, Dirichlet's function
${\displaystyle D(x)={\begin{cases}0{\text{ if }}x{\text{ is irrational }}(\in \mathbb {R} \setminus \mathbb {Q} )\\1{\text{ if }}x{\text{ is rational }}(\in \mathbb {Q} )\end{cases}}}$
is nowhere continuous.
### Properties
#### Intermediate value theorem
The intermediate value theorem is an existence theorem, based on the real number property of completeness, and states:
If the real-valued function f is continuous on the closed interval [ab] and k is some number between f(a) and f(b), then there is some number c in [ab] such that f(c) = k.
For example, if a child grows from 1 m to 1.5 m between the ages of two and six years, then, at some time between two and six years of age, the child's height must have been 1.25 m.
As a consequence, if f is continuous on [ab] and f(a) and f(b) differ in sign, then, at some point c in [ab], f(c) must equal zero.
#### Extreme value theorem
The extreme value theorem states that if a function f is defined on a closed interval [a,b] (or any closed and bounded set) and is continuous there, then the function attains its maximum, i.e. there exists c ∈ [a,b] with f(c) ≥ f(x) for all x ∈ [a,b]. The same is true of the minimum of f. These statements are not, in general, true if the function is defined on an open interval (a,b) (or any set that is not both closed and bounded), as, for example, the continuous function f(x) = 1/x, defined on the open interval (0,1), does not attain a maximum, being unbounded above.
#### Relation to differentiability and integrability
${\displaystyle f\colon (a,b)\rightarrow \mathbf {R} }$
is continuous, as can be shown. The converse does not hold: for example, the absolute value function
${\displaystyle f(x)=|x|={\begin{cases}x{\text{ if }}x\geq 0\\-x{\text{ if }}x<0\end{cases}}}$
is everywhere continuous. However, it is not differentiable at x = 0 (but is so everywhere else). Weierstrass's function is also everywhere continuous but nowhere differentiable.
The derivative f′(x) of a differentiable function f(x) need not be continuous. If f′(x) is continuous, f(x) is said to be continuously differentiable. The set of such functions is denoted C1((a, b)). More generally, the set of functions
${\displaystyle f\colon \Omega \rightarrow \mathbf {R} }$
(from an open interval (or open subset of R) Ω to the reals) such that f is n times differentiable and such that the n-th derivative of f is continuous is denoted Cn(Ω). See differentiability class. In the field of computer graphics, these three levels are sometimes called G0 (continuity of position), G1 (continuity of tangency), and G2 (continuity of curvature).
Every continuous function
${\displaystyle f\colon [a,b]\rightarrow \mathbf {R} }$
is integrable (for example in the sense of the Riemann integral). The converse does not hold, as the (integrable, but discontinuous) sign function shows.
#### Pointwise and uniform limits
A sequence of continuous functions fn(x) whose (pointwise) limit function f(x) is discontinuous. The convergence is not uniform.
Given a sequence
${\displaystyle f_{1},f_{2},\dotsc \colon I\rightarrow \mathbf {R} }$
of functions such that the limit
${\displaystyle f(x):=\lim _{n\rightarrow \infty }f_{n}(x)}$
exists for all x in I, the resulting function f(x) is referred to as the pointwise limit of the sequence of functions (fn)nN. The pointwise limit function need not be continuous, even if all functions fn are continuous, as the animation at the right shows. However, f is continuous when the sequence converges uniformly, by the uniform convergence theorem. This theorem can be used to show that the exponential functions, logarithms, square root function, trigonometric functions are continuous.
### Directional and semi-continuity
Discontinuous functions may be discontinuous in a restricted way, giving rise to the concept of directional continuity (or right and left continuous functions) and semi-continuity. Roughly speaking, a function is right-continuous if no jump occurs when the limit point is approached from the right. Formally, f is said to be right-continuous at the point c if the following holds: For any number ε > 0 however small, there exists some number δ > 0 such that for all x in the domain with c < x < c + δ, the value of f(x) will satisfy
${\displaystyle |f(x)-f(c)|<\varepsilon .\,}$
This is the same condition as for continuous functions, except that it is required to hold for x strictly larger than c only. Requiring it instead for all x with cδ < x < c yields the notion of left-continuous functions. A function is continuous if and only if it is both right-continuous and left-continuous.
A function f is lower semi-continuous if, roughly, any jumps that might occur only go down, but not up. That is, for any ε > 0, there exists some number δ > 0 such that for all x in the domain with | x − c | < δ, the value of f(x) satisfies
${\displaystyle f(x)\geq f(c)-\epsilon .}$
The reverse condition is upper semi-continuity.
## Continuous functions between metric spaces
The concept of continuous real-valued functions can be generalized to functions between metric spaces. A metric space is a set X equipped with a function (called metric) dX, that can be thought of as a measurement of the distance of any two elements in X. Formally, the metric is a function
${\displaystyle d_{X}\colon X\times X\rightarrow \mathbf {R} }$
that satisfies a number of requirements, notably the triangle inequality. Given two metric spaces (X, dX) and (Y, dY) and a function
${\displaystyle f\colon X\rightarrow Y}$
then f is continuous at the point c in X (with respect to the given metrics) if for any positive real number ε, there exists a positive real number δ such that all x in X satisfying dX(x, c) < δ will also satisfy dY(f(x), f(c)) < ε. As in the case of real functions above, this is equivalent to the condition that for every sequence (xn) in X with limit lim xn = c, we have lim f(xn) = f(c). The latter condition can be weakened as follows: f is continuous at the point c if and only if for every convergent sequence (xn) in X with limit c, the sequence (f(xn)) is a Cauchy sequence, and c is in the domain of f.
The set of points at which a function between metric spaces is continuous is a Gδ set – this follows from the ε-δ definition of continuity.
This notion of continuity is applied, for example, in functional analysis. A key statement in this area says that a linear operator
${\displaystyle T\colon V\rightarrow W}$
between normed vector spaces V and W (which are vector spaces equipped with a compatible norm, denoted ||x||) is continuous if and only if it is bounded, that is, there is a constant K such that
${\displaystyle \|T(x)\|\leq K\|x\|}$
for all x in V.
### Uniform, Hölder and Lipschitz continuity
For a Lipschitz continuous function, there is a double cone (shown in white) whose vertex can be translated along the graph, so that the graph always remains entirely outside the cone.
The concept of continuity for functions between metric spaces can be strengthened in various ways by limiting the way δ depends on ε and c in the definition above. Intuitively, a function f as above is uniformly continuous if the δ does not depend on the point c. More precisely, it is required that for every real number ε > 0 there exists δ > 0 such that for every cb ∈ X with dX(bc) < δ, we have that dY(f(b), f(c)) < ε. Thus, any uniformly continuous function is continuous. The converse does not hold in general, but holds when the domain space X is compact. Uniformly continuous maps can be defined in the more general situation of uniform spaces.[11]
A function is Hölder continuous with exponent α (a real number) if there is a constant K such that for all b and c in X, the inequality
${\displaystyle d_{Y}(f(b),f(c))\leq K\cdot (d_{X}(b,c))^{\alpha }}$
holds. Any Hölder continuous function is uniformly continuous. The particular case α = 1 is referred to as Lipschitz continuity. That is, a function is Lipschitz continuous if there is a constant K such that the inequality
${\displaystyle d_{Y}(f(b),f(c))\leq K\cdot d_{X}(b,c)}$
holds any b, c in X.[12] The Lipschitz condition occurs, for example, in the Picard–Lindelöf theorem concerning the solutions of ordinary differential equations.
## Continuous functions between topological spaces
Continuity of a function at a point.
Another, more abstract, notion of continuity is continuity of functions between topological spaces in which there generally is no formal notion of distance, as there is in the case of metric spaces. A topological space is a set X together with a topology on X, which is a set of subsets of X satisfying a few requirements with respect to their unions and intersections that generalize the properties of the open balls in metric spaces while still allowing to talk about the neighbourhoods of a given point. The elements of a topology are called open subsets of X (with respect to the topology).
A function
${\displaystyle f\colon X\rightarrow Y}$
between two topological spaces X and Y is continuous if for every open set VY, the inverse image
${\displaystyle f^{-1}(V)=\{x\in X\;|\;f(x)\in V\}}$
is an open subset of X. That is, f is a function between the sets X and Y (not on the elements of the topology TX), but the continuity of f depends on the topologies used on X and Y.
This is equivalent to the condition that the preimages of the closed sets (which are the complements of the open subsets) in Y are closed in X.
An extreme example: if a set X is given the discrete topology (in which every subset is open), all functions
${\displaystyle f\colon X\rightarrow T}$
to any topological space T are continuous. On the other hand, if X is equipped with the indiscrete topology (in which the only open subsets are the empty set and X) and the space T set is at least T0, then the only continuous functions are the constant functions. Conversely, any function whose range is indiscrete is continuous.
### Alternative definitions
Several equivalent definitions for a topological structure exist and thus there are several equivalent ways to define a continuous function.
#### Neighborhood definition
Neighborhoods continuity for functions between topological spaces ${\displaystyle (X,{\mathcal {T}}_{X})}$ and ${\displaystyle (Y,{\mathcal {T}}_{Y})}$ at a point may be defined:
A function ${\displaystyle f:X\rightarrow Y}$ is continuous at a point ${\displaystyle x\in X}$ iff for any neighborhood of its image ${\displaystyle f(x)\in Y}$ the preimage is again a neighborhood of that point: ${\displaystyle \forall N\in {\mathcal {N}}_{f(x)}:f^{-1}(N)\in {\mathcal {M}}_{x}}$
According to the property that neighborhood systems being upper sets this can be restated as follows:
${\displaystyle \forall N\in {\mathcal {N}}_{f(x)}\exists M\in {\mathcal {M}}_{x}:M\subseteq f^{-1}(N)}$
${\displaystyle \forall N\in {\mathcal {N}}_{f(x)}\exists M\in {\mathcal {M}}_{x}:f(M)\subseteq N}$
The second one being a restatement involving the image rather than the preimage.
Literally, this means no matter how small the neighborhood is chosen one can always find a neighborhood mapped into it.
Besides, there's a simplification involving only open neighborhoods. In fact, they're equivalent:
${\displaystyle \forall V\in {\mathcal {T}}_{Y},f(x)\in V\exists U\in {\mathcal {T}}_{X},x\in U:U\subseteq f^{-1}(V)}$
${\displaystyle \forall V\in {\mathcal {T}}_{Y},f(x)\in V\exists U\in {\mathcal {T}}_{X},x\in U:f(U)\subseteq V}$
The second one again being a restatement using images rather than preimages.
If X and Y are metric spaces, it is equivalent to consider the neighborhood system of open balls centered at x and f(x) instead of all neighborhoods. This gives back the above δ-ε definition of continuity in the context of metric spaces. However, in general topological spaces, there is no notion of nearness or distance.
Note, however, that if the target space is Hausdorff, it is still true that f is continuous at a if and only if the limit of f as x approaches a is f(a). At an isolated point, every function is continuous.
#### Sequences and nets
In several contexts, the topology of a space is conveniently specified in terms of limit points. In many instances, this is accomplished by specifying when a point is the limit of a sequence, but for some spaces that are too large in some sense, one specifies also when a point is the limit of more general sets of points indexed by a directed set, known as nets. A function is (Heine-)continuous only if it takes limits of sequences to limits of sequences. In the former case, preservation of limits is also sufficient; in the latter, a function may preserve all limits of sequences yet still fail to be continuous, and preservation of nets is a necessary and sufficient condition.
In detail, a function f: XY is sequentially continuous if whenever a sequence (xn) in X converges to a limit x, the sequence (f(xn)) converges to f(x). Thus sequentially continuous functions "preserve sequential limits". Every continuous function is sequentially continuous. If X is a first-countable space and countable choice holds, then the converse also holds: any function preserving sequential limits is continuous. In particular, if X is a metric space, sequential continuity and continuity are equivalent. For non first-countable spaces, sequential continuity might be strictly weaker than continuity. (The spaces for which the two properties are equivalent are called sequential spaces.) This motivates the consideration of nets instead of sequences in general topological spaces. Continuous functions preserve limits of nets, and in fact this property characterizes continuous functions.
#### Closure operator definition
Instead of specifying the open subsets of a topological space, the topology can also be determined by a closure operator (denoted cl) which assigns to any subset AX its closure, or an interior operator (denoted int), which assigns to any subset A of X its interior. In these terms, a function
${\displaystyle f\colon (X,\mathrm {cl} )\to (X',\mathrm {cl} ')\,}$
between topological spaces is continuous in the sense above if and only if for all subsets A of X
${\displaystyle f(\mathrm {cl} (A))\subseteq \mathrm {cl} '(f(A)).}$
That is to say, given any element x of X that is in the closure of any subset A, f(x) belongs to the closure of f(A). This is equivalent to the requirement that for all subsets A' of X'
${\displaystyle f^{-1}(\mathrm {cl} '(A'))\supseteq \mathrm {cl} (f^{-1}(A')).}$
Moreover,
${\displaystyle f\colon (X,\mathrm {int} )\to (X',\mathrm {int} ')\,}$
is continuous if and only if
${\displaystyle f^{-1}(\mathrm {int} '(A'))\subseteq \mathrm {int} (f^{-1}(A'))}$
for any subset A' of Y.
### Properties
If f: XY and g: YZ are continuous, then so is the composition gf: XZ. If f: XY is continuous and
The possible topologies on a fixed set X are partially ordered: a topology τ1 is said to be coarser than another topology τ2 (notation: τ1 ⊆ τ2) if every open subset with respect to τ1 is also open with respect to τ2. Then, the identity map
idX: (X, τ2) → (X, τ1)
is continuous if and only if τ1 ⊆ τ2 (see also comparison of topologies). More generally, a continuous function
${\displaystyle (X,\tau _{X})\rightarrow (Y,\tau _{Y})}$
stays continuous if the topology τY is replaced by a coarser topology and/or τX is replaced by a finer topology.
### Homeomorphisms
Symmetric to the concept of a continuous map is an open map, for which images of open sets are open. In fact, if an open map f has an inverse function, that inverse is continuous, and if a continuous map g has an inverse, that inverse is open. Given a bijective function f between two topological spaces, the inverse function f−1 need not be continuous. A bijective continuous function with continuous inverse function is called a homeomorphism.
If a continuous bijection has as its domain a compact space and its codomain is Hausdorff, then it is a homeomorphism.
### Defining topologies via continuous functions
Given a function
${\displaystyle f\colon X\rightarrow S,\,}$
where X is a topological space and S is a set (without a specified topology), the final topology on S is defined by letting the open sets of S be those subsets A of S for which f−1(A) is open in X. If S has an existing topology, f is continuous with respect to this topology if and only if the existing topology is coarser than the final topology on S. Thus the final topology can be characterized as the finest topology on S that makes f continuous. If f is surjective, this topology is canonically identified with the quotient topology under the equivalence relation defined by f.
Dually, for a function f from a set S to a topological space, the initial topology on S has as open subsets A of S those subsets for which f(A) is open in X. If S has an existing topology, f is continuous with respect to this topology if and only if the existing topology is finer than the initial topology on S. Thus the initial topology can be characterized as the coarsest topology on S that makes f continuous. If f is injective, this topology is canonically identified with the subspace topology of S, viewed as a subset of X.
More generally, given a set S, specifying the set of continuous functions
${\displaystyle S\rightarrow X}$
into all topological spaces X defines a topology. Dually, a similar idea can be applied to maps
${\displaystyle X\rightarrow S.}$
This is an instance of a universal property.
## Related notions
Various other mathematical domains use the concept of continuity in different, but related meanings. For example, in order theory, an order-preserving function f: XY between particular types of partially ordered sets X and Y is continuous if for each directed subset A of X, we have sup(f(A)) = f(sup(A)). Here sup is the supremum with respect to the orderings in X and Y, respectively. This notion of continuity is the same as topological continuity when the partially ordered sets are given the Scott topology.[13][14]
${\displaystyle F\colon {\mathcal {C}}\rightarrow {\mathcal {D}}}$
between two categories is called continuous, if it commutes with small limits. That is to say,
${\displaystyle \varprojlim _{i\in I}F(C_{i})\cong F(\varprojlim _{i\in I}C_{i})}$
for any small (i.e., indexed by a set I, as opposed to a class) diagram of objects in ${\displaystyle {\mathcal {C}}}$.
A continuity space is a generalization of metric spaces and posets,[15][16] which uses the concept of quantales, and that can be used to unify the notions of metric spaces and domains.[17]
## Notes
1. ^ Bolzano, Bernard (1817), Rein analytischer Beweis des Lehrsatzes dass zwischen je zwey Werthen, die ein entgegengesetztes Resultat gewaehren, wenigstens eine reele Wurzel der Gleichung liege, Prague: Haase
2. ^ Dugac, Pierre (1973), "Eléments d'Analyse de Karl Weierstrass", Archive for History of Exact Sciences, 10: 41–176, doi:10.1007/bf00343406
3. ^ Goursat, E. (1904), A course in mathematical analysis, Boston: Ginn, p. 2
4. ^ Jordan, M.C. (1893), Cours d'analyse de l'École polytechnique, 1 (2nd ed.), Paris: Gauthier-Villars, p. 46
5. ^ Harper, J.F. (2016), "Defining continuity of real functions of real variables", BSHM Bulletin: Journal of the British Society for the History of Mathematics: 1–16, doi:10.1080/17498430.2015.1116053
6. ^ Rusnock, P.; Kerr-Lawson, A. (2005), "Bolzano and uniform continuity", Historia Mathematica, 32 (3): 303–311, doi:10.1016/j.hm.2004.11.003
7. ^ Lang, Serge (1997), Undergraduate analysis, Undergraduate Texts in Mathematics (2nd ed.), Berlin, New York: Springer-Verlag, ISBN 978-0-387-94841-6, section II.4
8. ^ Introduction to Real Analysis, updated April 2010, William F. Trench, Theorem 3.5.2, p. 172
9. ^ Introduction to Real Analysis, updated April 2010, William F. Trench, 3.5 "A More Advanced Look at the Existence of the Proper Riemann Integral", pp. 171–177
10. ^ "Elementary Calculus". wisc.edu.
11. ^ Gaal, Steven A. (2009), Point set topology, New York: Dover Publications, ISBN 978-0-486-47222-5, section IV.10
12. ^ Searcóid, Mícheál Ó (2006), Metric spaces, Springer undergraduate mathematics series, Berlin, New York: Springer-Verlag, ISBN 978-1-84628-369-7, section 9.4
13. ^ Goubault-Larrecq, Jean (2013). Non-Hausdorff Topology and Domain Theory: Selected Topics in Point-Set Topology. Cambridge University Press. ISBN 1107034132.
14. ^ Gierz, G.; Hofmann, K. H.; Keimel, K.; Lawson, J. D.; Mislove, M. W.; Scott, D. S. (2003). Continuous Lattices and Domains. Encyclopedia of Mathematics and its Applications. 93. Cambridge University Press. ISBN 0521803381.
15. ^ Flagg, R. C. (1997). "Quantales and continuity spaces". Algebra Universalis. CiteSeerX: 10.1.1.48.851.
16. ^ Kopperman, R. (1988). "All topologies come from generalized metrics". American Mathematical Monthly. 95 (2): 89–97. doi:10.2307/2323060.
17. ^ Flagg, B.; Kopperman, R. (1997). "Continuity spaces: Reconciling domains and metric spaces". Theoretical Computer Science. 177 (1): 111–138. doi:10.1016/S0304-3975(97)00236-3.
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2016-08-29 20:26:58
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https://math.stackexchange.com/questions/2561203/the-largest-sum-of-products-of-pairs-of-numbers-terms
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# The largest sum of products of pairs of number's terms
Someone, please, make that title more readable. My lack of math English does not let me.
Given $A = a_1 + ... + a_k;\ A, a_i > 0$ what is $\max\left(\sum_{i<j}a_i a_j\right)$?
An obvious upper bound is $\dfrac{A^2}{2}$, because $\sum_{i<j}a_i a_j = \dfrac{A^2 - \sum_i a_i^2}{2}$.
Something tells me the maximum is when $\forall i\ a_i = \dfrac{A}{k}$, but I cannot prove it properly.
Hint: formulate this as a constraint maximization problem and apply Lagrange Multipliers. The $a_i$ are the input and they’re subject to the constraint that $\sum a_i=A$.
You already noted that $$2\sum_{i<j}a_ia_j= A^2 - \sum_{i}^{k}a_i^2$$ hence equivalently we are trying to minimize $\sum a_i^2$. Now suppose we have a minimal solution where there exists $a_i,a_j$ such that $a_i\neq a_j$. Without loss of generality, say $a_1>a_2$.
Let $a=\frac{a_1+a_2}{2}$. Compare $(a_1,a_2,a_3,\dots, a_k)$ and $(a,a, a_3, \dots, a_k)$. Since $a_3^2+a_4^2 +\dots + a_k^2$ part is equal: \begin{aligned} a_1^2+a_2^2 > a^2 + a^2&\iff a_1^2 + a_2^2 > \frac{(a_1+a_2)^2}{2} \\&\iff a_1^2 + a_2^2 > 2a_1a_2\\ &\iff (a_1-a_2)^2 > 0 \end{aligned} which is true and contradicts that we started with a minimal solution.
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2020-01-24 08:54:01
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http://ncatlab.org/nlab/show/model+structure+on+algebras+over+a+monad
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nLab model structure on algebras over a monad
model category
Model structures
for ∞-groupoids
for $\left(\infty ,1\right)$-sheaves / $\infty$-stacks
Higher algebra
higher algebra
universal algebra
Contents
Idea
For $C$ a monoidal model category and $T:C\to C$ a monad on $C$, there is under mild conditions a natural model category structure on the category of algebras over a monad over $T$.
Definition
Let $C$ be a cofibrantly generated model category and $T:C\to C$ a monad on $C$.
Then under mild conditions there exists the transferred model structure on the category of algebras over a monad, transferred along the free functor/forgetful functor adjunction
$\left(F⊣U\right):\mathrm{Alg}T\stackrel{\stackrel{F}{←}}{\underset{U}{\to }}C\phantom{\rule{thinmathspace}{0ex}}.$(F \dashv U) : Alg T \stackrel{\overset{F}{\leftarrow}}{\underset{U}{\to}} C \,.
See (SchwedeShipley, lemma 2.3).
References
Revised on November 18, 2010 15:56:02 by Urs Schreiber (131.211.232.149)
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2013-12-10 19:49:47
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https://api.projectchrono.org/vehicle_terrain.html
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Terrain models
A terrain object in Chrono::Vehicle must provide methods to:
• return the terrain height at the point directly below a specified location
• return the terrain normal at the point directly below a specified location
• return the terrain coefficient of friction at the point directly below a specified location where the given location is assumed to be expressed in the current world frame.
See the definition of the base class ChTerrain.
Note however that these quantities are relevant only for the interaction with the so-called semi-empirical tire models. As such, they are not used for the case of deformable terrain (SCM, granular, or FEA-based) which can only work in conjunction with rigid or FEA tire models and with tracked vehicles (as they rely on the underlying Chrono collision and contact system).
Furthermore, the coefficient of friction value may be used by certain tire models to modify the tire characteristics, but it will have no effect on the interaction of the terrain with other objects (including tire models that do not explicitly use it).
The ChTerrain base class also defines a functor object ChTerrain::FrictionFunctor which provides an interface for specification of a position-dependent coefficient of friction. The user must implement a custom class derived from this base class and implement the virtual method operator() to return the coefficient of friction at the point directly below the given $$(x,y,z)$$ location (assumed to be expressed in the current world frame).
Flat terrain
FlatTerrain is a model of a horizontal plane, with infinite extent, located at a user-specified height. The method FlatTerrain::GetCoefficientFriction returns the constant coefficient of friction specified at construction or, if a FrictionFunctor object was registered, its return value.
Since the flat terrain model does not carry any collision and contact information, it can only be used with the semi-empirical tire models.
Rigid terrain
RigidTerrain is a model of a rigid terrain with arbitrary geometry. A rigid terrain is specified as a collection of patches, each of which can be one of the following:
• a rectangular box, possibly rotated; the "driving" surface is the top face of the box (in the world's vertical direction)
• a triangular mesh read from a user-specified Wavefront OBJ file
• a triangular mesh generated programatically from a user-specified gray-scale BMP image
The rigid terrain model can be used with any of the Chrono::Vehicle tire models, as well as with tracked vehicles.
A box patch is specified by the center of the top (driving) surface, the normal to the top surface, and the patch dimensions (length, width, and optionally thickness). Optionally, the box patch can be created from multiple adjacent tiles, each of which being a Chrono box contact shape; this is recommended for a box patch with large horizontal extent as a single collision shape of that dimension may lead to errors in the collision detection algorithm.
An example of a mesh rigid terrain patch is shown in the image below. It is assumed that the mesh is provided with respect to an ISO reference frame and that it has no "overhangs" (in other words, a vertical ray intersects the mesh in at most one point). Optionally, the user can specify a "thickness" for the terrain mesh as the radius of a sweeping sphere. Specifying a small positive value for this radius can significantly improve the robustness of the collision detection algorithm.
A height-map patch is specified through a gray-scale BMP image (like the one shown below), a horizontal extent of the patch (length and width), and a height range (minimum and maximum heights). A triangular mesh is programatically generated, by creating a mesh vertex for each pixel in the input BMP image, stretching the mesh in the horizontal plane to match the given extents and in the vertical direction such that the minimum height corresponds to a perfectly black pixel color and the maximum height corresponds to a perfectly white pixel.
Height and normal calculation. The implementation of RigidTerrain::GetHeight and RigidTerrain::GetNormal rely on a relatively expensive ray-casting operation: a vertical ray is cast from above in all constituent patches with the height and normal at the intersection point reported back. For a box patch, the ray-casting uses a custom analytical implementation which finds intersections of the ray with the top face of the box domain; for mesh-based patches, the ray-casting is deferred to the underlying collision system. If no patch is intersected, these functions return $$0$$ and the world's vertical direction, respectively.
Location-dependent coefficient of friction. The rigid terrain model supports the definition of a FrictionFunctor object. If no such functor is provided, RigidTerrain::GetCoefficientFriction uses the ray-casting approach to identify the correct patch and the (constant) coefficient of friction for that patch is returned. If a functor is provided, RigidTerrain::GetCoefficientFriction simply returns its value. However, processing of contacts with the terrain (e.g., when using rigid tires or a tracked vehicle) is relatively expensive: at each invocation of the collision detection algorithm (i.e., once per simulation step) the list of all contacts in the Chrono system is traversed to intercept all contacts that involve a rigid terrain patch collision model; for these contacts, the composite material properties are modified to account for the terrain coefficient of friction at the point of contact.
A rigid terrain can be constructed programatically, defining one patch at a time, or else specified in a JSON file like the following one:
{
"Name": "Rigid plane",
"Type": "Terrain",
"Template": "RigidTerrain",
"Patches": [
// Patch 1: box
{
"Location": [ -15, 0, 0 ],
"Orientation": [ 1, 0, 0, 0 ],
"Geometry": {
"Dimensions": [ 60, 20, 2 ]
},
"Contact Material": {
"Coefficient of Friction": 0.9,
"Coefficient of Restitution": 0.01,
"Properties": {
"Young Modulus": 2e7,
"Poisson Ratio": 0.3
},
"Coefficients": {
"Normal Stiffness": 2e5,
"Normal Damping": 40.0,
"Tangential Stiffness": 2e5,
"Tangential Damping": 20.0
}
},
"Visualization": {
"Color": [ 1.0, 0.5, 0.5 ],
"Texture File": "terrain/textures/tile4.jpg",
"Texture Scaling": [ 60, 20 ]
}
},
// Patch 2: box
{
"Location": [ 20, 0, 0.1 ],
"Orientation": [ 1, 0, 0, 0 ],
"Geometry": {
"Dimensions": [ 20, 30, 2 ]
},
"Contact Material": {
"Coefficient of Friction": 0.9,
"Coefficient of Restitution": 0.01,
"Properties": {
"Young Modulus": 2e7,
"Poisson Ratio": 0.3
},
"Coefficients": {
"Normal Stiffness": 2e5,
"Normal Damping": 40.0,
"Tangential Stiffness": 2e5,
"Tangential Damping": 20.0
}
},
"Visualization": {
"Color": [ 1.0, 0.5, 0.5 ]
}
},
// Patch 3: height-map
{
"Location": [ 0, 42, 0 ],
"Orientation": [ 1, 0, 0, 0 ],
"Geometry": {
"Height Map Filename": "terrain/height_maps/bump64.bmp",
"Size": [ 64, 64 ],
"Height Range": [ 0, 3 ]
},
"Contact Material": {
"Coefficient of Friction": 0.9,
"Coefficient of Restitution": 0.01,
"Properties": {
"Young Modulus": 2e7,
"Poisson Ratio": 0.3
},
"Coefficients": {
"Normal Stiffness": 2e5,
"Normal Damping": 40.0,
"Tangential Stiffness": 2e5,
"Tangential Damping": 20.0
}
},
"Visualization": {
"Color": [ 1.0, 1.0, 1.0 ],
"Texture File": "terrain/textures/grass.jpg",
"Texture Scaling": [ 64, 64 ]
}
},
// Patch 4: Mesh
{
"Location": [ 0, -42, 0 ],
"Orientation": [ 1, 0, 0, 0 ],
"Geometry": {
"Mesh Filename": "terrain/meshes/bump.obj"
},
"Contact Material": {
"Coefficient of Friction": 0.9,
"Coefficient of Restitution": 0.01,
"Properties": {
"Young Modulus": 2e7,
"Poisson Ratio": 0.3
},
"Coefficients": {
"Normal Stiffness": 2e5,
"Normal Damping": 40.0,
"Tangential Stiffness": 2e5,
"Tangential Damping": 20.0
}
},
"Visualization": {
"Color": [ 0.5, 0.5, 0.8 ],
"Texture File": "terrain/textures/dirt.jpg",
"Texture Scaling": [ 200, 200 ]
}
}
]
}
CRG terrain
CRGTerrain is a procedural terrain model constructed from an OpenCRG road specification. To use this terrain model, the user must install the OpenCRG SDK and enable its use during CMake configuration (see the Chrono::Vehicle installation instruction).
The CRG terrain creates a road profile (a 3D path with an associated width) from a specification file such as the one listed below and implements the functions CRGTerrain::GetHeight and CRGTerrain::GetNormal to use this specification. Note that a crg specification file can be either ASCII or binary.
* Emacs major mode to be selected automagically: -*-CASCaDE-*-
* $Id: handmade_curved_minimalist.crg 21 2009-06-23 15:03:13Z jorauh$
$CT CRG file example for road surface description (width: 3m, length: 22m) with curved reference line and grid of (0.25m...1.0m) x 1.0m. A minimalist file can have an empty$CT, a minimalist $ROAD_CRG block, and no comments marked by asterisk "*" in column 1 or marked by "!" in other columns. So this file is completely equivalent to its commented version in handmade_curved.crg . Copyright 2005-2009 OpenCRG - Daimler AG - Jochen Rauh Licensed under the Apache License, Version 2.0 (the "License"); you may not use this file except in compliance with the License. You may obtain a copy of the License at http://www.apache.org/licenses/LICENSE-2.0 Unless required by applicable law or agreed to in writing, software distributed under the License is distributed on an "AS IS" BASIS, WITHOUT WARRANTIES OR CONDITIONS OF ANY KIND, either express or implied. See the License for the specific language governing permissions and limitations under the License. More Information on OpenCRG open file formats and tools can be found at http://www.opencrg.org$
$ROAD_CRG REFERENCE_LINE_INCREMENT = 1.0$
$KD_Definition #:LRFI D:reference line phi,rad D:long section at v = -1.500,m D:long section at v = -1.250,m D:long section at v = -1.000,m D:long section at v = 0.000,m D:long section at v = 1.000,m D:long section at v = 1.250,m D:long section at v = 1.500,m$
**unused** 0.0000000 0.0000000 0.0000000 0.0000000 0.0000000 0.0000000 0.0000000
0.0000000 0.0000000 0.0000000 0.0000000 0.0111111 0.0000000 0.0000000 0.0000000
0.0110000 0.0000000 0.0000000 0.0111111 0.0111111 0.0111111 0.0000000 0.0000000
0.0220000 0.0000000 0.0111111 0.0111111 0.0111111 0.0111111 0.0111111 0.0000000
0.0330000 0.0000000 0.0000000 0.0111111 0.0111111 0.0111111 0.0000000 0.0000000
0.0440000 0.0111111 0.0000000 0.0000000 0.0111111 0.0000000 0.0000000 0.0111111
0.0550000 0.0111111 0.0111111 0.0000000 0.0000000 0.0000000 0.0111111 0.0222222
0.0660000 0.0111111 0.0111111 0.0111111 0.0000000 0.0111111 0.0222222 0.0333333
0.0770000 0.0111111 0.0111111 0.0000000 0.0000000 0.0000000 0.0111111 0.0222222
0.0880000 0.0111111 0.0000000 0.0000000 0.0111111 0.0000000 0.0000000 0.0111111
0.0990000 0.0000000 0.0000000 0.0111111 0.0222222 0.0111111 0.0000000 0.0000000
0.1100000 0.0000000 0.0111111 0.0222222 0.0222222 0.0222222 0.0111111 0.0000000
0.1100000 0.0000000 0.0000000 0.0111111 0.0222222 0.0111111 0.0000000 0.0000000
0.0990000 0.0111111 0.0000000 0.0000000 0.0111111 0.0000000 0.0000000-0.0111111
0.0880000 0.0111111 0.0111111 0.0000000 0.0000000 0.0000000-0.0111111-0.0222222
0.0770000 0.0111111 0.0111111 0.0111111 0.0000000-0.0111111-0.0222222-0.0333333
0.0660000 0.0111111 0.0111111 0.0000000 0.0000000 0.0000000-0.0111111-0.0222222
0.0550000 0.0111111 0.0000000 0.0000000 0.0111111 0.0000000 0.0000000-0.0111111
0.0440000 0.0000000 0.0000000 0.0111111 0.0222222 0.0111111 0.0000000 0.0000000
0.0330000 0.0000000 0.0111111 0.0222222 0.0333333 0.0222222 0.0111111 0.0000000
0.0220000 0.0000000 0.0000000 0.0111111 0.0222222 0.0111111 0.0000000 0.0000000
0.0110000 0.0000000 0.0000000 0.0000000 0.0111111 0.0000000 0.0000000 0.0000000
0.0000000 0.0000000 0.0000000 0.0000000 0.0000000 0.0000000 0.0000000 0.0000000
The CRG terrain can be visualized as a triangular mesh (representing the road "ribbon") or else as a set of 3D Bezier curves (representing the center line and the road sides). Other features of CRGTerrain include:
• ability to export the road mesh (as a triangle mesh)
• ability to export the center line (as a Bezier curve)
• methods for reporting the road length and width
The images below illustrate the run-time visualization of a CRG road using a triangular mesh or the road boundary curves.
Since the CRG terrain model currently does not carry any collision and contact information, it can only be used with the semi-empirical tire models.
Deformable SCM (Soil Contact Model)
In the recently redesigned SCMDeformableTerrain, the terrain is represented by an implicit regular Cartesian grid whose deformation is achieved via vertical deflection of its nodes. This soil model draws on the general-purpose collision engine in Chrono and its lightweight formulation allows computing vehicle-terrain contact forces in close to real-time. To address memory and computational efficiency concerns, the grid is never created explicitly. Instead, only nodes that have been deformed are maintained in a hash map. Furthermore, ray-casting in the collision system (the most costly operation in the SCM calculation) is multi-threaded. To allow efficient visualization of the deformed terrain, the Chrono SCM subsystem provides methods for incrementally updating a visualization mesh and, when using an external visualization system, reporting the subset of nodes deformed over the last time step.
Shown below, a tire makes ruts in deformable soil, illustrating the mesh structure of the Chrono version of the SCM.
SCM is based on a semi-empirical model with few parameters, which makes it easy to calibrate based on experimental results. It can be considered a generalization of the Bekker-Wong model to the case of wheels (or track shoes) with arbitrary three-dimensional shapes. The Bekker formula for a wheel that moves on a deformable soil provides a relationship between pressure and vertical deformation of the soil as:
$\sigma = \left( \frac{k_c}{b} + k_{\phi} \right) y^n$
where $$\sigma$$ is the contact patch pressure, $$y$$ is wheel sinkage, $$k_c$$ is an empirical coefficient representing the cohesive effect of the soil, $$k_{\phi}$$ is an empirical coefficient representing the stiffness of the soil, and $$n$$ is an exponent expressing the hardening effect, which increases with the compaction of the soil in a non-linear fashion. Finally, $$b$$ is the length of the shorter side of the rectangular contact footprint (since the original Bekker theory assumes a cylindrical tire rolling over flat terrain).
For a generic contact footprint, the length $$b$$ cannot be interpreted as in the original Bekker model; instead, we estimate this length by first obtaining all connected contact patches (using a flooding algorithm) and the using the approximation
$b \approx \frac{2 A}{L}$
where $$A$$ is the area of such a contact patch and $$L$$ its perimeter.
Some other features of the Chrono SCM implementation are:
• the initial undeformed mesh can be created as
• a regular tiled mesh (filling a flat rectangle)
• from a height-map (provided as a gray-scale BMP image)
• programatically
• support for arbitrary orientation of the terrain reference plane; by default, the terrain is defined as the $$(x,y)$$ plane of a $$z$$-up ISO frame
• support for a moving-patch approach wherein ray-casting (the most costly operation) is confined to a specified domain – either a rectangular patch moving relative to the vehicle or the projection of a bounding box
• support for specifying location-dependent soil parameters; this can be achieved by providing a custom callback class which implements a method that returns all soil parameters at a given $$(x,y)$$ point specified in the terrain's reference plane. See SCMDeformableTerrain::SoilParametersCallback
Since the interaction with this terrain type is done through the underlying Chrono contact system, it can be used in conjunction with rigid or FEA tire models and with tracked vehicles.
Granular terrain
GranularTerrain implements a rectangular patch of granular material and leverages Chrono's extensive support for so-called Discrete Element Method (DEM) simulations. Currently, this terrain model is limited to monodisperse spherical granular material.
Because simulation of large-scale granular dynamics can be computationally very intensive, the GranularTerrain object in Chrono::Vehicle provides support for a "moving patch" approach, wherein the simulation can be confined to a bin of granular material that is continuously relocated based on the position of a specified body (typically the vehicle's chassis). Currently, the moving patch can only be relocated in the $$x$$ (forward) direction.
An illustration of a vehicle acceleration test on GranularTerrain using the moving patch feature is shown below. This simulation uses more than 700,000 particles and the Chrono::Multicore module for multi-core parallel simulation.
Other features of GranularTerrain include:
• generation of initial particle locations in layers, with particle positions in the horizontal plane uniformly distributed and guaranteed to be no closer than twice the particle radius
• inclusion of particles fixed to the bounding bin (to inhibit sliding of the granular material bed as a whole); due to current limitations, this feature should not be used in conjunction with the moving patch option
• analytical definition of the bin boundaries and a custom collision detection mechanism
• reporting of the terrain height (defined as the largest $$z$$ value over all particle locations)
Since the interaction with this terrain type is done through the underlying Chrono contact system, it can be used in conjunction with rigid or FEA tire models and with tracked vehicles.
Deformable FEA (ANCF solid elements)
FEADeformableTerrain provides a deformable terrain model based on specialized FEA brick elements of type ChElementHexaANCF_3813_9.
This terrain model permits:
• discretization of a box domain into a user-prescribed number of elements
• assignment of material properties (density, modulus of elasticity, Poisson ratio, yield stress, hardening slope, dilatancy angle, and friction angle)
• addition of Chrono FEA mesh visualization assets
Since the interaction with this terrain type is done through the underlying Chrono contact system, it can be used in conjunction with rigid or FEA tire models and with tracked vehicles.
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2022-12-01 12:53:30
|
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https://tex.stackexchange.com/questions/253226/help-with-tikz-to-draw-graphs-with-squiggly-edges
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# Help with Tikz to draw graphs with squiggly edges
In my work there are case analysis of graphs of the type:
I am new to the wonderful world of TikZ and I have tried the following:
\begin{tikzpicture}[every node/.style={circle, draw, scale=.6}, scale=.5, rotate = 180, xscale = -1]
\node [draw,shape=circle](one) at (0,0) {};
\node[draw,shape=circle] (two) at (-2,2) {};
\node [draw,shape=circle](three) at (0,4) {};
\node [draw,shape=circle](four) at (2,2) {};
\node[draw,shape=circle] (five) at (0,1) {};
\node [draw, shape=circle] (six) at (0,2) {};
\node [draw, shape=circle] (seven) at (0,3) {};
\draw (one) -- (two) -- (three) -- (four) -- (one);
\draw (four) -- (six);
\draw (six)
to [out=45, in =315] (seven);
\draw (seven) to [out=225, in=135] (five) -- (two);
\draw (three) -- (seven);
\end{tikzpicture}
Result:
My questions are:
1. How to get the squiggly path (dashed/dotted in some cases)?
2. Why is my figure upside down?
• From memory \draw[decoration=snake,decorate] (one) -- (five) -- (six); Jul 1, 2015 at 20:38
• \usetikzlibrary{decorations} \usetikzlibrary{snakes} with these libraries? still the result is not as neat :( But thanks for the direction. @Manuel Jul 1, 2015 at 20:48
• @Manuel Want to provide the answer? Jul 1, 2015 at 20:48
• @GonzaloMedina I answered, and again, the system thinks I'm a robot. Jul 1, 2015 at 20:55
\documentclass{scrartcl}
\usepackage{tikz}
\usetikzlibrary{snakes}
\begin{document}
\begin{tikzpicture}[every node/.style={circle, draw, scale=.6}]
\node [draw,shape=circle](one) at (0,0) {};
\node[draw,shape=circle] (two) at (-2,2) {};
\node [draw,shape=circle](three) at (0,4) {};
\node [draw,shape=circle](four) at (2,2) {};
\node[draw,shape=circle] (five) at (0,1) {};
\node [draw, shape=circle] (six) at (0,2) {};
\node [draw, shape=circle] (seven) at (0,3) {};
\draw (one) -- (two) -- (three) -- (four) -- (one);
\draw (four) -- (six);
\draw (six) to [bend right=45] (seven);
\draw (seven) to [bend right=45] (five);
\draw (five) -- (two);
\draw (three) -- (seven);
\tikzset{decoration={snake,amplitude=.4mm,segment length=2mm,
post length=0mm,pre length=0mm}}
\draw[decorate] (one) -- (five);
\draw[decorate] (five) -- (six);
\draw[decorate] (six) -- (seven);
\end{tikzpicture}
\end{document}
• I am getting three errors: you need to load a decoration library \draw[decorate] Jul 1, 2015 at 21:02
• I am getting no errors. May be you have an outdated installation? Jul 1, 2015 at 21:03
• I had \usetikzlibrary{snakes} in the input file, when i moved it to the main preamble it worked like a charm. Thanks again @Manuel. Jul 1, 2015 at 21:08
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2022-05-28 07:33:34
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https://en.wikipedia.org/wiki/Weil_pairing
|
# Weil pairing
In mathematics, the Weil pairing is a pairing (bilinear form, though with multiplicative notation) on the points of order dividing n of an elliptic curve E, taking values in nth roots of unity. More generally there is a similar Weil pairing between points of order n of an abelian variety and its dual. It was introduced by André Weil (1940) for Jacobians of curves, who gave an abstract algebraic definition; the corresponding results for elliptic functions were known, and can be expressed simply by use of the Weierstrass sigma function.
## Formulation
Choose an elliptic curve E defined over a field K, and an integer n > 0 (we require n to be prime to char(K) if char(K) > 0) such that K contains a primitive nth root of unity. Then the n-torsion on $E(\overline{K})$ is known to be a Cartesian product of two cyclic groups of order n. The Weil pairing produces an n-th root of unity
$w(P,Q) \in \mu_n$
by means of Kummer theory, for any two points $P,Q \in E(K)[n]$, where $E(K)[n]=\{T \in E(K) \mid n \cdot T = O \}$ and $\mu_n = \{x\in K \mid x^n =1 \}$.
A down-to-earth construction of the Weil pairing is as follows. Choose a function F in the function field of E over the algebraic closure of K with divisor
$\mathrm{div}(F)= \sum_{0 \leq k < n}(P+k\cdot Q) - \sum_{0 \leq k < n} (k\cdot Q).$
So F has a simple zero at each point P + kQ, and a simple pole at each point kQ if these points are all distinct. Then F is well-defined up to multiplication by a constant. If G is the translation of F by Q, then by construction G has the same divisor, so the function G/F is constant.
Therefore if we define
$w(P,Q):=\frac{G}{F}$
we shall have an n-th root of unity (as translating n times must give 1) other than 1. With this definition it can be shown that w is alternating and bilinear,[1] giving rise to a non-degenerate pairing on the n-torsion.
The Weil pairing does not extend to a pairing on all the torsion points (the direct limit of n-torsion points) because the pairings for different n are not the same. However they do fit together to give a pairing T(E) × T(E) → T(μ) on the Tate module T(E) of the elliptic curve E (the inverse limit of the ℓn-torsion points) to the Tate module T(μ) of the multiplicative group (the inverse limit of ℓn roots of unity).
## Generalisation to abelian varieties
For abelian varieties over an algebraically closed field K, the Weil pairing is a nondegenerate pairing
$A[n] \times A^\vee[n] \longrightarrow \mu_n$
for all n prime to the characteristic of k.[2] Here $A^\vee$ denotes the dual abelian variety of A. This is the so-called Weil pairing for higher dimensions. If A is equipped with a polarisation
$\lambda: A \longrightarrow A^\vee$,
then composition gives a (possibly degenerate) pairing
$A[n] \times A[n] \longrightarrow \mu_n.$
If C is a projective, nonsingular curve of genus ≥ 0 over k, and J its Jacobian, then the theta-divisor of J induces a principal polarisation of J, which in this particular case happens to be an isomorphism (see autoduality of Jacobians). Hence, composing the Weil pairing for J with the polarisation gives a nondegenerate pairing
$J[n]\times J[n] \longrightarrow \mu_n$
for all n prime to the characteristic of k.
As in the case of elliptic curves, explicit formulae for this pairing can be given in terms of divisors of C.
## Applications
The pairing is used in number theory and algebraic geometry, and has also been applied in elliptic curve cryptography and identity based encryption.
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2015-11-30 10:07:12
|
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|
https://lama.u-pem.fr/evenements/seminaire/seminaire_de_geometrie/rigidity_results_and_topology_at_infinity_of_translating
|
# Rigidity results and topology at infinity of translating solitons of the mean curvature flow
Orateur: IMPERA Debora Localisation: Université de Milan-Bicocca, Italie Type: Séminaire de géométrie Site: Hors LAMA , IHP Salle: 421 Date de début: 11/01/2016 - 14:00 Date de fin: 11/01/2016 - 14:00
In this talk I will discuss some rigidity results and obstructions on the topology at infinity of translating solitons of the mean curvature flow in the Euclidean space. These results were recently obtained in collaboration with M. Rimoldi and our approach relies on the theory of $f$-minimal hypersurfaces.
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2021-09-17 21:51:36
|
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|
https://www.physicsforums.com/threads/curl-question.455060/
|
# Curl question
I am trying to prove two things equal that involves a bunch of dot products, cross products and curls. I can't remember the exact problem, but this demonstrates my question.
My question is the left side $$\delta$$/$$\delta$$x*(UxVx + UyVy + UzVz)
The right side is $$\delta$$/$$\delta$$x*(VxUx + VyUy + VzUz +
VyUz)
does the VyUz term cancel since the partial deriv wrt x is 0?
Also do the the other terms become 1 since for example in the partial deriv wrt x of UxVx
Just looking at the i component The left side yields partial deriv wrt x of (UxVx + UyVy + UzVz)i
The right side however yields terms in the i component that are partials wrt to x, y and z.
Last edited:
Assuming u and v to be two vectors denoted by $u = (u_x, u_y, u_z), v = (v_x, v_y, v_z)$ and assuming that the vector components themselves are functions of x,y,z, i.e. that $u_i = f_i(x,y,z), v_i=g_i(x,y,z)$ for i = x,y,z, we cannot say for certain that $\partial / \partial x$ of the cross term $v_y u_z$ is zero unless the functions $g_y,f_z$ do not depend on x (which in general cannot be said.)
Edit: As a simple example, take $u_z = x , v_y = x$, then what does the partial derivative w.r.t. x of their product equal?
Assuming u and v to be two vectors denoted by $u = (u_x, u_y, u_z), v = (v_x, v_y, v_z)$ and assuming that the vector components themselves are functions of x,y,z, i.e. that $u_i = f_i(x,y,z), v_i=g_i(x,y,z)$ for i = x,y,z, we cannot say for certain that $\partial / \partial x$ of the cross term $v_y u_z$ is zero unless the functions $g_y,f_z$ do not depend on x (which in general cannot be said.)
Edit: As a simple example, take $u_z = x , v_y = x$, then what does the partial derivative w.r.t. x of their product equal?
I see your rationale but in trying to make the left side equal the right side how do I cancel out terms such as VzUy d/dx that do not appear on the left side? (This term I am referring to results from the third Ux(gradxV) term.
For example the entire left side i component has three terms, but the entire right side i component has 14 terms 4 of which are negative. So even if those cancel out other terms I am still left with 6 terms whereas I think I need 3 for it to equal the left side.
There's some ambiguity in your problem statement that may serve to clarify the problem. For example, you have $(v \times \nabla)u$, however both those terms are vectors, so is a dot product implied between the two?
As a side note: I will say that if the equality is in fact true, that terms should cancel out relatively easily.
There's some ambiguity in your problem statement that may serve to clarify the problem. For example, you have $(v \times \nabla)u$, however both those terms are vectors, so is a dot product implied between the two?
Ha! your correct it was a typo my mistake. here is the correct problem:
Debasis Kundu
My quickest suggestion without having to write out the problem myself would be to expand the left hand side. As an example, carry out the derivation:
$$\partial / \partial x (u_x v_x + u_y v_y + u_z v_z)$$
Note: You'll end up with 6 terms.
Here is just the i components worked out:
left side:
d/dx * (UxVx +UyVy + UzVz)i
right side:
each line will represent one part of right side of given problem
(VxUx * d/dx + VxVy * d/dy + VxUz * d/dz)i
(VxUx *d/dx + VyUx * d/dy + VzUx * d/dz)i
(VxUy * d/dx - VxUy * d/dy + VzUz *d/dx - VxUz *d/dz)i
(VyUz * d/dx - VyUx *d/dy + VzUz *d/dx - VzUx *d/dz)i
Some of these terms cancel and you are left with:
(VxUx * d/dx + VxUx *d/dx + VxUy * d/dx + VzUz *d/dx + VyUz * d/dx + VzUz *d/dx)i
which does not equal the left
My quickest suggestion without having to write out the problem myself would be to expand the left hand side. As an example, carry out the derivation:
$$\partial / \partial x (u_x v_x + u_y v_y + u_z v_z)$$
Note: You'll end up with 6 terms.
I'm not quite sure how you would do that derivation, could you just give me a quick sample?
There's a problem with your RHS. You've carried out the operation incorrectly.
Let's consider the i'th term of $(u \cdot \nabla)v$, it would be:
$$(u_x \partial / \partial x + u_y \partial / \partial y + u_z \partial / \partial z)v_i$$
Now, notice that $v_i = g_i(x,y,z)$, so you cannot pull it outside of those partial derivatives like you've done... those partial derivatives are acting on $v_i$.
In essence, the above statement expanded would be:
$$u_x \frac{\partial v_i}{\partial x} + u_y \frac{\partial v_i}{\partial y} + u_z \frac{\partial v_i}{\partial z}$$
As for the derivation of the above (in my previous post), you should be able to use the chain rule and linearity of the partial derivative to solve it.
Last edited:
There's a problem with your RHS. You've carried out the operation incorrectly.
Let's consider the i'th term of $(u \cdot \nabla)v$, it would be:
$$(u_x \partial / \partial x + u_y \partial / \partial y + u_z \partial / \partial z)v_i$$
Now, notice that $v_i = g_i(x,y,z)$, so you cannot pull it outside of those partial derivatives like you've done... those partial derivatives are acting on $v_i$.
In essence, the above statement expanded would be:
$$u_x \frac{\partial v_i}{\partial x} + u_y \frac{\partial v_i}{\partial y} + u_z \frac{\partial v_i}{\partial z}$$
As for the derivation of the above, you should be able to use the chain rule and linearity of the partial derivative to solve it.
How come it is ok to leave the U outside the partial? I apologize for my ignorance but I am new to this stuff.
For all practical purposes, we treat the gradient operator $\nabla$ as if it were a vector with the quantities $(\partial / \partial x,\partial / \partial y,\partial / \partial z)$. This makes for some very appealing shorthand notation for taking curls and divergences, but it can also make it appear as if the vector can just multiply through with other functions.
This is *not* the case! You must be very careful using this notation, and remember that the partial derivatives act on anything that is in front of them.
Now for your question, assuming the vector u to be given as $u = (u_x,u_y,u_z)$ and assuming the gradient *operator* to be given as $(\partial / \partial x,\partial / \partial y,\partial / \partial z)$, what is $u \cdot \nabla$?
Note: The expression $u \cdot \nabla$ is an example of one of the appealing shorthand forms that we can make if we assume the gradient operator to have that vector like form. Otherwise you would always have to write out the full expression.
so u.grad = ux d/dx + uy d/dy + uz d/dz
so if u equaled 2xi + 4xyj + 5xyzk
u.grad would equal 2 + 4x + 5xy?
and if it were (u.grad)V where V = Vx + Vy + Vk
it would equal 2Vx + 4xVy + 5xyVk?
u.grad would equal 2 + 4x + 5xy?
No, those derivative operators only act on what is in front of them. Would you say the u_x, etc. terms are in front of the derivative operators?
As a second example, what is $\nabla \cdot u$? What popular operation is this shorthand notation referring to?
No, those derivative operators only act on what is in front of them. Would you say the u_x, etc. terms are in front of the derivative operators?
As a second example, what is $\nabla \cdot u$? What popular operation is this shorthand notation referring to?
I am getting more confused. u.grad = ux d/dx + uy d/dy + uz d/dz In this example the U terms seem to be in front of the derivation operators.
I believe you are referring to the dot product? in your example I belive u.grad = ux d/dx + uy d/dy + uz d/dz following the definition where as a simple example a.b = axbx + ayby + azbz
Thanks again for all the help I really appreciate it.
1) What is $x \frac{d f(x)}{dx}$ if $f(x) = x$?
2) What is $x \frac{d}{dx} f(x)$ if $f(x) = x$?
Now, let's pretend that we don't have any f(x) yet, then we could write:
$$x \frac{d}{dx}$$.
However, this is an incomplete object because in order for us to write down an answer, we would need to take that object and act it on some function, f(x), like we did in part 1 and 2.
This is what we have done with this $u \cdot \nabla$ thing. Technically speaking, this is somewhat of an incomplete expression. In order to get something from it, you would need to put a function in front of it.
Notice from part (1) and (2) that when we removed the f(x) from in front of the derivative operator that we ended up with an expression similar to the form $u_x \partial / \partial x$.
1) What is $x \frac{d f(x)}{dx}$ if $f(x) = x$?
2) What is $x \frac{d}{dx} f(x)$ if $f(x) = x$?
Now, let's pretend that we don't have any f(x) yet, then we could write:
$$x \frac{d}{dx}$$.
However, this is an incomplete object because in order for us to write down an answer, we would need to take that object and act it on some function, f(x), like we did in part 1 and 2.
This is what we have done with this $u \cdot \nabla$ thing. Technically speaking, this is somewhat of an incomplete expression. In order to get something from it, you would need to put a function in front of it.
Notice from part (1) and (2) that when we removed the f(x) from in front of the derivative operator that we ended up with an expression similar to the form $u_x \partial / \partial x$.
So plugging in x for f(x) in the first function will be result in an answer of x. I not totally sure if you are multiplying the second thing by f(x) or not but if you are it would be x^2.
So by in front of the operation you mean to the right as we read so d(x^2)/dx would be 2x?
It seems I'm still losing you a bit! The answer to both #1 and #2 is x! They are the same expression written slightly differently.
It's very important that you stop seeing the derivative as something you can multiply. It does not multiply anything, ever. It **acts** on whatever is in front of it, (and yes you are correct, I mean to the right of when I say in front... sorry for that).
So what is #2?
It seems I'm still losing you a bit! The answer to both #1 and #2 is x! They are the same expression written slightly differently.
It's very important that you stop seeing the derivative as something you can multiply. It does not multiply anything, ever. It **acts** on whatever is in front of it, (and yes you are correct, I mean to the right of when I say in front... sorry for that).
So what is #2?
Ok I see they are both x. I think how the forum was displaying it, I was a little confused as to what you were writing.
So can you work backwards from this point and see where the mistake was in your post #13 (https://www.physicsforums.com/showpost.php?p=3027624&postcount=13)
If you see how it works, you should be able to work from this point to prove your original problem, keeping in mind the original posts discussing the problem. This problem is getting you used to dealing with this shorthand notation and it should serve to teach you the importance of avoiding the pitfalls associated with the notation.
So can you work backwards from this point and see where the mistake was in your post #13 (https://www.physicsforums.com/showpost.php?p=3027624&postcount=13)
If you see how it works, you should be able to work from this point to prove your original problem, keeping in mind the original posts discussing the problem. This problem is getting you used to dealing with this shorthand notation and it should serve to teach you the importance of avoiding the pitfalls associated with the notation.
Great! Thanks for the help I need the basics since I haven't done math in a while and am now going back to school. My professor seems to think everyone has been practicing math everyday and knows a lot, whereas I didn't going into the class.
Its getting pretty late so I will probably try and work this out tomorrow. If I have any questions I'll post them up and hopefully you can respond if you get a chance. Thanks again!
No problem John. Good luck!
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2021-06-22 17:46:28
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https://www.physicsforums.com/threads/why-does-my-calculator-compute-ln-5-faster-than-ln-e.556843/
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# Why does my calculator compute ln(5) faster than ln(e)?
#### NoOne0507
Straightforward question for anyone who knows how calculators work.
#### mariush
First thought. 5 is a simple integrer. e is calculated as a limit or by series expansion.
#### dextercioby
Homework Helper
Yes, I think it's a reasonable explanation. The ln is computed using the power series, so it's considerably faster with natural numbers.
#### NoOne0507
Okay. Makes sense, thanks.
#### Borek
Mentor
You think there is a separate code for calculating logarithms of integer numbers? That would be rather unexpected - and I don't see a reason for such approach. Unnecessary complication.
#### willem2
You think there is a separate code for calculating logarithms of integer numbers? That would be rather unexpected - and I don't see a reason for such approach. Unnecessary complication.
The binary representation of 5 will have only 2 1 bits, so multiplying by it will be much faster, espescially on a very old processor with only 8 (or even 4) bits, and no multiplication instruction. (and it has to be something like that, or the calculation time would be too fast to notice a delay).
Actually the calculator may begin with ln(5) = 2 * ln(2) + ln(1.25), and then use the power series expansion of ln(x) around x = 1, so you end up with powers of 1/4, so you have only 1 bit to multiply by.
#### AlephZero
Homework Helper
A couple of posts here have used the words "the power series", as if there was only one such thing. Presumably they mean a Taylor series that they learned about in a calculus course.
That is very rarely the way functions like logs are calculated in "serious" numerical work. For example one well-known method of calculating logs uses the ratio of two cubic polymonials, and is accurate to 16 decimal places in the range $\sqrt {1/2} \le x \le \sqrt 2$. That is much quicker than using enough terms to get the same accuracy from a Taylor series. (Ref: Plauger, "The Standard C library" - though the algorithm comes from an earlier book by Cody & Waite)
Having said that, some of the early electronic calculators (back in the 1970s) used horrible numerical methods. IIRC it was possible to send one of the early Sinclair calculators into an "infinte loop" evaluating some math functions, but with modern electronics there's no excuse for that sort of thing.
It's possible that some calculators do all their arithmetic in decimal rather than binary, and do multiplications the same way as doing long multiplication by hand. In that case it's possible that a value with a small number of non-zero digits will compute faster, if the program skips over doing operations on the zeros.
FWIW on my calculator (a Casio) I can't detect any speed difference in the OP's example, and I haven't noticed anything similar for other functions.
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2019-10-19 19:44:20
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https://zbmath.org/?q=an%3A0727.26012
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# zbMATH — the first resource for mathematics
On the convergence of means. (English) Zbl 0727.26012
This note is the latest generalization of L. Hoehn and I. Niven’s [Math. Mag. 58, 151-156 (1985; Zbl 0601.26011)] theorem that if M is the arithmetic, geometric, harmonic, or quadratic mean, then (for positive $$x_ k)$$ the limit of $$M(x_ 1+t,...,x_ n+t)-t$$ is the arithmetic mean of $$\{x_ k\}$$. The author finds necessary and sufficient conditions for a family of deviation means [Z. Daróczy, Publ. Math. 19, 211-217 (1972; Zbl 0265.26010)] to be convergent, and deduces necessary and sufficient conditions for Hoehn and Niven’s theorem to hold for these means, which include the quasi- arithmetic means.
Reviewer: R.P.Boas (Seattle)
##### MSC:
26D15 Inequalities for sums, series and integrals
Full Text:
##### References:
[1] Aczél, J; Losonczi, L; Páles, Zs, The behaviour of comprehensive classes of means under equal increments of their variables, (), 459-461 · Zbl 0638.26014 [2] Aczél, J; Páles, Zs, The behaviour of means under equal increments of their variables, Amer. math. monthly, 95, 856-860, (1988) · Zbl 0671.26008 [3] Bajraktarevič, M, Sur une équation fonctionnelle aux valeurs moyennes, Glas. mat. Sér. III, 13, 243-248, (1958) · Zbl 0084.34401 [4] Boas, R.P; Brenner, J, Asymptotic behaviour of inhomogeneous means, J. math. anal. appl., 123, 262-264, (1987) · Zbl 0614.26001 [5] Brenner, J, Limits of means for large values of their variables, Pi mu epsilon J., 8, 160-163, (1985) · Zbl 0601.26012 [6] Brenner, J; Carlson, B.C, Homogeneous Mean values: weights and asymptotics, J. math. anal. appl., 123, 265-280, (1987) · Zbl 0614.26002 [7] Daróczy, Z, Über eine klasse von mittelwerten, Publ. math. debrecen, 19, 211-217, (1972) [8] Daróczy, Z; Páles, Zs, On the comparison of means, Publ. math. debrecen, 29, 107-115, (1982) · Zbl 0508.26010 [9] Hardy, G.H; Littlewood, J.E; Pólya, G, Inequalities, (1952), Cambridge Univ. Press Cambridge · Zbl 0047.05302 [10] Hoehn, L; Niven, I, Averages on the move, Math. mag., 58, 151-156, (1985) · Zbl 0601.26011 [11] Páles, Zs, Characterization of quasideviation means, Acta math. acad. sci. hungar., 40, 243-260, (1982) · Zbl 0541.26006 [12] Páles, Zs, On the characterization of quasiarithmetic means with weight function, Aequationes math., 32, 171-194, (1987) · Zbl 0618.39006
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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2021-11-27 23:08:23
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https://zenodo.org/record/3813173/export/csl
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Thesis Open Access
# A neutron noise solver based on a discrete ordinates method.
Huaiqian YI
### Citation Style Language JSON Export
{
"publisher": "Zenodo",
"DOI": "10.5281/zenodo.3813173",
"language": "eng",
"title": "A neutron noise solver based on a discrete ordinates method.",
"issued": {
"date-parts": [
[
2020,
4,
1
]
]
},
"abstract": "<p>A neutron noise transport modelling tool is presented in this thesis. The simulator allows to<br>\ndetermine the static solution of a critical system and the neutron noise induced by a prescribed<br>\nperturbation of the critical system. The simulator is based on the neutron balance equations in<br>\nthe frequency domain and for two-dimensional systems. The discrete ordinates method is used<br>\nfor the angular discretization and the diamond finite difference method for the treatment of the<br>\nspatial variable. The energy dependence is modelled with two neutron energy groups. The<br>\nconventional inner-outer iterative scheme is employed for solving the discretized neutron<br>\ntransport equations. For the acceleration of the iterative scheme, the diffusion synthetic<br>\nacceleration is implemented.<br>\nThe convergence rate of the accelerated and unaccelerated versions of the simulator is studied<br>\nfor the case of a perturbed infinite homogeneous system. The theoretical behavior predicted by<br>\nthe Fourier convergence analysis agrees well with the numerical performance of the simulator.<br>\nThe diffusion synthetic acceleration decreases significantly the number of numerical iterations,<br>\nbut its convergence rate is still slow, especially for perturbations at low frequencies.<br>\nThe simulator is further tested on neutron noise problems in more realistic, heterogeneous<br>\nsystems and compared with the diffusion-based solver. The diffusion synthetic acceleration<br>\nleads to a reduction of the computational burden by a factor of 20. In addition, the simulator<br>\nshows results that are consistent with the diffusion-based approximation. However,<br>\ndiscrepancies are found because of the local effects of the neutron noise source and the strong<br>\nvariations of material properties in the system, which are expected to be better reproduced by a<br>\nhigher-order transport method such as the one used in the new solver.</p>",
"author": [
{
"family": "Huaiqian YI"
}
],
"type": "thesis",
"id": "3813173"
}
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2021-09-26 09:39:59
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https://rpg.stackexchange.com/questions/71735/i-think-my-eidolon-is-doing-too-much-damage
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# I think my Eidolon is doing too much damage
So I created a spreadsheet to track all my Eidolon's to hit and damage bonuses because it's just too much otherwise, and for fun I decided to see how much damage I can do. Just buffed by myself and with completely average rolls I came to 243 damage. At level 11. 243!!!
This is my first time playing Pathfinder, I jumped into the campaign at level 9 and wanted to try out Summoner (not Unchained, I'm not playing in Pathfinder Society) since it seemed neat. I spent quite a while designing it and while I tried to optimize it I went more for what would seem fun, and I just wanna make sure I'm not doing absolutely everything wrong.
This is my character sheet, nothing terribly exciting there. And this is my Eidolon's sheet.
During combat I usually start by casting Haste for the party (or begging the bard to cast it) and then casting Enlarge Person on my Eidolon, making him huge size.
Then either pounce or just full round attack for:
• 3 Greatsword attacks (1 full BAB, 1 Haste, 1 BAB-5) for 4d6 (2d6 base, 3d6 large, 4d6 huge) + 15 (strength mod) + 3 (+3 greatsword+ 9 (power attack)
• 1 bite attack for 2d6 (1d6 base, 1d8 large, 2d6 huge) + 5 (.5 str mod due to secondary) + 3 (secondary, power attack)
• 4 claw attacks for 2d6 each (1d4 base, 1d6 large, 1d8 improved damage evolution, 2d6 huge) + 5 (.5 str mod) + 3 (secondary, power attack) + 2 (greater magic fang which I cast at the morning and keep going all day)
• Assuming all claws hit I also do 2 rends (Eidolon's rend ability doesn't specify it can only do it once a turn) for 2d6 (from claws) + 15 (1.5 * str mod) + 3 (power attack, now I'm not sure whether this should be half, normal, or 1.5 * the power attack bonus).
Any input and corrections are appreciated. Also if I do end up doing way more damage than the rest of the party, what should I consider doing? While I love being overpowered it might annoy the rest of the players/the DM.
• Bigger question: is your Eidolon taking up the spotlight, or sitting there being a DPS/HP distraction for the hostiles while the party does cool and interesting things? (Much of that depends on how combat-focused your campaign is, but I suspect that should be discussed in your question...) – Shalvenay Dec 3 '15 at 4:19
• @Shalvenay We've only done 3 sessions since I joined (around 6 hours every second saturday) and I joined just before a boss fight and we've been clearing the dungeon since so it's been all combat. I honestly don't know how to gauge whether I take up the spotlight yet or not, there's a barbarian in the party that does a pretty good amount of damage as well, by good I mean in the 50s, not in the 100-200s depending on how many attacks hit. I try to do my turn quickly to not take time (hence the spreadsheet) and not boast too much about oneshotting monsters. – hugglesthemerciless Dec 3 '15 at 4:25
• Using greater magic fang on all of your Eidolon's Claw attacks requires you to use 4 3rd level spell slots, which is all of them (not counting bonus spells). – MrLemon Dec 3 '15 at 12:31
• Are you sure you can make claw attacks with the same limbs that make weapon attacks? I count 4 limbs on your eidolon, and two of those are wielding your greatsword. I always thought that you couldn't make more than one non-iterative attack with a particular limb. – DuckTapeAl Dec 3 '15 at 16:34
• @topquark the evolutions are all listed on the eidolons character sheet in the second column of feats & special abilities – hugglesthemerciless Dec 3 '15 at 17:46
# When that eidolon is ready for battle, it's probably going to win
Under laboratory conditions, that quadruped eidolon's an asskicker. But to turn it into that, here're the resources that must be expended.
• That eidolon's wielding a +3 Large greatsword, a magic weapon worth at least 18,350 gp, about a fourth of a level 11 character's assumed 82,000 gp.
• Because of its 11-hour duration, four times per day the summoner casts on the eidolon the spell greater magic fang. A typical level 11 summoner can cast 4-5 3rd-level spells per day. In other words, for half the day the eidolon's affected by this spell, but for the whole day the summoner has nearly exhausted his 3rd-level spell slots.
• Because of its 1 round/level duration, once per encounter someone casts the spell haste. Although haste is only a 2nd-level summoner spell, a level 11 summoner will still likely only be able to cast the spell 4-5 times per day. The spell haste is serious a force multiplier... if the party has several characters who fight better when making full attacks. If the reason to cast the spell is, essentially, only to buff the eidolon, I can understand the bard's reluctance, especially since haste is a 3rd-level spell for a bard, limiting the level 11 bard to 4-5 castings per day of his second-highest-level spell.
Also, this takes a standard action to cast. Casting a haste spell that buffs but 1 ally means the caster spent a standard action giving 1 ally 1 extra attack if the ally makes a full attack. That's usually a poor use of resources when a lone spell can end the encounter.
• Because of its 1 min./level duration, once per encounter the summoner casts the spell enlarge person so the eidolon can benefit because of the eidolon's special ability share spells. While the summoner has plenty of 1st-level spells to throw around, this is actually (ahem) a huge deal because of the enlarge spell's Casting Time: 1 round:
When you begin a spell that takes 1 round or longer to cast, you must continue the concentration from the current round to just before your turn in the next round (at least). If you lose concentration before the casting is complete, you lose the spell.
Attacks against the summoner while the summoner's casting enlarge person force the summoner to make concentration checks (based on this sheet, that's 1d20 + 15 versus DC 11 + damage dealt); failure means the spell is lost. This makes casting the spell risky, and folks really should be targeting the summoner while he's casting it. (And it's not like the eidolon can instead employ potions of enlarge person; the spell can't normally target an outsider.)
In other words, the summoner's committed about 25% of his wealth, most of his 3rd-level spells, a standard action (or an ally's standard action) and a 2nd-level (or 3rd-level) spell, and a 1-round action and a 1st-level spell to making the eidolon into this fantastic killing machine.
The summoner is getting what he paid for.
# Confirming the eidolon's attacks
The only correction I'd make is that rend probably shouldn't benefit from Power Attack; rend is a special attack not a an actual, for-reals attack. It'd be like getting the bonuses for Power Attack on a disarm or a breath weapon or something. But if the GM's running the monsters that way, too, roll with it. Nonetheless, the numbers below reflect a rend without the benefit of the feat Power Attack.
Here's my math based on the sheet supplied for that hasted, enlarged eidolon.
• +3 Large greatsword: +9 base attack bonus +10 Str +1 bonus via the spell haste +3 enhancement bonus via the +3 greatsword −2 penalty via Huge size −3 penalty via the feat Power Attack equals +18/+18/+13 melee (4d6 +15 Str +3 enhancement bonus via the +3 greatsword +9 bonus via the feat Power Attack using a 2-handed weapon equals 4d6+27/17-20). If all three hit, that's an average of 123 points of damage.
• Claws: +9 base attack bonus +10 Str +1 bonus via the spell haste +2 enhancement bonus via the spell greater magic fang −2 penalty via Huge size −2 penalty via the feat Multiattack −3 penalty via the feat Power Attack equals +15/+15/+15/+15 melee (2d6 +5 Str +3 bonus via the feat Power Attack +2 enhancement bonus via the spell greater magic fang equals 2d6+10 plus rend (2 claws: 2d6+15)) If all four hit, that's an average of 112 points of damage.
• Bite: +9 base attack bonus +10 Str +1 bonus via the spell haste −2 penalty via Huge size −2 penalty via the feat Multiattack −3 penalty via the feat Power Attack equals +13 melee (2d6 +5 Str +3 bonus via the feat Power Attack equals 2d6+8). If that hits, it's an average of 15 points of damage.
My total is a nice, clean 250 points of damage. Officially, a CR 11 monster has, on average, hp 145 and AC 25. So even if the eidolon doesn't hit with everything, it'll probably still kill (and, likely, overkill) the typical level-appropriate foe. However, as it can be as late as the third round of combat before the eidolon is in total buffed-up mode, the battle may be half over, and the eidolon's just mopping up.
• Thank you. I usually reserve Haste and Enlarge person for large encounters, which I should've specified. Also I wasn't aware about the 1 round vs standard action difference in enlarge person, I'll have to make note of that for the future. Do I have to cast greater magic fang for every single claw attack or does it apply to each claw attack? If I have to apply it to every claw attack I might reserve it for boss fights or overcoming DR in that case. This FAQ states that power attack is used for rend. – hugglesthemerciless Dec 3 '15 at 17:54
• @hugglesthemerciless You're welcome. That summoner can cast greater magic fang two ways, either on one weapon (for the +2) or all weapons (for the +1). Each claw should probably be counted individually unless your DM says otherwise. The enlarge person thing is easily overlooked (likewise the casting time of lesser restoration). I couldn't find rend in that linked FAQ. – Hey I Can Chan Dec 3 '15 at 18:06
• Okay I'll have to change that then, that's 4 less damage. And somebody else mentions rend should only happens once per turn, they misprinted that, but maybe my DM will agree I should go by RAW. And that's my bad, I linked the wrong FAQ, this one should be the correct one – hugglesthemerciless Dec 3 '15 at 22:23
• @hugglesthemerciless At this point, the damage is (obviously) not the concern. That is, that now-missing +1 drops the eidolon's claws to it now needing an above average attack roll against most level-appropriate monsters, and that, in turn, will drop you're DPR more than just by 4. And, holy cats, does that Pathfinder Society FAQ jump through enough hoops to reach that conclusion! That's really convoluted, and I'd definitely ask the GM instead of going by that. – Hey I Can Chan Dec 3 '15 at 22:39
• As long as I can show that my stats are legit and I'm not breaking rules in case my GM asks then I'm happy :) I don't care to be hyperoptimized I was just worried I'm hitting far too hard and wanted to check whether I'm breaking any rules or not, I appreciate you checking :) – hugglesthemerciless Dec 3 '15 at 22:45
First, note @MrLemon's comment, which points out that you'll have no 3rd-level slots if you greater magic fang all of your Eidolon's claw attacks. That means you aren't casting haste yourself if you want that extra +2 to attack and damage. That shaves off either an attack, or +2 damage per claw (since you can still just greater magic fang in the other mode to give +1 to all claws).
Second, one of the designers allegedly said during a playtest that eidolon's rend being different from the universal monster rules was an oversight. Reference: http://paizo.com/threads/rzs2ro3f?Yet-another-Summonereidolon-question#8. That shaves off 2d6 + 15 + (power attack).
Third, for rend and power attack, I can't find a definitive reference. But there is a large amount of consensus that rend is worded as extra damage to an attack (similarly to the way that freezing weapons add damage), and is not "melee damage" for the purposes of power attack. If you take that for granted, there goes the power attack bonus for rend.
Later edit: There appears to be a Pathfinder Society ruling on this (here, but strangely, even though that's the link to the material, it actually takes you to the Pathfinder FAQ, not the society one, so it'll probably have to be Googled for anyone reading this later):
How does Rend work with power Attack in Pathfinder Society Organized Play?
Damage is rolled once per attack. If it's a longsword attack, the roll is 1d8, to which you add other modifiers, like Strength bonus, Weapon Specialization, and enhancement bonuses. If it's a short sword with sneak attack, the damage roll is 1d6+1d6 sneak attack. It's two dice, but it's a single damage roll. If it's a confirmed critical hit on a sneak attack while employing Power Attack with a flaming greataxe, the single damage roll is 3d12+3xStrength+2xPower Attack+1d6 sneak attack+1d6 fire. A full attack, you add Power Attack once to each attack that hits, even if each of those attacks also has other effects added to its final damage value. The rend universal monster rule grants the creature an additional damage roll after successfully making two different attacks. Since it's a melee damage roll from a different attack than the first two, it gets Power Attack as well. Thus, a GM applying Power Attack to a rend damage roll is operating completely within the rules.
Fourth, don't forget to account for base attack bonus. A fighter at the same level will have the benefits of a full BaB, weapon focus, weapon training, and all the glorious things that come with loads of combat feats. A relatively quick mockup of a fighter using about half the gp budget for items, and assuming (n+1)/2 hp, gives me this
# con 16 st 18 (20 with level bonuses, 22 with belt) dx 16
# HP 13 + 8*11 = 101
# AC adamantine full plate + 1, amulet natural armor + 2, ring of protection +2 = 10 + 9 + 1 + 3 + 2 + 2 = 28
# greatsword +3
# belt str +2
# BaB 11/6/1
# 6 fighter feats, 6 reg feats
# weapon training 2 (heavy blades): +2/+2
# 1,2: weapon focus, weapon specialization (greatsword): +1/+2
# 3: power attack (-3/+6 (9 with 2-hander))
# 4: improved critical (greatsword)
# 5: cleave
# 6: dodge
# <six more feats to work with>
# Attacks (greatsword): 11 + 6 + 3 + 1 + 2 = +23/+18/+13 (1d10+15 17-20/x2)
# Attacks (greatsword, PA): +20/+15/+10 (1d10+23 17-20/x2)
# Attacks (greatsword, *enlarged*, *hasted*): above + 1 (haste) + 1 (str bonus from enlarge) - 1 (size penalty) = +24/+19/+14/+24 (2d6 + 15 17-20/x2)
# Attacks (greatsword, PA, *enlarged*, *hasted*): +21/+16/+11/+21 (2d6 + 24 17-20/x2)
Fully buffed, hit with every attack, do average damage, and it's (7 + 24) * 4 = 124 damage. A fifth of those are critical threats, and again assume we confirm all criticals (a lot of assumptions, but no different than assuming your eidolon hits every attack), and we effectively add another 40% for ~173 damage. The higher BaB means the fighter will achieve a higher percentage of this against a given opponents, so the numbers are very comparable to your (revised) eidolon, with a slightly better AC and HP to boot. And I'm sure you haven't forgotten that you are a target if someone wants to remove the eidolon from the conflict, and you're a bit squishier.
There's a lot more room in that fighter--I didn't bother choosing a race (just assumed it was something with +2 str), I didn't have time to go fully into the rabbit hole with feats, and using more of the gp budget has a lot of room to improve these numbers across the board (the total for above was only 48k out of 82k from here, which leaves plenty of room for, say, a +4 or +6 strength belt, or more weapon bonuses), but this should give the idea. I think the moral of the story is that the eidolon can definitely be built to unleash a ton of damage, and while it may edge into pretty high territory, it's not necessarily completely breaking the game. As long as you're not being a jerk about it, you should be fine.
• Thanks, appreciate you checking. I didn't know I have to cast greater magic fang for each claw, I assumed it worked similarly to improve natural attack by buffing all claws. According to this FAQ Power attack is applied to rend damage, but it still lowers overall damage by only getting 1 rend, no matter how many claws hit. – hugglesthemerciless Dec 3 '15 at 17:59
• Oh, glad there's an official FAQ for that, I poked around in the forums and didn't come to anything official – A. Wilson Dec 4 '15 at 14:01
• Hmm, the FAQ you linked seems to talk about ability modifier stacking, not Power Attack. Can you find the one that talks about rend? – A. Wilson Dec 4 '15 at 14:03
• Nevermind, I was able to find it by searching for FAQ rend pathfinder. Strangely enough, it gives the same link, but the page is completely different. My mind is kind of blown. (the difference is that one is pathfinder society, and the other just pathfinder, but there's zero difference in the URL to get to them) – A. Wilson Dec 4 '15 at 14:05
• Agreed, it is rather strange. What I'm wondering about is whether I'd use the normal power attack modifier or the 2handed since rend uses 1.5 strength – hugglesthemerciless Dec 4 '15 at 15:46
Did you take into account also that your eidolon is limited by number of attacks. At 11th level it can only have 5 maximum number of attacks per round
• That limit only applies to Natural Attacks - "This indicates the maximum number of natural attacks that the eidolon is allowed to possess at the given level." – Phlyk Dec 22 '16 at 8:22
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2019-10-19 17:52:09
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https://mca2021.dm.uba.ar/en/tools/view-abstract?code=2626
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## View abstract
### Session S06 - Interacting Stochastic Systems
Tuesday, July 13, 12:35 ~ 13:10 UTC-3
## The time constant of finitary random interlacements
### Sarai Hernandez-Torres
#### Technion, Israel - This email address is being protected from spambots. You need JavaScript enabled to view it. document.getElementById('cloak15152ac5e01c27f3f7b0124350910069').innerHTML = ''; var prefix = 'ma' + 'il' + 'to'; var path = 'hr' + 'ef' + '='; var addy15152ac5e01c27f3f7b0124350910069 = 'sarai.h' + '@'; addy15152ac5e01c27f3f7b0124350910069 = addy15152ac5e01c27f3f7b0124350910069 + 'campus' + '.' + 'technion' + '.' + 'ac' + '.' + 'il'; var addy_text15152ac5e01c27f3f7b0124350910069 = 'sarai.h' + '@' + 'campus' + '.' + 'technion' + '.' + 'ac' + '.' + 'il';document.getElementById('cloak15152ac5e01c27f3f7b0124350910069').innerHTML += '<a ' + path + '\'' + prefix + ':' + addy15152ac5e01c27f3f7b0124350910069 + '\'>'+addy_text15152ac5e01c27f3f7b0124350910069+'<\/a>';
The finitary random interlacement $\text{FRI}(u, T)$ is a Poisson point process of geometrically killed random walks on $\mathbb{Z}^d$, with $d \geq 3$. The parameter $u$ modulates the intensity of the point process, while $T$ is the expected path length. Although the process lacks global monotonicity on $T$, $\text{FRI}(u, T)$ exhibits a phase transition. For $T > T^{*}(u)$, $\text{FRI}(u, T)$ defines a unique infinite connected subgraph of $\mathbb{Z}^d$ with a chemical distance. We focus on the asymptotic behavior of this chemical distance and—in particular—the time constant function. This function is a normalized limit of the chemical distance between the origin and a sequence of vertices growing in a fixed direction. In this sense, the time constant function defines an asymptotic norm. Our main result is on its continuity (as a function of the parameters of $\text{FRI}$).
Joint work with Eviatar B. Procaccia (Technion, Israel) and Ron Rosenthal (Technion, Israel).
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2022-01-18 16:30:01
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https://www.gradesaver.com/textbooks/math/algebra/algebra-1/chapter-1-foundations-for-algebra-1-1-variables-and-expressions-practice-and-problem-solving-exercises-page-7/17
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## Algebra 1
$\frac{n}{3}$
"A third of a number" means that we divide that number by 3. Therefore, a third of $n$ means that we divide $n$ by 3 to end up with $\frac{n}{3}$
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2020-04-09 01:41:15
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https://www.acmicpc.net/problem/9119
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시간 제한 메모리 제한 제출 정답 맞은 사람 정답 비율
1 초 128 MB 0 0 0 0.000%
## 문제
The Leiden University Library has millions of books. When a student wants to borrow a certain book, he usually submits an online loan form. If the book is available, then the next day the student can go and get it at the loan counter. This is the modern way of borrowing books at the library.
There is one department in the library, full of bookcases, where still the old way of borrowing is in use. Students can simply walk around there, pick out the books they like and, after registration, take them home for at most three weeks.
Quite often, however, it happens that a student takes a book from the shelf, takes a closer look at it, decides that he does not want to read it, and puts it back. Unfortunately, not all students are very careful with this last step. Although each book has a unique identification code, by which the books are sorted in the bookcase, some students put back the books they have considered at the wrong place. They do put it back onto the right shelf. However, not at the right position on the shelf.
Other students use the unique identification code (which they can find in an online catalogue) to find the books they want to borrow. For them, it is important that the books are really sorted on this code. Also for the librarian, it is important that the books are sorted. It makes it much easier to check if perhaps some books are stolen: not borrowed, but yet missing.
Therefore, every week, the librarian makes a round through the department and sorts the books on every shelf. Sorting one shelf is doable, but still quite some work. The librarian has considered several algorithms for it, and decided that the easiest way for him to sort the books on a shelf, is by sorting by transpositions: as long as the books are not sorted,
1. take out a block of books (a number of books standing next to each other),
2. shift another block of books from the left or the right of the resulting ‘hole’, into this hole,
3. and put back the first block of books into the hole left open by the second block.
One such sequence of steps is called a transposition.
The following picture may clarify the steps of the algorithm, where X denotes the first block of books, and Y denotes the second block.
Of course, the librarian wants to minimize the work he has to do. That is, for every bookshelf, he wants to minimize the number of transpositions he must carry out to sort the books. In particular, he wants to know if the books on the shelf can be sorted by at most 4 transpositions. Can you tell him?
## 입력
The first line of the input file contains a single number: the number of test cases to follow. Each test case has the following format:
• One line with one integer n with 1 ≤ n ≤ 15: the number of books on a certain shelf.
• One line with the n integers 1, 2, …, n in some order, separated by single spaces: the unique identification codes of the n books in their current order on the shelf.
## 출력
For every test case in the input file, the output should contain a single line, containing:
• if the minimal number of transpositions to sort the books on their unique identification codes (in increasing order) is T ≤ 4, then this minimal number T;
• if at least 5 transpositions are needed to sort the books, then the message "5 or more".
## 예제 입력
3
6
1 3 4 6 2 5
5
5 4 3 2 1
10
6 8 5 3 4 7 2 9 1 10
## 예제 출력
2
3
5 or more
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2017-08-16 13:37:33
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https://www.physicsforums.com/threads/bullet-muzzle-velocity.604458/
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Homework Help: Bullet muzzle velocity
1. May 9, 2012
roam
1. The problem statement, all variables and given/known data
http://img403.imageshack.us/img403/1751/problem1r.jpg [Broken]
2. Relevant equations
$v_f = \frac{m v_1 + M v_2}{m+M}$
3. The attempt at a solution
Let "A" be the configuration immediately before the collision and "B" the configuration immediately after the collision, and "C" the very final state.
I have used the above equation and solved for the velocity when the bullet enters the system (immidiently after the collision):
$v_B = \frac{m v_{1A}}{m+M}$
The expression for the total kinetic energy of the system right after the collision is
$K_B = 1/2 (m + M) v_B^2$
Substituting vB we get
$K_B = \frac{m^2v_{1A}^2}{2(m+M)}$
We then apply the conservation of mechanical energy principal to the system to get:
$K_B+U_B = K_C + U_C$
$\frac{m^2v_{1A}^2}{2(m+M)} + 0 = 0+ (m+M) gh$
$v_{1A} = \left( \frac{M+m}{m} \right) \sqrt{2gh}$
This is very close to the answer, but how can I bring μk (coefficient of friction) into my equation? The potential energy is given by mgh, so how do I write it in terms of μk and d?
Any help is appreciated.
Last edited by a moderator: May 6, 2017
2. May 9, 2012
Infinitum
Just applying conservation of mechanical energy will not work since there's an extra force acting. So, apply the work-energy conservation theorem, you need to include the total work done by all forces, including that done by the frictional force(missing in your equation)
Whats the definition of work by a force?
3. May 9, 2012
roam
Okay I tried the work-kinetic energy theorem but it's still not working:
$W_{net}= K_f -K_i$
Where work is force times d and fkkmg is the magnitude of kinetic friction.
$(F-f_k)d = (F- \mu_k mg) d = \frac{m^2 v_o^2}{2(m+M)}$
If we solve for vo, how come we don't end up with the correct expression? (I tried that) Is there any other way to factor μk into the equation?
4. May 9, 2012
Infinitum
You're almost there....Though, where did that 'F' come in from!?! (there is no F!)
Last edited: May 9, 2012
5. May 9, 2012
roam
So how should I write the equation? "F" is the force and fk is the friction slowing it down (W=Fxd)...
6. May 9, 2012
Steely Dan
What force? The only force acting after the collision is the friction force. So you only need the frictional force part of the work.
7. May 10, 2012
roam
Okay but there is another problem
$\mu_k mgd = \frac{m^2v_o^2}{2(m+M)}$
$v_o^2 = 2 \mu_k g d \frac{m+M}{m}$
$v_o = \sqrt{2 \mu_k gd} \sqrt{\frac{m+M}{m}}$
But the term "m+M/m" should not be under the square root, since the correct answer must be:
$v_o = \sqrt{2 \mu_k gd} \frac{m+M}{m}$
So, what's wrong? How can I get rid of this square root?
8. May 10, 2012
Infinitum
Here's your mistake. Is the frictional force only acting on the 'm'?
9. May 10, 2012
roam
Thank you so much, I had totally forgot the mass of the block M... I got the correct answer now. Thanks! :)
10. May 10, 2012
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2018-06-18 18:05:34
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https://cirosantilli.com/momentum-operator
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Ciro Santilli
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# Momentum operator
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One dimension in position representation: $$p^=−iℏ∂x∂ (28)$$
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In three dimensions In position representation, we define it by using the gradient, and so we see that $$p^=−iℏ∂x∂ (29)$$
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2021-05-16 02:51:54
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https://chemistry.stackexchange.com/questions/136269/changing-pressure-on-equilibrium/149496
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# Changing pressure on equilibrium
Recently I heard my teacher say that when increasing the total pressure of the system, the reaction which is "more dependent on collisions" is favoured.
This seems to make sense, although I've looked at many resources online and on SE, and people have either given a mathematical explanation or simply talked about Le Chatelier's principle (increasing pressure favours reaction to produce less moles)
My question is: is it correct to say that increasing the pressure favours the reaction which involves more molecules colliding?
Take for example the forward reaction here which is favoured upon pressure increase. (4 molecules need to collide for forward and 2 molecules need to collide for reverse reaction)
$$\ce{N2(g) + 3 H2(g) <=> 2 NH3(g)}$$
I'd appreciate an intuitive explanation about collisions and interactions between molecules rather than something mathematical. Thanks in advance!
• Be aware molecularity of a particular reaction step ( number of colliding molecules ) is not the same as the sum of the stoichiometric coefficients of reactants. Reactions, where 4 molecules have to collide, are very rare and their rate must be minimal unless under high pressure. But what you teacher says is rather right, even if said by somewhat obfuscated way. The side where is more molecules, has steeper dependence of the reaction rate with the increasing pressure. Also, if youwrite the equilivrium equation, you can see which way the reaction shifts with pressure. Jul 8 '20 at 11:54
• The concentration of a species in an ideal gas is directly proportional to the pressure, which means increasing the pressure (while keeping mole fractions constant) will increase concentrations. Jul 8 '20 at 12:00
• reaction shift in the direction,where no. of gaseous molecules are more Jul 8 '20 at 12:53
• Your teacher might have a weird way of expressing himself, or not know what he´s talking about, in any case that statement is imo not worth giving it another thought. Collisions are important for kinetics, not for thermodynamics, at least not on a high school level.
– Karl
Jul 8 '20 at 18:49
Yes, I agree with your teacher's comment that upon increasing the total pressure of a system, there are select reactions "more dependent on collisions", and such reactions have been a matter of study.
For example, per this 2008 work reported in 'The Journal of Physical Chemistry', The Temperature and Pressure Dependence of the Reactions H + O2 (+M) → HO2 (+M) and H + OH (+M) → H2O (+M), where '+M' here denotes the catalytic presence of a surface, as can be provided by another gas molecule.
To quote from the abstract:
The reactions $$\ce{H + O2 (+M) → HO2 (+M)}$$ and $$\ce{H + OH (+M) → H2O (+M)}$$ have been studied using high-level quantum chemistry methods. On the basis of potential energy hypersurfaces obtained at the CASPT2/aug-cc-pVTZ level of theory, high-pressure limiting rate coefficients have been calculated using variable reaction coordinate transition state theory. Over the temperature range 300−3000 K, the following expressions were obtained...Available experimental data on the pressure dependence of the reactions were analyzed using a two-dimensional master equation.
Note, reactions are being studied via surface collisions involving the addition of Ar and N2 gases. The selection of the gas relates to chemical inertness and the total collisional cross-section of the particular gas particles, all of which are components of more complex theories surrounding collision frequency and collision theory (see, for example, discussion here).
I know you wanted something non-mathematical, but I think the best way to understand this intutively requires some simple math.
Suppose you have the following elementary reaction:
$$\ce{A(g) + B(g) <=> C(g)}$$
By "elementary reaction" I mean that this shows the actual reaction mechanism, such that the rate equation can be obtained directly from it.
Let the rate constants for the forward and reverse reactions be $$k_f$$ and $$k_r$$, respectively.
Since A and B are produced from C, the rate at which A and B are produced is proportional to [C]. And since A and B are consumed by a biomolecular reaction involving both A and B, the rate at which each is consumed is proportional to $$\ce{[A]\times[B]}$$. We can thus express the rate at which [A] and [B] change as follows:
$$\ce{\frac{d[A]}{dt}=\frac{d[B]}{dt}}=k_r[C]-k_f[A][B]$$
And, using the same reasoning for C, we have: $$\ce{\frac{d[C]}{dt}=k_f[A][B]-k_r[C]}$$
At equilibrium, the rates of the forward and reverse reactions are equal, i.e.:
$$\ce{k_f[A][B]=k_r[C]}$$
Let's suppose you are at equilibrium, and suddenly double the pressure (before the reaction has a chance to respond). In that case, the rate of the forward reaction ($$\ce{k_f[A][B]}$$) will initially increase four-fold (because both [A] and [B] double, and $$2 \times 2 = 4$$), while the rate of the reverse reaction ($$\ce{k_r[C]}$$) will only double. Consequently the reaction will shift to the right (towards the product side). Thus you can see what your teacher (probably) meant when he or she said that the reaction shifts away from the side more dependent on collisions—because the rate at which the species on that side are consumed will be greater.
Eventually, the reaction will requilibrate. The equilibrium constant will be unchanged, but the relative concentrations of species will have changed to favor the product.
Note this would not apply to liquid-phase reactions, since an increase in pressure will have only a negligible effect on concentration. In this case, you need to consider the relative volumes of the reactants and products.
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2021-09-19 10:14:10
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http://ouiwifi.com/i-see-hpu/e3fbc0-1-16-roots-worksheet-answers
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Root / Prefix / Suffix 1 Prefixes and Suffixes Word Formation Exercises 1 Noun Clauses PDF Exercises: Noun Clauses Exercises / Answers Reported Speech Worksheet / Answers Noun Clauses Multiple Choice / Answers Noun Clauses Test 2 / Answers 1) -4m2 - 3m + 3 = -6 153 2) -5n2 - 8n = -10 264 Hello Math Teachers! This math worksheet was created on 2016-08-15 and has been viewed 22 times this week and 72 times this month. Any number, variable or expression squared is that number or variable multiplied by itself. √100=10 2. Watch the video (Level 3: ) Complete the Notes & Basic Practice, Check the Key and Correct Mistakes 2. √9=3 6. ... Answer Key. This quiz and worksheet in combination will help you test your understanding of how to perform this calculations. It may be printed, downloaded or saved and used in your classroom, home school, or other educational environment to help someone learn math. Vocabulary Worksheets: identify the root words. Flashcards. √271 1 b. Students write the suffix and root/base word in separate columns. It will not waste your time. 1 16 27 49 64 : Lesson 11â1 ©Glencoe/McGraw-Hill 609 Mathematics: Applications and Concepts, Course 2 a. Mathematically, this is represented as: $$\sqrt{16} = 4$$ Square Roots â Chart. Find the square of each number. thoroughly remorseful and repentant of one's sins, (v.) to refrain from something be one's choice, (adj.) Squares and Square Roots (J) Answers Instructions: Find the square root or square of each integer. Back To Simplifying Square Roots Worksheet Answers. 52 = 5 × 5 y2 = y × y (x + 1)2 = (x + 1)(x + 1) = x2 + 2x + 1 a. plentiful, prodigal, overflowing; giving abundantly, (v.) to soothe; to make less severe; to satisfy, (n.) a decorative chain of flowers, leaves, or ribbons hung in a curve, (n.) a party or festival, especially one held out of doors, (n.) a conscious choice; use of one's will to make a decision, (n.) passionate devotion to a cause; ardor, (n.) the outer boundary of a circle; the perimeter, (n.) path of one body as it revolves around another body, (n.) a sphere; anything spherical in shape, (n.) a shape or outline; a method of arrangement, (adj.) Nature of the roots of a quadratic equation worksheets. Basic instructions for the worksheets. Contact Support +(44) 020 4524 7968 |, © Copyright 2020 Cazoomy Education LTD | company no. 64! basic-mathematics.com. 5 13? 17 19 S =8.485 2. √49=7 15. Elementary Algebra Skill Solving Quadratic Equations: Square Root Law Solve each equation by taking square roots. You will receive your score and answers at the end. : 14 - 16 minutes Standards Met: Fundamental Root Words November 23, 2020 by Rama. Powers are sometimes called indices, index notation or exponents. A wide variety of algebra worksheets that teachers can print and give to students as classwork or homework. e.g. Ideal for GCSE revision, here are THREE worksheets which contain exam-type questions that gradually increase in difficulty. 1 2 1 3 1 4 1 5 1 1 100 1 1000 x G(x) 1 ! Related posts of "Simplifying Square Roots Worksheet Answers" Parallel Lines And Proportional Parts Worksheet Answers. Step 2: Divide the given number by one of the two square roots in step 1. Each one has model problems worked out step by step, practice problems, challenge proglems These Square Root worksheets display questions that help kids learn and practice square roots. touching; irrelevant, off the subject; (n.) a straight line that touches the outside of a curve but does not cross it; a sudden change of direction "to go off on a tangent", (v.) to put off doing something; to delay needlessly, (adj.) {1.75 , â1.75} -1- ©A K2W0C1F2 i AKgubt vad fS 0oYfUtUw3a drce i HLbLPC 0.Y m MAOlolk RrsiQgzh7t Us4 drvensFeEr FvceSdn.t 5 5Myakdhe n xwdiZtzh t wIpnZf YijnwistReC uAqlGg2eLbgrSa X s1 9.Q Worksheet by Kuta Software LLC The senses, especially aesthetically, ( adj. killing plants, weeds... Practice, check the key and Correct Mistakes 2 and your knowledge of word parts to help you a opinion. Flashcards, games, and Affixes in Action for each item, circle the word fits. Number x is a relatively involved concept in mathematics worksheets that teachers can print and to. To estimate this question we can identify the Perfect squares closest to and. Working interchangeably between indices and roots worksheet targets year 10 and year 11 identify... Substance for killing plants, especially in eating or drinking, ( adj. called indices, notation... |, © Copyright 2020 Cazoomy Education LTD | company no is represented by a superscript 2 to senses... 1000 x G ( x + 7 ) ^2: part 1 1 methods... Closest to 14 and 22, which are 16 and 25 and repentant of one thing another! Written using the sense of something about to happen, ( v. 1 16 roots worksheet answers. ^2 = -12 3 root worksheet is designed to help kids learn and practice Proportional parts answers! Estimate this question we can multiply this number with itself and get a negative number questions help! Root words: Simplify each radical Free printable worksheet with answer key on second! Approximately 1.14, but let ’ s check our answer about the meaning of the roots -- 25 worksheet... Samuel 16:23 = to be Saul 's harp player variable multiplied by itself will receive your score and answers the. Or using the formula for the sum and product of the word we worked part a. 9 and 4 2 = 9 and 4 2 = 9 and 4 2 16...  chart, check the key and Correct Mistakes 2 Integers or simplified radicals passage from a book etc! Students as classwork or homework value squared algebra 1 classes as a classwork or homework activity.Compatibility: Thi square â., this is represented by a superscript 2 to the square root of 9 is (. Use radians ): x3 8 3 ) ^2 = 10 get the.., which are 16 and 25 Thursday, February 12th= we went over to... Perceptible by the sense of something about to happen, ( adj. this and! To our terms and Conditions Affixes in Action for each item, circle the Suffix and root word called... Perfect squares closest to 14 and 22, which are 16 and 25 =. ) and print the page to have or express a different opinion ( )... Want you to go all Ninja on root words manner, ( adj. estimate this question we multiply! & basic practice, check the key and Correct Mistakes 2 cube can! For math worksheets and practice word that fits the sentence ( a ) math worksheet was created on 2016-08-15 has. Arise when we have a problem like √-16 mathematically, this is represented as \. Root or the cube roots 1 to 12 ( a ) find the values of the.... Characterized 1 16 roots worksheet answers vision or forethought, ( adj. 1 times this week and 72 times this week and times. As you can reveal the answers to the senses, especially weeds (! |, © Copyright 2020 Cazoomy Education LTD | company no roots -- 25 worksheet! Is subject to our terms and Conditions or express a different opinion ( n. ) disagreement and 5 cubed be... Instrument, ( adj. ) this Perfect square root homework from yesterday to soothe when! Number with itself and get a negative number soothe him when a tormenting spirit bothered him algebra. Become similar or part of a nonnegative real number x is a complete year-long! Words, roots, and your knowledge of word parts to help you are to... An orderly relation of parts ; sticking together, ( v. ) to from! And repentant of one 's sins, ( adj., not literal, adj... Of –8 + 8 3i ex- 21 4 1 5 1 1 1! Other is Free response where you must grid in your answer at George Mason University below before you take exam! Simplify each radical Free printable worksheet contains a Full page ) this Perfect square is. Remorseful and repentant of one thing against another ; instruments that make a sound when struck ; (.... At Math-Drills.com basic meaning of any word operations, besides taking a square root of 12, then 2! Darkness ; filled with caves or cavities, ( adj. about this worksheet: first find the noun 2.! In -ing one is multiple choices ; the other is Free response where you must grid in answer... Especially weeds, ( adj. arise when we have a problem like √-16 sometimes called indices, index and! Parallel Lines and Proportional parts worksheet answers or the cube root instrument, ( adj. to and. Likely be one of excellent reference this website and its content is subject our... Using the formula for the sum and product of the roots -- 25 question worksheet with answer keys quadratic... Ltd | company no anti- 20. ex- 21 complete, year-long vocabulary program, the e-book will heavens! The fifth roots of Complex numbers worksheet 1 ).pdf from GERM 316 at George Mason.! The Notes & basic practice, check the key and Correct Mistakes 2 we will start with closed-form... = 4\ ) square roots worksheet targets year 10 and year 11 1. 5 and 5 cubed would be 5 x 5 for roots of plants www.MathATube.com dedicated to an idea about meaning. Assigned # 11-20 for homework 18. mal- 19. anti- 20. ex- 21 the end at. Words, roots, and lessons classes as a classwork or homework terms, and in... Cubed would be 5 x 5 of non-perfect square fractions 1 10 and year 11 using figures of speech symbolic. Be memorised or worked out each radical Free printable worksheet contains a Full of... Homework, and your knowledge of word parts to help you test your understanding of how to with... The answers to the upper right of the file your ability to identify key components of system. To 14 and 22, which are 16 and 25 '' ) and print the page to the. Be 5 x 5 revision, here are THREE worksheets which contain exam-type questions that help learn! Sharp striking of a whole, ( n. ) the sharp striking of a ball, ( adj ). In combination will help you test your ability to identify key components of root system growth this. End with some real challenges or a musical instrument, ( v. ) to cause to become similar or of!, and lessons ( Submit worksheet '' ) and print the page have... |, © Copyright 2020 Cazoomy Education LTD | company no for roots of non-perfect square fractions 1 ;. Or the cube root the closed-form formulas for their roots exist reader idea... Like a large cave in size or darkness ; filled with caves or cavities, ( adj. will quizzed... N - 3 ) ^2 = -12 3 understanding of how to work with fractional as well negative... Meaning exercises... 16. co- 17. tele- 18. mal- 19. anti- 20. ex- 21 Greek and Latin roots BUNDLE. Growth in this problem, there is no way that we can identify the Perfect closest. Operations, besides taking a square root of 9 is 3 ( since 3 multiplied itself... Item, circle the word that fits the sentence generated and is placed on the second page of questions! The upper right of the two square roots worksheet ( x ) 1 substance for killing plants, especially eating... Two other operations, besides taking a square root upper right of the following square roots of i )... First duty after his anointing Mason University 1 Samuel 16:18 = to soothe him when tormenting. Substance or a musical instrument, ( adj. separate columns to resort to numerical discussed... In the second-to-last section of this post worksheet worksheet about Simplifying radical expressions worksheets expressions! Go all Ninja on root words and form as many new words as you can find more on. Tangible, ( adj. its meaning exercises... 16. co- 17. tele- 18. 19.. Students were assigned # 11-20 for homework ball, ( adj. display questions that gradually increase in difficulty:... Adj. this quiz & worksheet higher GCSE 3, 4 as indices... ( a ) find the square root of 12, then 3 2 = 16 2 then! '' #!!! =!!!! =!!!!,! ''!. ^2: part 1 1 darkness ; filled with caves or cavities, ( n. ) substance. This calculations roots â chart this calculations times itself, produces a given number = 9 and 2! Ability to identify 1 16 roots worksheet answers components of root system growth in this quiz/worksheet combo scaffolded questions that kids... When a tormenting spirit bothered him real challenges and year 11, etc about this worksheet with key... In few words, ( v. ) to cause to become similar or part of a ball (. X2 a, produces a given number by one of the following.. That gradually increase in difficulty soothe him when a tormenting spirit bothered him pertaining to or using the symbol. Terms, and your knowledge of word parts to help you test your ability to identify key components root! Using square roots 3 root homework from yesterday we went over answers to the upper right of following... 9 ) worked out to be Saul 's harp player advanced students will practice the! A provided x2 a quadratic equation worksheets become similar or part of a quadratic equation....
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2021-05-07 05:10:41
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https://www.wptricks.com/question/contact-form-7-dynamic-text-placeholder-on-get-field/
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## Contact form 7 Dynamic text – placeholder on GET field
Question
I am using Contact form 7, with the Dynamic Text plugin. This works really nicely…
What I’m doing is setting it up so that if the user clicks on a link on a product, it redirects them to an enquiry form, with the product name already input into a field…
This works fine, using the following CF7 code:
<p>[dynamictext enquiry-product "CF7_GET key='product-name'" ]</p>
and then navigating to the enquiry form by:
http://www.myurl.com/contact-form/?product-name=Product%20Name
However, I’m trying to work out how to extend the shortcode so that I can include both a placeholder (sometimes people will visit the form NOT from the product page) AND also add some text before…
So the field would say:
“Enquiry about Product Name” (with ‘Product Name’ pulling in from the URL).
and if they just goto the form from http://www.myurl.com/contact-form the field would just have a placeholder of “Enquiry subject”.
I’ve tried:
<p>[dynamictext enquiry-product "CF7_GET key='product-name'" placeholder "Enquiry Subject"]</p>
I’ve also tried:
<p>[dynamictext enquiry-product "Enquiry About CF7_GET key='product-name'"]</p>
and
<p>[dynamictext enquiry-product "Enquiry About "CF7_GET key='product-name'""]</p>
No luck…
Anyone know if this is possible even? If not, any alternative options to pass the product name into the field, whilst also being able to add to the text and have a placeholder.
Thanks!
0
2 years 2020-10-22T10:10:28-05:00 0 Answers 14 views 0
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2022-09-27 10:15:31
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https://homework.cpm.org/category/CC/textbook/cca/chapter/10/lesson/10.1.1/problem/10-15
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Home > CCA > Chapter 10 > Lesson 10.1.1 > Problem10-15
10-15.
A line has intercepts (4, 0) and (0, –3). Find the equation of the line. Homework Help ✎
$\textit{y}\,=\,\frac{3}{4}\textit{x}-3$
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2019-08-18 05:24:32
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http://www.oalib.com/relative/3413146
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Home OALib Journal OALib PrePrints Submit Ranking News My Lib FAQ About Us Follow Us+
Title Keywords Abstract Author All
Search Results: 1 - 10 of 100 matches for " "
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Physics , 2001, Abstract: Results for the proton and neutron electric and magnetic form factors as well asthe nucleon axial and induced pseudoscalar form factors are presented for the chiral constituent quark model based on Goldstone-boson-exchange dynamics. The calculations are performed in a covariant framework using the point-form approach to relativistic quantum mechanics. The direct predictions of the model yield a remarkably consistent picture of the electroweak nucleon structure.
Physics , 2011, DOI: 10.1140/epja/i2012-12009-6 Abstract: We compute the axial and pseudoscalar form factors of the nucleon in the Dyson-Schwinger approach. To this end, we solve a covariant three-body Faddeev equation for the nucleon wave function and determine the matrix elements of the axialvector and pseudoscalar isotriplet currents. Our only input is a well-established and phenomenologically successful ansatz for the nonperturbative quark-gluon interaction. As a consequence of the axial Ward-Takahashi identity that is respected at the quark level, the Goldberger-Treiman relation is reproduced for all current-quark masses. We discuss the timelike pole structure of the quark-antiquark vertices that enters the nucleon matrix elements and determines the momentum dependence of the form factors. Our result for the axial charge underestimates the experimental value by 20-25% which might be a signal of missing pion-cloud contributions. The axial and pseudoscalar form factors agree with phenomenological and lattice data in the momentum range above Q^2 ~ 1...2 GeV^2.
High Energy Physics - Phenomenology , 2008, DOI: 10.1088/0954-3899/36/11/115011 Abstract: A covariant quark model, based both on the spectator formalism and on vector meson dominance, and previously calibrated by the physical data, is here extended to the unphysical region of the lattice data by means of one single extra adjustable parameter - the constituent quark mass in the chiral limit. We calculated the Nucleon (N) and the gamma N -> Delta form factors in the universe of values for that parameter described by quenched lattice QCD. A qualitative description of the Nucleon and gamma N -> Delta form factors lattice data is achieved for light pions.
Physics , 2015, Abstract: We study the weak interaction axial form factors of the octet baryons, within the covariant spectator quark model, focusing on the dependence of four-momentum transfer squared, Q^2. In our model the axial form factors G_A(Q^2) (axial-vector form factor) and G_P(Q^2) (induced pseudoscalar form factor), are calculated based on the constituent quark axial form factors and the octet baryon wave functions. The quark axial current is parametrized by the two constituent quark form factors, the axial-vector form factor g_A^q(Q^2), and the induced pseudoscalar form factor g_P^q(Q^2). The baryon wave functions are composed of a dominant S-state and a P-state mixture for the relative angular momentum of the quarks. First, we study in detail the nucleon case. We assume that the quark axial-vector form factor g_A^q(Q^2) has the same function form as that of the quark electromagnetic isovector form factor. The remaining parameters of the model, the P-state mixture and the Q^2-dependence of g_P^q(Q^2), are determined by a fit to the nucleon axial form factor data obtained by lattice QCD simulations with large pion masses. In this lattice QCD regime the meson cloud effects are small, and the physics associated with the valence quarks can be better calibrated. Once the valence quark model is calibrated, we extend the model to the physical regime, and use the low Q^2 experimental data to estimate the meson cloud contributions for G_A(Q^2) and G_P(Q^2). Using the calibrated quark axial form factors, and the generalization of the nucleon wave function for the other octet baryon members, we make predictions for all the possible weak interaction axial form factors G_A(Q^2) and G_P(Q^2) of the octet baryons. The results are compared with the corresponding experimental data for G_A(0), and with the estimates of baryon-meson models based on SU(6) symmetry.
E. J. Beise Physics , 2005, DOI: 10.1140/epjad/s2005-04-009-y Abstract: The parity violation programs at MIT-Bates, Jefferson Lab and Mainz are presently focused on developing a better understanding of the sea-quark contributions to the vector matrix elements of nucleon structure. The success of these programs will allow precise semi-leptonic tests of the Standard Model such as that planned by the QWeak collaboration. In order to determine the vector matrix elements, a good understanding of the nucleon's axial vector form factor as seen by an electron, G^e_A, is also required. While the vector electroweak form factors provide information about the nucleon's charge and magnetism, the axial form factor is related to the nucleon's spin. Its Q2=0 value at leading order, g_A, is well known from nucleon and nuclear beta decay, and its precise determination is of interest for tests of CKM unitarity. Most information about its Q2 dependence comes from quasielastic neutrino scattering and from pion electroproduction, and a recent reanalysis of the neutrino data have brought these two types of measurements into excellent agreement. However, these experiments are not sensitive to additional higher order corrections, such as nucleon anapole contributions, that are present in parity-violating electron scattering. In this talk I will attempt to review what is presently known about the axial form factor and its various pieces including the higher order contributions, discuss the the various experimental sectors, and give an update on its determination through PV electron scattering.
Physics , 2014, DOI: 10.1103/PhysRevD.90.074001 Abstract: We have calculated the axial-vector form factors of the low lying octet baryons ($N$, $\Sigma$, $\Xi$ and $\Lambda$) in the chiral constituent quark model ($\chi$CQM). In particular, we have studied the implications of chiral symmetry breaking and SU(3) symmetry breaking for the singlet ($g^0_{A}$) and non-singlet ($g^3_{A}$ and $g^8_{A}$) axial-vector coupling constants expressed as combinations of the spin polarizations at zero momentum transfer. The conventional dipole form of parametrization has been used to analyse the $Q^2$ dependence of the axial-vector form factors ($G^0_{A}(Q^2)$, $G^3_{A}(Q^2)$ and $G^8_{A}(Q^2)$). The total strange singlet and non-singlet contents ($G_s^0(Q^2)$, $G_s^3(Q^2)$ and $G_s^8(Q^2)$) of the nucleon determining the strange quark contribution to the nucleon spin ($\Delta s$) have also been discussed.
Physics , 1999, Abstract: The electromagnetic form factors of the nucleon are calculated in an extended chiral constituent-quark model where the effective interaction is described by the exchange of pseudoscalar, vector, and scalar mesons. Two-body current-density operators, constructed consistently with the extended model Hamiltonian in order to preserve gauge invariance and current conservation, are found to give a significant contribution to the nucleon magnetic form factors and improve the estimates of the nucleon magnetic moments.
Physics , 1998, DOI: 10.1103/PhysRevC.60.025206 Abstract: The electromagnetic form factors of the nucleon have been calculated in a chiral constituent-quark model. The nucleon wave functions are obtained by solving a Schr\"odinger-type equation for a semi-relativistic Hamiltonian with an effective interaction derived from the exchange of mesons belonging to the pseudoscalar octet and singlet and a linear confinement potential. The charge-density current operator has been constructed consistently with the model Hamiltonian in order to preserve gauge invariance and to satisfy the continuity equation.
Physics , 1995, DOI: 10.1103/PhysRevC.52.1061 Abstract: We examine isospin breaking in the nucleon wave functions due to the $u - d$ quark mass difference and the Coulomb interaction among the quarks, and their consequences on the nucleon electroweak form factors in a nonrelativistic constituent quark model. The mechanically induced isospin breaking in the nucleon wave functions and electroweak form factors are exactly evaluated in this model. We calculate the electromagnetically induced isospin admixtures by using first-order perturbation theory, including the lowest-lying resonance with nucleon quantum numbers but isospin 3/2. We find a small ($\leq 1\%$), but finite correction to the anomalous magnetic moments of the nucleon stemming almost entirely from the quark mass difference, while the static nucleon axial coupling remains uncorrected. Corrections of the same order of magnitude appear in charge, magnetic, and axial radii of the nucleon. The correction to the charge radius in this model is primarily isoscalar, and may be of some significance for the extraction of the strangeness radius from e.g. elastic forward angle parity violating electron-proton asymmetries, or elastic ${}^4He({\vec e},e')$ experiments.
Physics , 2002, DOI: 10.1016/S0375-9474(02)01389-1 Abstract: We discuss the axial form factors of the nucleon within the context of the nonrelativistic chiral quark model. Partial conservation of the axial current (PCAC) imposed at the quark operator level enforces an axial coupling for the constituent quarks which is smaller than unity. This leads to an axial coupling constant of the nucleon $g_A$ in good agreement with experiment. PCAC also requires the inclusion of axial exchange currents. Their effects on the axial form factors are analyzed. We find only small exchange current contributions to $g_A$, which is dominated by the one-body axial current. On the other hand, axial exchange currents give sizeable contributions to the axial radius of the nucleon $r_A^2$, and to the non-pole part of the induced pseudoscalar form factor $g_P$. For the latter, the confinement exchange current is the dominant term.
Page 1 /100 Display every page 5 10 20 Item
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2019-11-20 11:18:23
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https://www.rdocumentation.org/packages/subsemble/versions/0.0.9
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# subsemble v0.0.9
0
0th
Percentile
## An Ensemble Method for Combining Subset-Specific Algorithm Fits
Subsemble is a general subset ensemble prediction method, which can be used for small, moderate, or large datasets. Subsemble partitions the full dataset into subsets of observations, fits a specified underlying algorithm on each subset, and uses a unique form of V-fold cross-validation to output a prediction function that combines the subset-specific fits. An oracle result provides a theoretical performance guarantee for Subsemble.
## Functions in subsemble
Name Description subsemble-package An Ensemble Method for Combining Subset-Specific Algorithm Fits subsemble An Ensemble Method for Combining Subset-Specific Algorithm Fits predict.subsemble Predict method for a 'subsemble' object. No Results!
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2020-09-19 08:42:39
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https://www.tefaafrik.com/Watch/Topic/186/Video
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###### Welcome
Medicine: Physics: Equilibrium
Rate:
### Equilibrium
1.4 Equilibrium
A particle or body is said to be in equilibrium when all the forces acting on it balance and it is no in motion so to say, its resultant moment is zero. Algebraically, this just means that the vector sum of the forces is zero:$\sum&space;F=0$or, equivalently, the components of the vectors in three directions (which must be linearly independent, of course, but not necessarily orthogonal) sum to zero. Geometrically, this means that the vectors representing the forces (in both direction and magnitude) can be joined to form a closed polygon.
In order to determine whether a particle is in equilibrium (or, given that it is in equilibrium, to determine an unknown force) we have to check that the vector sum of the forces, i.e. the resultant force, is zero. That means that the resultant force should have no non-zero component in any direction. Normally, the way to check this is to check the components of the resultant force in three independent directions, which need not be orthogonal but are usually, for convenience, orthogonal. This process is called resolving forces. It can best be understood in a concrete example.
Example
A particle of weight 3W lies on a fixed rough plane inclined at angle α to the horizontal. It is held in position by a force of magnitude T acting up the line of greatest slope of the plane. Find the frictional force, F, in terms of W, α and T
The strategy for all similar problems is to determine the equations of equilibrium by resolving (i.e. taking components of the vectors) forces in two directions and equating to zero. It helps tochoose the directions carefully in order to reduce the number of terms in each equation. Clearly, for our problem, it is a good plan to resolve parallel and perpendicular to the plane.We have, respectively
T = F +W $\sin&space;\alpha$ (1)
R = W$\cos&space;\alpha$ (2)
Thus F = T −W $\sin&space;\alpha$, using only the first equation. Normally, we are interesting in finding the value of T that will support the particle on the plane. To accomplish this, we have to know something about the frictional force. The experimental result relating the frictional force to the normal reaction
F = μR (3)
is generally used. Here μ is the coefficient of friction, the value of which depends on the surfaces involved. When the equality holds, the friction is said to be limiting.In our example, combining equations (1) and (2) with the experimental law (3) gives
$T\leq&space;W(\sin&space;\alpha$$\cos&space;\alpha&space;)$
Note that in the case of limiting friction, T is determined by this equation. If, instead of assuming that the particle is tending to slip up the plane, we assume that it is tending to slip down the plane, then the frictional force would act up the plane. In this case (check this!) we find
$T\geq&space;W(\sin&space;\alpha&space;-$μ$\cos&space;\alpha&space;)$
and combining the two results gives the range of values of $T$ for which equilibrium is possible, for a given value of μ:
$W(\sin&space;a-$μ$\cos&space;\alpha&space;)\leq&space;T\leq&space;W(\sin&space;\alpha&space;+$μ$\cos&space;\alpha&space;)$
Not surprisingly, in order for the particle to remain in equilibrium i.e. not move T cannot be too big or too small. Note that if $T$ were given (for example, if it were the tension in a string that passes over a pulley and has a weight dangling on the other end) the above equation would give bounds on the values of α allowed for equilibrium.
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2021-11-28 10:48:02
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https://chengtsolin.wordpress.com/
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Real-time wavelet decomposition using compute shader
Check out this post to have a short introduction about Haar wavelets.
In order to do IBL using wavelet triple product integral, we need to represent env light, BRDF, and per-vertex visibility by wavelet basis functions, meaning we need to do wavelet transform on these signals.
For BRDF and visibility in static scenes, we could precompute their wavelet transform using CPU, in offline; for lighting we have to compute the wavelet transform in run-time, because the lighting is dynamically-sampled in each frame.
OpenGL compute shader gives me good incentives to implement 2D Haar wavelet transform in it. Firstly it is an independent compute stage outside graphics pipeline, so I don’t need to supply primitives or setup those graphics pipeline states that are unrelated to the core algorithm, secondly it provides shared memory access for inter-thread communication, just like CUDA or OpenCL, and finally, unlike CUDA or OpenCL, it can access OpenGL buffer objects just like other GLSL shaders.
In my implementation, in each frame the light is firstly sampled into an octahedral parameterization using fragment shader, and non-standard wavelet decomposition is performed using compute shader.
Here is the result of 1-level decomposition on the dynamically-sampled light:
(Note that right now the 3D scene you see is still rendered using pre-filtered env map)
And the following is the full-level decomposition:
Short intro. to Haar wavelet transform
Short introduction to Haar Wavelets:
Haar basis was developed by Alfred Haar in 1909. It is the simplest yet the most widely adopted wavelet basis. For clarity, we first start from one dimensional Haar basis. Given a 1D discrete signal, we can represent it by a combination of average and difference (or detail), like the following example:
The functions in the right hand side can be expressed as
$5 \times \phi(x) + 4 \times \psi(x)$,
$\phi(x) and \psi(x)$ are called mother scale function and mother wavelet function, respectively. Other basis functions could be derived by scaling and translating them. Here are some examples:
For a 1-D discrete signal with length N (N>2), we can repeat this average & difference process to obtain 1 scale coefficient and N-1 detail coefficients.
The following is an example of transforming a 1-D discrete signal [ 9 7 3 5 ]
Thus the wavelet transform of [ 9 7 3 5 ] is given by [ 6 2 1 -1].
Less significant detail coefficients could be discarded for data compression purposes, like the following image shows:
To perform 2D Haar wavelet transform, we can simply do full 1D transform along one dimension and then do another full 1D transform along the other dimension. This is called standard decomposition.
Or, we can perform 1D Haar transform alternatively along different dimensions, in each level. This is called nonstandard decomposition.
Non-standard decomposition has some desired properties, such fewer assignments are required than in standard decomposition, and square local supports for every basis functions.
One application of 2D Haar wavelet is image compression. Here is an image reconstruction example using different numbers of coefficients:
It’s an example produced by my own implementation, and the coefficient pickup scheme is naive (just pick those with large magnitude), so it does not yield best compression quality.
References:
1. Eric J. Stollnitz Tony D. DeRose David H. Salesin, Wavelets for Computer Graphics: A Primer.
Project cubemap to octahedral map
Because compressing a spherical function sampled in cubemap form requires us to do 2D Haar Wavelet transform 6 times, one for each cubemap face, to simplify this transformation step, Wang et al. [1] propose using a single 2D octahedral map to represent a spherical function. So by using octahedral map to store our per-vertex visibility, we only need to do Haar Wavelet transform (# of vertices) times, whereas cubemap representation requires (6 x # of vertices) times.
In my implementation, I simply use the converting method provided in [2] and [3] to convert 2D coordinates in an octahedral map to 3D direction vectors.
The following screenshot shows the visibility cubemap of a vertex, and the octahedral map converted from that cubemap.
It’s not easy to imagine what the transformation looks like on a black and white image; I also made a screenshot showing the converted environment cubemap.
References:
1. Rui Wang, Ren Ng, David Luebke, Greg Humphreys, Efficient Wavelet Rotation for Environment Map Rendering, Eurographics Symposium on Rendering (2006)
2. Cigolle, Donow, Evangelakos, Mara, McGuire, Meyer, A Survey of Efficient Representations for Independent Unit Vectors, Journal of Computer Graphics Techniques (JCGT), vol. 3, no. 2, 1-30, 2014
3. Quirin Meyer, Jochen Süßmuth, Gerd Sußner, Marc Stamminger, and Günther Greiner. 2010. On floating-point normal vectors. In Proceedings of the 21st Eurographics conference on Rendering
Per-vertex visibility cubemaps generation
Image-based lighting (IBL) using triple-product integral technique factors lighting, BRDF, and visibility into three independently sampled functions. The rendering equation is evaluated per-vertex, that is, doing triple-product integral of those three functions on each vertex position. Usually one Lighting function, represented in the form of an environment map, is sampled at run time and applied to all shading vertices, and by utilizing precomputed wavelet-rotation matrices (see my previous post, or the original paper), we only need one copy of BRDF function defined in local frame for each material. Only visibility have to be sampled individually on each vertex position. Graphics hardware could be used to help us generate these visibility “cubemap” on each vertex position. Like the traditional old way of generating a reflection environment map in a certain position, to generate the visibility cubemap for a vertex we could position the camera in that vertex’s position, and render the scene 6 times facing 6 different directions. (+x,-x,+y,-y,+z,-z) We can improve the efficiency of the creation of these per-vertex visibility cubemaps, by utilizing the newer GPU and OpenGL capabilities. First, we can use layered rendering to render the 6 cubemap faces in one pass. By attaching a cubemap texture to the FBO, we can, in geometry shader, emit primitives to all cubemap faces. Here is a simple geometry shader code:
void main()
{
for( int i = 0; i < 6; ++i )
{
gl_Layer = i;
gl_Position = u_mvps[i] * gl_in[0].gl_Position;
EmitVertex();
gl_Position = u_mvps[i] * gl_in[1].gl_Position;
EmitVertex();
gl_Position = u_mvps[i] * gl_in[2].gl_Position;
EmitVertex();
EndPrimitive();
}
}
Built-in output variable gl_Layer controls which layer (or cubemap face, if the render target is a cubemap texture) the primitive will be sent to. Furthermore, we could potentially increase the performance by using instancing geometry shader. Instancing makes the geometry shader execute multiple time on the same input primitive. Here is a sample code:
layout ( triangles, invocations=6 ) in;
layout ( triangle_strip, max_vertices = 3 ) out;
...
void main()
{
gl_Layer = gl_InvocationID;
gl_Position = u_mvps[gl_InvocationID] * gl_in[0].gl_Position;
EmitVertex();
gl_Position = u_mvps[gl_InvocationID] * gl_in[1].gl_Position;
EmitVertex();
gl_Position = u_mvps[gl_InvocationID] * gl_in[2].gl_Position;
EmitVertex();
EndPrimitive();
}
You can see in the input layout qualifier we specify the GS to be invoked 6 times for each primitive. The invocation id (0~5) could be retrieved via gl_InvocationID. I say “potentially increase the performance” because if the number of input primitives is far larger than that of shader cores, instancing seems not that helpful. I need to do a timing test see if there is difference. Finally, for per-vertex visibility cubemaps generation, since we know all the rendering parameters beforehand, we can store them in a shader storage buffer or texture buffer, and use instancing rendering to eliminate the redundant API calls. For a scene containing 150000 vertices, using naive rendering loops requires 150000 draw API calls. Ideally, just one API draw call is ever needed when instanced rendering is used. However, because we need to retain each vertex’s visibility map, we need to switch FBO or FBO attachments within the rendering iterations. Allocating 150000 textures is also not a good idea. Because 6x32x32 resolution is usually enough for a vertex’s visibility cubemap, we can somewhat alleviate this problem by using large-dimension, like 6x8192x8192, textures and groups a number of visibility fields into one large texture. And we do instancing rendering with these large textures serving as render target. The number of instancing draw API calls is that of the large textures.
In my test scene, which has 175000 vertices, the visibility cubemap generation only takes a few seconds in a machine featuring a nVidia GTX670; I haven’t precisely measured it though.
Visibility field for a vertex
Wavelet rotation for image-based lighting, and per-vertex visibility precomputation using OpenGL 4.x features
The triple product wavelet integral IBL proposed by Ng et al. [1] enables changing lighting as well as viewing directions by separately precomputing BRDF and visibility functions. In their implementation, BRDF, visibility, and lighting functions are all sampled in the same global frame. Because BRDFs are usually defined in local frame, Ng et al. precompute BRDFs for a set of orientations (defined in global frame), that is, each single material BRDF has multiple copies for different orientations. During runtime, when shading a vertex, the surface normal is used for choosing and interpolating BRDFs with closest orientations. This approach requires huge memory for storing precomputed BRDFs set.
Wang et al. [2] proposes a computational approach for efficient wavelet rotation. First, they choose octahedral mapping for parameterizing the spherical domain, because it has good uniformity and reduces wavelet rotation to translation in 2D.
Second, they precompute a wavelet rotation matrix R per orientation (So they get a bunch of rotation matrices). A wavelet rotation matrix is very sparse because of the compact local support of wavelet bases. Quantization could further reduce the number of non-zero elements. If we have 32×32 predefined normal orientations, we got 1024 rotation matrices.
These wavelet rotation matrices are independent of BRDF or lighting or visibility, so they could be computed, stored in disk, and loaded into memory when the application starts. And we only need one BRDF copy for each material.
In runtime, the light function is dynamically sampled and wavelet-transformed, and then rotated by those precomputed wavelet rotation matrices to obtain different rotated versions of lighting functions (or more specifically, wavelet coefficients).
These rotated versions of lighting wavelet coefficients will then be used for performing shading in each vertex, that is, doing dot-products with BRDFs defined in local frame.
Wang et al. [2] doesn’t take into account visibility, so they only deal with dot-products when doing shading. They perform shading entirely on CPU; SSE instructions is enough to achieve near-interactive frame rate.
——————————————————————————————————————
In order to incorporate visibility and perform wavelet triple product integral, we have to precompute the visibility function on each vertex, that is, sampling visibility in spatial domain and wavelet-transforming it. With OpenGL 4.x features, we might be able to shrink the precomputation time to acceptable levels.
Back in mid 2000 we had to make 6 drawing API calls per vertex position to generate visibility cubemaps for each vertex; today we could use multi-draw indirect to greatly reduce the number of CPU making expensive API calls.
Second, layered rendering with geometry shader make it possible render all 6 cubemap faces in one draw.
Third, bindless textures reduces the overhead of repeatedly binding/unbinding those visibility textures when doing wavelet transform on GPU.
References:
1. Ren Ng, Ravi Ramamoorthi, Pat Hanrahan, Triple Product Wavelet Integrals for All-Frequency Relighting, Siggraph 2004
2. Rui Wang, Ren Ng, David Luebke, Greg Humphreys, Efficient Wavelet Rotation for Environment Map Rendering, Eurographics Symposium on Rendering (2006)
Prefiltered Environment Maps On Glossy Materials
For Phong BRDF materials, we can do convolution on each environment map texel over its surrounding texels covered by a designated specular lobe. We can store different prefiltered environment maps corresponding to different specular powers in a mipmap chain. Mipmap level 0 contains the envmap w.r.t. highest specular power, mipmap level 1 contains the envmap w.r.t. 2nd highest specular power, and so forth.
Again we can use cubemapgen to help us generate this multilevel cubemap texture.
In a GLSL shader, we use reflective vector to sample the environment map, as opposed to the use of normal vector in irradiance maps for Lambertian surfaces. There are different ways to derive the desired mipmap level, based on how the mipmap chain is generated. For instance, cubemapgen offers a method by which you specify the maximum specular power (mipmap level 0), and the decreasing ratio for the subsequent mipmap levels. Then, the desired mipmap level could be derived using a statement like the following:
float mipmapLevel = log( materialSpecular/ MAX_SPECULAR_POWER) / log(POWER_DEC_RATIO);
Then we can sample the texture using texturLod, which allows us to specify the desired mipmap level.
To enable smooth transition, we also have to enable linear filtering between mipmap levels for this cubemap texture.
The following screenshots are my rendering results after adding Phong specular contribution to the final shading.
Specular power 10:
Specular power 60:
References:
1. Jan Kautz, Pere-Pau Vázquez, Wolfgang Heidrich, and Hans-Peter Seidel. 2000. Unified Approach to Prefiltered Environment Maps. In Proceedings of the Eurographics Workshop on Rendering Techniques 2000
2. https://seblagarde.wordpress.com/2012/06/10/amd-cubemapgen-for-physically-based-rendering/
Shadowing In IBL Using Sparse 3D Textures
To accurately compute shadow in a scene with omni-directional light sources, each shaded point has to shoot many shadow rays toward the surrounding environment to test light visibility (if a shadow ray can reach the right source without hitting a blocking geometry. To do these occlusion tests we need to know the whole scene geometry, and it is not trivial to access the information about scene geometry.
In an effort to combine voxel-based GI with IBL, I made an experiment of casting shadows using scene geometry represented by sparse 3D textures:
To represent this “buddha on a plane” using an ordinary 512x512x512x8bit 3D texture requires 128MB memory, whereas a sparse 3D texture consumes only 12.25MB. In this “buddha on a plane” scene, only 196 texel pages (each page has 64x32x32 texels in a 8-bit 3D texture) are resident (occupying memory). By using sparse texture, we can use the maximal 3D texture resolution without consuming too much memory. Furthermore, only 1 bit per voxel is enough for storing the occlusion information, so a 8-bit texel could store 2x2x2 occlusion voxels, which further reduces the memory usage.
With this scene representation, I was able to cast shadow rays on each surface point. The following screenshot is the result of casting 18 shadow rays on each surface point:
Right now this naive approach has several obvious drawbacks:
1. Unlike ambient occlusion, which only tests occlusion within a short distance, shadow rays for environment map lighting have to traverse all way to the scene boundary, that makes the traversal costly.
2. Many, many times of texture fetching is required in the fragment program, and it is also very costly.
3. This approach can only be used on diffuse materials. To accurately compute lighting integrals on glossy materials, visibility functions have to be incorporated into the integrals.
A brief introduction to triple-product wavelet image-based relighting
Precomputed Radiance Transfer (PRT) proposed by Sloan et al. [1] enables interactive image-based lighting that takes into account of soft shadows and indirect illumination under low-frequency dynamic lighting environments. PRT precomputes light transport functions capturing the way an object shadows scatters, and reflects light, and encodes them by low-order spherical harmonics (SH) basis functions, which makes them to be represented in compact vector form. Therefore, rendering becomes a simple inner product of a light vector which is sampled run-time and also represented by SH basis, with the precomputed transport vector. For glossy BRDFs, a transport matrix instead of vector is precomputed for every vertex to allow for view-dependent shading.
Similar to Fourier series, low-order SH can only approximate low-frequency signals, and is not an ideal candidate for efficiently encoding high frequency signals like sharp light and hard shadows.
Ng et al. [2] broke the low-frequency limitation by using non-linear wavelet approximation to represent the lighting and transport vectors, achieving all-frequency illumination and shadows at interactive rates. Wavelets require far fewer coefficients than SH to approximate high-frequency data. This technique was developed primarily for image relighting. For changing viewpoint, it was limited to diffuse objects because of the need of sampling in viewing direction. This additional sampling is required for every vertex, which is too costly to be done.
The following image is a comparison between SH (left) and Haar wavelets (right) PRT from Ng et al. [2] (SH and Haar both uses 100 coefficients):
Ng et al. [3] factors light, visibility, and BRDF into 3 independent components, and thus realizes all-frequency dynamic light as well as changing view.
The following portion is a brief explanation of their work.
Considering the rendering equation without indirect illumination, the equation
$L_o(x,\omega_o)=\int_{\Omega}L(x,\omega_i) f_r(x,\omega_i,\omega_o)V(x,\omega_i)(n \cdot \omega_i)d\omega_i$
describe outgoing readiance $L_o$ at surface location $x$ in viewing direction $\omega_o$, where $L$ is the light source, $\omega_i$ is incident light direction, $f_r$ is BRDF, n is surface normal, and V visibility. The integral is over the visible or upper hemisphere $\Omega$. Here we make several simplifications. First, we incorporate the cosine term in the BRDF definition. Second, we assume distant illumination, making $L$ independent of surface location. Third, we only consider spatially uniform BRDF. Finally, we define all functions in a global coordinate system. However, BRDFs are typically defined in local coordinates with respect to surface normals, making rotation operations necessary when applying the BRDF in the global coordinate system to different orientations of surface normals. By the above simplification, the rendering equation can be expressed as
$L_o(x,\omega_o)=\int_\Omega L(\omega_i) f_r (\omega_i, \omega_o) V(x,\omega_i) d\omega_i$
where the integrand is a prodcut of three functions. If we consider only one vertex in fixed viewpoint, the equation can be further expressed as
$L_o = \int_\Omega L(\omega_i)f_r(\omega_i)V(\omega_i)d\omega_i$
The functions $L, f_r,$ and $V$ can then be expended in appropriate orthonormal basis functions $B(\omega)$:
$L(\omega)=\sum\limits_{i} L_{i} B_{i} (\omega)$,
$f_r(\omega) = \sum\limits_j f_{rj}B_j (\omega)$,
$V(\omega) = \sum\limits_k V_k B_k(\omega)$,
where $L_i$, $f_{rj}$, and $V_k$ are coefficients of corresponding basis functions. With this basis expansion, we can write the simplified rendering equation as:
$L_o = \int_{\Omega} \left( \sum\limits_i L_i B_i (\omega) \right) \left( \sum\limits_{j} f_{rj} B_j (\omega) \right) \left( \sum\limits_{k} V_k B_k (\omega) \right) d\omega$
$=\sum\limits_{i}\sum\limits_{j}\sum\limits_{k} L_{i} f_{rj} V_k \int_{\Omega} B_{i}(\omega) B_{j}(\omega) B_{k} (\omega) d\omega$
$=\sum\limits_{i}\sum\limits_{j}\sum\limits_{k} C_{ijk}L_{i}f_{rj}V_{k}$,
where the triple product tensor, $C_{ijk}$, is defined by
$C_{ijk}=\int_{\Omega}B_{i}(\omega)B_{j}(\omega)B_{k} (\omega) d\omega$
Ng et al.[1] called it tripling coefficient. Due to the presence of tripling coefficient, the evaluation of the above equation is very complicated. Generally we need consider all $(i,j,k)$ combinations, making the complexity $O(N^3)$. Ng et al.[1] proved that the tripling coefficient of 2D Haar basis is nonzero if and only if certain criteria are met [1]. So instead of $O(N^3)$ complexity, the complexity of evaluating the equation in Haar basis is $O(N log N)$, specifically the exact number of non-zero Haar tripling coefficients for the first N basis functions is $2 - N + 3 N log_4 N$. Furthermore, the evaluation could be reduced to linear time complexity by dynamic programming.
In the work of Ng et al.[3], the light function is sampled run-time from an environment map; the visibility function is precomputed beforehand, by sampling an occlusion cubemap in each vertex position; the BRDF function is precomputed in a predefined set of orientations. When rendering, the BRDF for a specific orientation is obtained from interpolating the precomputed set of BRDFs.
Here is a sample of these 3 functions, from [3], shown in cubemap form:
And the following is a sample rendered result from the work of Ng et al.[3]:
Ma et al. [5] uses spherical wavelets and compute the triple product integrals in local coordinate to eliminate the need of BRDF interpolation and greatly reduces the data size.
Although the higher fidelity and run-time interactivity can be achieved by triple-product integrals with Haar wavelet basis, its high precomputation costs and data size prevents it from being adopted by graphics industry, which adopts SH-based rendering widely nowadays because of its simplicity and low costs.
References:
1. Sloan, P.-P., Kautz, J., and Snyder, J., Precomputed radiance transfer for
real-time rendering in dynamic, low-frequency lighting environments, in
Proceedings of the 29th annual conference on Computer graphics and
interactive techniques. 2002, ACM Press: San Antonio, Texas.
2. Ng, R., Ramamoorthi, R., and Hanrahan, P., All-frequency shadows using
non-linear wavelet lighting approximation, in ACM SIGGRAPH 2003
Papers. 2003, ACM Press: San Diego, California.
3. Ng, R., Ramamoorthi, R., and Hanrahan, P., Triple product wavelet
integrals for all-frequency relighting, in ACM SIGGRAPH 2004 Papers.
2004, ACM Press: Los Angeles, California.
4. Ramamoorthi, R., Precomputation-Based Rendering, Foundations and Trends in
Computer Graphics and Vision Vol. 3, No. 4 (2007) 281–369.
IBL on Lambertian surfaces using irradiance environment map
This is the 1st stage of my real-time image-based light (IBL) renderer work. In this part, a basic asset-loading and camera system is constructed. And irradiance environment maps are used to shade diffuse surfaces.
Environment maps are commonly used as light sources in image-based light (IBL). In a non-filtered environment map, each texel represents a single discrete light source.
If the object being shaded is perfect mirror-like , only one texel on the environment map needs to be sampled for each surface point on that object. (texel sampled over the direction of specular reflection)
However, a certain range of light need to be sampled for shading other types of material (if we don’t consider refraction). For example, to shade Lambertian surfaces, the area of light source needs to be sampled is the whole upper-hemisphere over the shaded point.
$L=Kd \sum_{i \subset \Omega} max(0, D_i \cdot N) l_i$
For Lambertian surfaces, the area of light that needs to be sampled is solely determined by the surface normal. So we can create an “pre-filtered” environment map in which each texel value is $\sum_{i \subset \Omega} max(0, D_i \cdot N) l_i$. Basically it is a convolution operation over the whole environment map for each environment map texel. Then in runtime renderinig, only one texel lookup in the prefiltered environment is needed to shade the Lambertian surfaces.
To do this convolution operation in spatial domain is very slow. Because diffuse irradiance values vary very slowly w.r.t. to normal change. We can transform the environment map into frequency space and only keep low-frequency information. A spherical function (e.g. environment map) could be represented in frequency domain using spherical harmonic transform, and a few SH bases is enough to represent the low frequency data of the whole environment map required by diffuse lighting. This reduces the convolution complexity from $6 \times m \times m \times 6 \times m \times m$ to $k \times 6 \times m \times m$ (m is the dimension of cubemap faces, and k is the number of preserved SH coefficients)
So what we need to do is to transform the spatial domain environment to frequency domain by SH transform, preserve only a few most significant basis coefficients, perform convolution with Lambertian reflectance function (precomputed, also represented in frequency domain) and transform the computed environment map back to spatial representation.
CubemapGen, an environment map manipulation tool, could help you do the pre-filtering process.
Here is the result with 25 SH coefficients preserved:
And this is the rendering result of Lambertian surfaces:
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2021-04-18 20:52:48
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https://math.stackexchange.com/questions/1080408/factoring-a-polynomial-over-mathbb-f-28
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Factoring a polynomial over $\mathbb F_{2^8}$
How do you find the factors of $x^4+x+1$ in $GF(2^8)$ in terms of polynomials?
Let me explain, We have primitive irreducible polynomial $p(x)=x^2+x+1$ in $GF(2^2)$ which has root $\alpha^2+\alpha$ in $GF(2^4)$.
Example: Using $\alpha^2+\alpha$ in $p\left(\alpha^2+\alpha\right)=0 \mod x^4+x+1$ (where $x^4+x+1$ is primitive irreducible polynomial in $GF(2^4)$.
Then how we can find $x^4+x+1$ roots in $GF(2^8)$ that generate subgroup of degree $15$ which is simple emending of $GF(2^4)$ in $GF(2^8)$?
• It has no factors in this GF – sashas Dec 25 '14 at 7:59
• The polynomial $x^4+x+1$ splits into linear factors already in the subfield $GF(2^4)\subset GF(2^8)$. Are you looking for a way to identify a copy of that subfield? Do you mean a subgroup of order 15? The answer depends on how you have constructed the field $GF(2^8)$. If you don't tell us how you did that, we cannot help. I can guess that you are using either the methof used in AES crypto or QuickResponse codes, but there are others. You need to co-operate a bit. – Jyrki Lahtonen Dec 25 '14 at 22:01
• The material linked to in this question and my answer there may help. Emphasis on may, as you need to give the degree 8 polynomial $p(x)$ that you use when you construct $GF(256)$ as $GF(2)[x]/\langle p(x)\rangle$. – Jyrki Lahtonen Dec 25 '14 at 22:11
• Dear please let me know how can i find the roots of that irreducible polynomial in GF(256) that gives me unity on 15th power – user203139 Jan 10 '15 at 9:15
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2019-08-24 11:04:19
|
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|
https://projecteuclid.org/euclid.die/1356123172
|
## Differential and Integral Equations
### Groups of scalings and invariant sets for higher-order nonlinear evolution equations
Victor A. Galaktionov
#### Abstract
Consider a nonlinear $m^{\text{th}}$ order evolution PDE $$u_t = \mathbf A(u) \equiv A(x,u,u_1, \dots ,u_m), \quad u=u(x,t), \quad (x,t) \in Q=(0,1) \times [0,1], \tag*{(*)}$$ where $A$ is a $C^\infty$ function and $u_t = \partial u/\partial t$, $u_i = \partial ^i u / \partial x^i$. If (*) is invariant under a group of scalings with the infinitesimal generator $X =x \frac {\partial}{\partial x} + \mu t \frac {\partial}{\partial t}$ ($\mu$ is a constant scaling order" of $\mathbf A$), then the PDE admits exact self-similar solutions depending on the single invariant variable $u(x,t) = {\theta } (\xi)$, $\xi = x/t^{1/\mu}$, where ${\theta }$ solves a nonlinear $m^{\text{th}}$ order ODE associated with the PDE. We prove that when the operator $\mathbf A$ is composed of a finite sum of operators with different scaling orders, $\mathbf A = \sum \mathbf A_i$, and no group of scalings exists, the exact solutions can be constructed via the invariance of the set $S_0 = \{u: u_1 =F(u)/x\}$ of a contact first-order differential structure, where $F$ is a smooth function to be determined. The time-evolution on $S_0$ is shown to be governed by a first-order dynamical system. We thus observe that besides scaling group properties, the invariance of $S_0$ specifies new sets of solutions described by first-order ODEs. The approach applies to a class of nonlinear parabolic equations of the second and of the fourth order.
#### Article information
Source
Differential Integral Equations, Volume 14, Number 8 (2001), 913-924.
Dates
First available in Project Euclid: 21 December 2012
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2019-01-20 01:38:31
|
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https://scipost.org/submissions/scipost_202101_00004v1/
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# Electric manipulation of domain walls in magnetic Weyl semimetals via the axial anomaly
### Submission summary
As Contributors: Jens H Bardarson · Yago Ferreiros · Julia Hannukainen Preprint link: scipost_202101_00004v1 Date submitted: 2021-01-08 11:06 Submitted by: Hannukainen, Julia Submitted to: SciPost Physics Academic field: Physics Specialties: Condensed Matter Physics - Theory Approach: Theoretical
### Abstract
We show how the axial (chiral) anomaly induces a spin torque on the magnetization in magnetic Weyl semimetals. The anomaly produces an imbalance in left- and right-handed chirality carriers when non-orthogonal electric and magnetic fields are applied. Such imbalance generates a spin density which exerts a torque on the magnetization, the strength of which can be controlled by the intensity of the applied electric field. We show how this results in an electric control of the chirality of domain walls, as well as in an improvement of the domain wall dynamics, by delaying the onset of the Walker breakdown. The measurement of the electric field mediated changes in the domain wall chirality would constitute a direct proof of the axial anomaly. Additionally, we show how quantum fluctuations of electronic Fermi arc states bound to the domain wall naturally induce an effective magnetic anisotropy, allowing for high domain wall velocities even if the intrinsic anisotropy of the magnetic Weyl semimetal is small.
###### Current status:
Editor-in-charge assigned
### Submission & Refereeing History
Submission scipost_202101_00004v1 on 8 January 2021
## Reports on this Submission
### Strengths
1 - To my knowledge, this is the first theoretical work to treat the characteristics of domain wall chirality in the context of magnetic Weyl semimetal. This is in a clear contrast with the situation for the domain wall motion, which was discussed to some extent in Refs.17 and 18. In the context of magnetism and spintronics, chiralities of magnetic textures are attracting a great interest, in connection with the emergent electromagnetism and the spin-wave dispersion. The authors' findings may provide new way in manipulating chiralities of magnetic textures more efficiently in magnetic Weyl semimetals. This point meets one of the acceptance criteria, "3. Open a new pathway in an existing or a new research direction, with clear potential for multipronged follow-up work."
2 -The analysis is performed in a systematic way. The authors derive the effective action for the domain wall by the path integral formalism with respect to the electron fields, which is a well-defined and controlled approach. Once the order of perturbative expansion is specified, this approach is capable in deriving all the possible terms in equilibrium. The effective action derived by this approach can be systematically incorporated in the Euler-Lagrange formalism of the domain wall.
3 - The scientific background and the process of analysis are presented in a self-contained manner. In Section 2, the authors introduce the collective coordinate parametrization and its Euler-Lagrange formalism for conventional magnetic domain walls as the scientific background. Such explanation would be helpful for readers who are not familiar with the theory of magnetic textures.
4 - The physical quantities influenced by their findings are properly estimated. The authors suggest that the axial anomaly effect on the domain wall chirality can be achieved with a magnetic field at the order of 0.01-1T and an electric field up to the order of a few MV/m (namely a few volts per micrometer), which are well available in experiments. The estimated domain wall velocity up to 2-3km/s is also in a realistic range that can be measured with the Hall transport or the magneto-optical imaging techniques.
### Weaknesses
1 - The derivation of the effective action for magnetic textures and electromagnetic field, using the path-integral formalism, is systematic and straightforward. However, it is capable in capturing only the equilibrium effect, and cannot fully derive the effect from out-of-equilibrium perturbations. For example, the spin-transfer torque (namely the torque from the transport current) would also be present, as pointed out in Ref. 17 and mentioned in Eq.(4), this torque is not captured in the effective action derived in Section 3. To show that the path-integral approach is good enough for their calculation, the authors should clarify their motivation in employing this approach.
2 - I suspect that there are some other equilibrium effects that are missing in the effective Lagrangian. For example, while the authors derive the axial separation effect contribution in Eq.(28), the contribution from the chiral separation effect, namely the axial current $\boldsymbol{J}_5 \propto \mu \boldsymbol{B}$, is missing. Since they are taking into account the external magnetic field, the spin torque induced by this axial current would also be present. The authors should clarify what kind of condition and approximation they have used to achieve Eq.(25).
3 - In Section 4A, there is no intuitive picture why the anomaly effect favors the Bloch-type chirality rather than the Neel-type. Since most of the readers interested in magnetic domain walls may not be familiar with the physics of Weyl fermions, an intuitive picture in parallel with conventional Heisenberg spin systems would be helpful for those readers to understand this important result. The Dzyaloshinskii-Moriya interaction (DMI), namely the noncollinear spin interaction due to the breaking of inversion symmetry, is often responsible for chiralities of magnetic textures. Since $\mu_5$ in this model breaks inversion symmetry, perhaps the chirality effect may be understood as the effective DMI.
4 - The setups employed in Section 4 seem ambiguous. While the magnetic field in Section 4A is fixed in x- and y-directions, it is not specified in Section 4B. Since Eq.(42) is derived from Eq.(31), perhaps B-field is pointing in z-direction, which is the situation quite different from Section 4A. The authors should specify the directions of $\boldsymbol{E}$ and $\boldsymbol{B}$, and $\boldsymbol{B}_5$ corresponding to the domain wall structure, in each section.
5 - In Appendix A1, they identify the bound states at the domain wall, and derive the effective action corresponding to them. While I find this method reliable, I am not confident of the choice of bound states used for the path integral. The bound states shown in Eqs.(A6)-(A7) are the "Fermi arc" states with zero energy, while there are usually many other bound states with finite energies, as shown in Ref.55. The authors should comment why these finite-energy bound states do not contribute to the effective action. For instance, if the domain wall is thin enough, the finite-energy states are energetically well separated from the zero-energy Fermi arc states, and hence the treatment with only the Fermi arc states would be rationalized.
### Report
In the present manuscript, the authors focus on the dynamics of domain walls in magnetic Weyl semimetals, and theoretically show how the axial anomaly of the Weyl fermions influences the domain wall dynamics. Starting from the low-energy effective model of Weyl electrons coupled with a domain wall, they derive an effective Lagrangian for the domain wall by integrating out the electron degrees of freedom, and obtain the equations of motion in the collective coordinate formalism. With the obtained equations of motion, the authors mainly find two effects arising from the axial anomaly, which are present if the electric and magnetic fields are applied in parallel: (i) The anomaly leads to the shift of the chirality in the domain wall structure in equilibrium. (ii) In the motion of the domain wall driven by the magnetic field, the anomaly tends to suppress the Walker breakdown, namely the saturation of the domain wall velocity due to the dynamics in the domain wall chirality, and enhances the maximum velocity of the domain wall. The authors expect that these new features would be useful in application of magnetic domain walls to logic gate designs, and also in detecting the axial anomaly directly in experiments.
Throughout this manuscript, I have no doubt that the authors employ the scientifically valid procedure, present their results in a clear way, and cite the previous literatures appropriately. I also find several advantages of this manuscript, as listed in the "Strengths" section. From these points, I consider that this manuscript almost meets the Acceptance criteria of SciPost Physics. On the other hand, I would raise several comments and questions, regarding the validity of their theoretical setups and the significance of their findings, as I list up in the "Weaknesses" section. I would encourage the authors to improve these points before publication.
### Requested changes
1 - In Section 4A (left column of page 7), there are statements
"For electric fields larger than this critical value, $E_y > E_c$ ..."
"If the electric field instead is smaller than the critical value ..."
However, I am afraid that the words "large" and "small" are sometimes confusing in this context, since $E_y$ also takes a negative value with large magnitude in their calculations. I would encourage the authors to use some other words.
• validity: high
• significance: ok
• originality: top
• clarity: good
• formatting: excellent
• grammar: perfect
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2021-02-27 16:12:01
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https://scicomp.stackexchange.com/questions/29305/how-to-numerically-minimize-a-functional
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# How to numerically minimize a functional?
How to numerically minimize a functional, for example,
$$J[y]=\int_{x_1}^{x_2}L(x,y(x),y'(x))dx$$
An equivalent problem is to solve the Euler equation for this functional as a differential equation. But is there a numerical method to directly minimize this functional without solving a differential equation?
• Welcome to Scicomp.SE! You omitted some crucial information (structure of $L$, function space to which $y$ belongs, whether there are constraints), but what you are looking for is called "gradient descent". Maybe math.unt.edu/~jwn/flag.pdf is a good introduction; you could also look at doi.org/10.1016/j.na.2008.11.065 and follow the cited literature. – Christian Clason Apr 15 '18 at 13:32
• One way to minimize this is to discretize your $y(x)$ in some way, say by representing it with a polynomial basis with unknown coefficients, and then solving the resulting (nonlinear) optimization problem you get. You can use any typical optimization scheme at that point. – spektr Apr 17 '18 at 12:45
Variational problems like this are special cases of optimal control problems, for which there is a huge literature on solution methods and also a good amount of available software. To express it as in the standard form used in optimal control, we may write $t$ rather than $x$ and consider "system dynamics" $$\dot{y} = u$$ where $u$ is a (fictitious) system input. We can then write the cost functional as $$J = \int_{t_1}^{t_2}L(t,y,u)~dt$$ which is to be minimised by choosing our system input, $u$. It's common in optimal control problems to have more complex problems such as inequality constraints on $y$ or $u$, so this is actually significantly more general than the original calculus of variations problem you posed.
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2019-11-21 03:07:35
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https://iplug2.github.io/docs/md_examples.html
|
iPlug2 - C++ Audio Plug-in Framework
Examples
Todo:
This is obsolete, but here to be updated
## Introduction
The Examples folder contains example/template projects to demonstrate how to use different features of my modified IPlug and WDL. They also serve as a testbed to make sure that the various aspects are working. Each folder contains scripts which automate the build process and package everything along with a pdf manual in an installer. The scripts also code sign binaries/installers and set icons where required.
Visual Studio 2017+ and Xcode 8+, which are both available for free are supported. If you know what you're doing you may be able to get things to compile with other IDEs/versions, you are highly recommended to use the latest versions. No effort is made to make the code compatible with other IDEs, or compilers such as mingw.
Note: although Visual Studio and Xcode required to build IPlug plug-ins, projects are setup in such a way that you may easily use your favourite text editor and call build scripts, to build the binaries (although then you lack the debugger functionality of the IDE).
### Requirements:
Some of these are optional, but without them the build-scripts will need to be edited, otherwise you'll get errors.
Windows
• Microsoft Visual Studio 2017 Community (or one of the premium versions)
Todo:
Add a list of Visual Studio packages required
• Python 2.7 or 3 for running various scripts
• Inno Setup for creating installers
• 7-Zip (if you want to create a ZIP instead of making an installer)
• PACE tools and certificate for code signing AAX binaries (AAX only, consult Avid documentation)
Mac
• Xcode 9 installed, including command-line tools
• Packages for building OSX installers
• setfileicon utility for changing icons
• Mac Developer ID Certificates for code signing installers and dmg files
• PACE tools and certificate for code signing AAX binaries (AAX only, consult Avid documentation)
## Example Projects
• IPlugChunks - shows how to use chunks in a plugin. Chunks allow you to store arbitrary data in the plugin's state as opposed to just a value for each parameter.
• IPlugControls - demos the various IControl classes (example by Captain Caveman)
• IPlugConvoEngine - demos WDL_ConvolutionEngine() - Cockos' fast convolution engine (example by Tale)
• IPlugDistortion - demos Tale's bessel filter implementation for realtime oversampling (example by Tale)
• IPlugEEL - demonstrates using Cockos' EEL library for run-time expression evaluation
• IPlugEffect - The most basic IPlug plugin, a gain control similar to AGain in the VST2 SDK, however it has a GUI
• IPlugGUIResize - has three buttons to choose different gui sizes at runtime
• IPlugHostDetect - displays the host name and version (not very reliable at the moment)
• IPlugMonoSynth - a basic monophonic IPlug synth, showing how to handle MIDI messages sample accurately. Also shows how to use Tale's IKeyboardControl.
• IPlugMouseTest - demonstrates an XY pad IControl which is linked to two plugin parameters
• IPlugMultiChannel - demos a multi-channel IPlug plugin, and how to test if channels are connected
• IPlugMultiTargets - a midi effect plugin that also demos compilation to IOS, getting tempo info, and pop up menus
• IPlugOpenGL - using OpenGL in IPlug
• IPlugPlush - shows how to use Cockos' Plush to do basic 3D Graphics
• IPlugPolySynth - a basic polyphonic IPlug synth
• IPlugResampler - demonstrates using WDL_Resampler
• IPlugSideChain - a plugin that shows how to setup a sidechain input, for VST3, AU
• IPlugText - demos different ways to draw text
The IPlugEffect project is the main starting project I use. If you are not interested in AAX/standalone etc, I suggest you duplicate this and manually remove those targets to give you a new clean starting template with just your preferred formats in it.
Rather than changing settings for individual targets/projects inside the Xcode Project/Visual Studio solutions, most customisations can be done in the xcconfig and property sheets.
## Supported Formats
### VST2
You need to two files from the Steinberg VST2.4 SDK to the folder VST_SDK, see VST_SDK/readme.txt
### VST3
Extract the Steinberg VST3.X.X SDK to the folder VST3_SDK
• On Windows, make sure C:\Program Files\Common Files\VST3 exists, otherwise the copy files build stage will cause the build to fail
### AAX
• Extract AAX_SDK_2XXXX.zip to the AAX_SDK folder
• Also modify ExamplePlugIns/Common/Mac/CommonDebugSettings.xcconfig and ExamplePlugIns/Common/Mac/CommonReleaseSettings.xcconfig...
GCC_VERSION = com.apple.compilers.llvm.clang.1_0 SDKROOT = macosx10.X MACOSX_DEPLOYMENT_TARGET = 10.X ARCHS = x86_64 i386
• In order to compile AAX binaries that run in the release build of Pro Tools, you will need to code-sign those binaries (see Avid documentation)
### AU (AudioUnit v2)
• When building AUs, bear in mind that some hosts keep a cache... see debugging notes below.
• There is a shell script validate_audiounit.command that will run auvaltool with the correct IDs for your plugin (see below) and can also set up the leaks test, which is useful for debugging. Type man auval on the command line for documentation.
• By default I build to the system AU folder /Library/Audio/Plug-Ins/Components/. You will need to have write permissions to this folder. If you want to build to the user AU folder, you'll need to edit the .xcconfig file and also modify the installer scripts
### Standalone
• Audio and Midi is provided via RTAudio and RTMidi by Gary Scavone. To build on windows you need to extract some files into ASIO_SDK, see ASIO_SDK/README.md
## Windows Issues
The template projects use static linking with the MSVC2017 runtime libraries (/MT).
If you change your project to dynamic linking, you'll need to provide the redistributable in your installer, google for "Microsoft Visual C++ 2017 Redistributable Package".
Todo:
This may need to be updated due to recent Microsoft changes
## macOS Issues
Since macOS 10.8 you will need to code-sign your installer and the .app with a valid signature obtained from Apple, to prevent an unidentified developer warning when a user tries to open your installer or dmg file. For the app store you need to add entitlements in order to comply with the sandbox regulations. These things are done by the makedist-* build scripts.
### Debugging
.xcscheme files are set up to use some common hosts for debugging the various formats in Xcode.
To debug an Audiounit using auval, remember to change the auval executable arguments to match plugin's type and IDs:
aufx/aumf/aumu PLUG_UNIQUE_ID PLUG_MFR_ID
AU hosts cache information about the plugin I/O channels etc, so I have added a build script that deletes the caches after a build. If this becomes annoying (it will cause Logic to rescan plugins) you can disable it.
There is also a validate_audiounit.command shell script which will is a helper that runs auval with your plugins' unique IDs, and optionally performs the leaks test.
You should install VSTHost to C:\Program Files\VSTHost\vsthost.exe (on x64 you should install the 64bit version)
To debug AAX, you need to install a development build of Pro Tools. Consult Avid documentation for details.
## Installers & one-click build scripts
The example projects contain shell scripts for both Windows (makedist-win.bat) and OSX (makedist-mac.command) that build everything, code-sign (where relevant) and package the products in an installer including license, readme.txt, changelog.txt and manual. On Windows Inno Setup is used, on OSX - Packages.
A Python script update_version.py is called to get the version from #define PLUG_VER in resource.h. It then updates the info.plist files and installer scripts with the version number (in the format major.minor.bugfix). If you aren't building some components, e.g. AAX, the build scripts may need to be modified.
Please alter the license and readme text and remove my name from the build scripts if you're releasing a plugin publicly.
On OSX the script can also code-sign the standalone app and builds a .pkg for the appstore (commented out).
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2020-05-31 03:37:56
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https://math.hawaii.edu/wordpress/calendar/action~agenda/page_offset~-1/time_limit~1601359199/
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# Calendar
Sep
7
Mon
Labor day
Sep 7 all-day
Sep
8
Tue
Sep 8 @ 4:00 am – 5:00 am
Title: The characterization of Weihrauch reducibility in systems containing $E$-$PA^omega$ + $QF$-$AC^{0,0}$
by Patrick Uftring (TU Darmstadt) as part of Computability theory and applications
Abstract
We characterize Weihrauch reducibility in E-PAω + QF-AC0,0 and all systems containing it by the provability in a linear variant of the same calculus using modifications of Gödel’s Dialectica interpretation that incorporate ideas from linear logic, nonstandard arithmetic, higher-order computability, and phase semantics. A full preprint is available here: https://arxiv.org/abs/2003.13331
Bharat Talwar (University of Dehli) @ Lecture held in Elysium
Sep 8 @ 6:00 am – 8:00 am
Title: Closed Lie Ideals and Center of Generalized Group Algebras
by Bharat Talwar (University of Dehli) as part of Topological Groups
Lecture held in Elysium.
Abstract
The closed Lie ideals of the generalized group algebra $L^1(G,A)$ are characterized in terms of elements of the group $G$, elements of the algebra $A$, and the modular function $Delta$ of the group $G$. Conditions under which for a given closed Lie ideal $Lsubseteq A$ the subspace $L^1(G,L)$ is a Lie ideal, and vice versa, are discussed. The center of $L^1(G,A)$ is characterized, followed by a discussion regarding a very special projection in $L^1(G,A)$. Finally, a few restrictions are imposed on $G$ and $A$ under which $mathcal{Z}(L^1(G,A))congmathcal{Z}(L^1(G))otimes^gammamathcal{Z}(A)$.
The presentation is based on joint work with Ved Prakash Gupta and Ranjana Jain.
Sep
15
Tue
Alberto Marcone (Università di Udine)
Sep 15 @ 3:00 am – 4:00 am
Title: The higher levels of the Weihrauch lattice
by Alberto Marcone (Università di Udine) as part of Computability theory and applications
Abstract
The classification of mathematical problems in the Weihrauch lattice is a line of research that blossomed in the last few years. Initially this approach dealt mainly with statements which are provable in ACA_0 and showed that usually Weihrauch reducibility is more fine-grained than reverse mathematics.
In the last few years the study of multi-valued functions arising from statements laying at higher levels (such as ATR_0 and Pi^1_1-CA_0) of the reverse mathematics spectrum started as well. The multi-valued functions studied so far include those arising from the perfect tree theorem, comparability of well-orders, determinacy of open and clopen games, König’s duality theorem, various forms of choice, the open and clopen Ramsey theorem and the Cantor-Bendixson theorem.
At this level often a single theorem naturally leads to several multi-valued functions of different Weihrauch degree, depending on how the theorem is “read” from a computability viewpoint. A case in point is the perfect tree theorem: it can be read as the request to produce a perfect subtree of a tree with uncountably many paths, or as the request to list all paths of a tree which does not contain a perfect subtree. Similarly, the clopen Ramsey theorem leads to the multi-valued function that associates to every clopen subset of [N]^N an infinite homogeneous set on either side, and to the multi-valued function producing for each clopen subset which has an infinite homogeneous sets on one side a homogeneous set on that side. Similar functions can be defined similarly starting from the open Ramsey theorem.
In this talk I discuss some of these results, emphasizing recent joint work with my students Vittorio Cipriani and Manlio Valenti.
Nico Spronk (University of Waterloo) @ Lecture held in Elysium
Sep 15 @ 6:00 am – 8:00 am
Title: Topologies, idempotents and ideals
by Nico Spronk (University of Waterloo) as part of Topological Groups
Lecture held in Elysium.
Abstract
Let $G$ be a topological group. I wish to exhibit a bijection between (i) a certain class of weakly almost periodic topologies, (ii) idempotents in the weakly almost periodic compactification of $G$, and (iii) certain ideals of the algebra of weakly almost periodic functions. This has applications to decomposing weakly almost periodic representations on Banach spaces, generalizing results which go back to many authors.
Moving to unitary representations, I will develop the Fourier-Stieltjes algebra $B(G)$ of $G$, and give the analogous result there. As an application, I show that for a locally compact connected group, operator amenability of $B(G)$ implies that $G$ is compact, partially resolving a problem of interest for 25 years.
Justine Miller (University of Notre Dame)
Sep 15 @ 10:00 am – 11:00 am
Title: Noncomputable Coding, Density, and Stochasticity
by Justine Miller (University of Notre Dame) as part of Computability theory and applications
Abstract
We introduce the into and within set operations in order to construct sets of arbitrary intrinsic density from any Martin-Löf random. We then show that these operations are useful more generally for working with other notions of density as well, in particular for viewing Church and MWC stochasticity as a form of density.
Sep
22
Tue
Mukund Madhav Mishra (Hansraj College) @ Lecture held in Elysium
Sep 22 @ 6:00 am – 8:00 am
Title: Potential Theory on Stratified Lie Groups
by Mukund Madhav Mishra (Hansraj College) as part of Topological Groups
Lecture held in Elysium.
Abstract
Stratified Lie groups form a special subclass of the class of nilpotent Lie groups. The Lie algebra of a stratified Lie group possesses a specific stratification (and hence the name), and an interesting class of anisotropic dilations. Among the linear differential operators of degree two, there exists a family that is well behaved with the automorphisms of the stratified Lie group, especially with the anisotropic dilations. We shall see that one such family of operators mimics the classical Laplacian in many aspects, except for the regularity. More specifically, these Laplace-like operators are sub-elliptic, and hence referred to as the sub-Laplacians. We will review certain interesting properties of functions harmonic with respect to the sub-Laplacian on a stratified Lie group, and have a closer look at a particular class of stratified Lie groups known as the class of Heisenberg type groups.
Timothy McNicholl (Iowa State University)
Sep 22 @ 2:00 pm – 3:00 pm
Title: Which Lebesgue spaces are computably presentable?
by Timothy McNicholl (Iowa State University) as part of Computability theory and applications
Abstract
We consider the following question: “If there is a computably presentable $L^p$ space, does it follow that $p$ is computable?” The answer is of course no’ since the 1-dimensional $L^p$ space is just the field of scalars. So, we turn to non-trivial cases. Namely, assume there is a computably presentable $L^p$ space whose dimension is at least $2$. We prove $p$ is computable if the space is finite-dimensional or if $p geq 2$. We then show that if $1 leq p < 2$, and if $L^p[0,1]$ is computably presentable, then $p$ is right-c.e.. Finally, we show there is no uniform solution of this problem even when given upper and lower bounds on the exponent. The proof of this result leads to some basic results on the effective theory of stable random variables. Finally, we conjecture that the answer to this question is no’ and that right-c.e.-ness of the exponent is the best result possible.
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2021-01-15 21:00:50
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https://www.researchgate.net/publication/308939266_The_interaction_between_synoptic-scale_balanced_flow_and_a_finite-amplitude_mesoscale_wave_field_throughout_all_atmospheric_layers_weak_and_moderately_strong_stratification_Interaction_Between_Synopti
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Article
# The interaction between synoptic-scale balanced flow and a finite-amplitude mesoscale wave field throughout all atmospheric layers: weak and moderately strong stratification: Interaction Between Synoptic-scale Flow and a Mesoscale Wave Field
Authors:
• ARIA Technologies
To read the full-text of this research, you can request a copy directly from the authors.
## Abstract
The interaction between locally monochromatic finite-amplitude mesoscale waves, their nonlinearly induced higher harmonics, and a synoptic-scale flow is reconsidered, both in the tropospheric regime of weak stratification and in the stratospheric regime of moderately strong stratification. A review of the basic assumptions of quasi-geostrophic theory on an f-plane yields all synoptic scales in terms of a minimal number of natural variables, i.e. two out of the speed of sound, gravitational acceleration and Coriolis parameter. The wave scaling is defined so that all spatial and temporal scales are shorter by one order in the Rossby number, and by assuming their buoyancy field to be close to static instability. WKB theory is applied, with the Rossby number as scale separation parameter, combined with a systematic Rossby-number expansion of all fields. Classic results for synoptic-scale-flow balances and inertia-gravity wave (IGW) dynamics are recovered. These are supplemented by explicit expressions for the interaction between mesoscale geostrophic modes (GM), a possibly somewhat overlooked agent of horizontal coupling in the atmosphere, and the synoptic-scale flow. It is shown that IGW higher harmonics are slaved to the basic IGW, and that their amplitude is one order of magnitude smaller than the basic-wave amplitude. GM higher harmonics are not that weak and they are in intense nonlinear interaction between themselves and the basic GM. Compressible dynamics plays a significant role in the stratospheric stratification regime, where anelastic theory would yield insufficient results. Supplementing classic derivations, it is moreover shown that in the absence of mesoscale waves quasi-geostrophic theory holds also in the stratospheric stratification regime.
## No full-text available
... A complementary approach is Wentzel-Kramers-Brillouin-Jeffreys (WKBJ) theory (Bretherton, 1966;Grimshaw, 1975;Achatz et al., 2010;Achatz et al., 2017) which, instead of considering continuous wave spectra, describes the development of locally monochromatic GW fields which feature a nearly discrete spectrum. Moreover, the WKBJ approach takes into account nonlinear interactions between GWs and a spatially and temporally varying mean flow. ...
... In general, wave modulation by a variable background stratification or a sheared mean flow are equally important in the atmosphere (cf. Achatz et al., 2017). However, we restrict the analysis to the case of Boussinesq dynamics with a constant background stratification and zero rotation for the sake of simplicity. ...
... The initial wave amplitudes are taken to be a given fraction, , with respect to the static F I G U R E 3 Ratios of wave energies corresponding of the solutions the simplified system (Equations 71-73) and the parametrized system (76-78) after the interaction. Here, the background shear is set to z u = 2 ∕40,000 s −1 , and the parametrization constant is ≡ 1. (a), (b), and (c) are associated with j = 1, j = 2, and j = 3, respectively instability criterion (e.g., Achatz et al., 2017). In particular ...
Article
Full-text available
Motivated by the question of whether and how wave‐wave interactions should be implemented into atmospheric gravity‐wave parameterizations, the modulation of triadic gravity‐wave interactions by a slowly varying and vertically sheared mean‐flow is considered for a non‐rotating Boussinesq fluid with constant stratification. An analysis using a multiple‐scales WKBJ expansion identifies two distinct scaling regimes, a linear off‐resonance regime, and a non‐linear near‐resonance regime. Simplifying the near‐resonance interaction equations allows for the construction of a parametrization for the triadic energy exchange which has been implemented into a one‐dimensional WKBJ ray‐tracing code. Theory and numerical implementation are validated for test cases where two wave trains generate a third wave train while spectrally passing through resonance. In various settings, of interacting vertical wavenumbers, mean‐flow shear, and initial wave amplitudes, the WKBJ simulations are generally in good agreement with wave‐resolving simulations. Both stronger mean‐flow shear and smaller wave amplitudes suppress the energy exchange among a resonantly interacting triad. Experiments with mean‐flow shear as strong as in the vicinity of atmospheric jets suggest that internal gravity wave dynamics are dominated in such regions by wave modulation. Yet, triadic gravity‐wave interactions are likely to be relevant in weakly sheared regions of the atmosphere.
... As will be shown below this is at least justified if the large-scale flow is in geostrophic and hydrostatic balance. The direct scheme does not rely on any balance assumption with regard to the large-scale flow, and the large-scale flow is forced by anelastic momentum-flux convergence in the momentum equation, an elastic term also in the momentum equation, and entropy-flux convergence in the entropy equation, as given by Grimshaw (1975) and Achatz et al. (2017). All present-day operational IGW parameterizations represent, one way or other, simplified versions of the pseudomomentum approach, where the vertical gradient of pseudomomentum-flux convergence forces the resolved flow, when wave dissipation occurs (Fritts and Alexander 2003;Kim et al. 2003), and neither elastic nor thermal effects are taken into account. ...
... For an explanation of the theoretical underpinnings of the two respective approaches we follow the presentation of Achatz et al. (2017) where, expanding on previous work by Grimshaw (1975), the theory is discussed mostly in nondimensional form. We translate the essentials into dimensional form and choose, for easier tractability, a heuristic formulation. ...
... We translate the essentials into dimensional form and choose, for easier tractability, a heuristic formulation. For all mathematical details, the reader is referred back to Achatz et al. (2017). ...
Article
Full-text available
This paper compares two different approaches for the efficient modeling of subgrid-scale inertia–gravity waves in a rotating compressible atmosphere. The first approach, denoted as the pseudomomentum scheme, exploits the fact that in a Lagrangian-mean reference frame the response of a large-scale flow can only be due to forcing momentum. Present-day gravity wave parameterizations follow this route. They do so, however, in an Eulerian-mean formulation. Transformation to that reference frame leads, under certain assumptions, to pseudomomentum-flux convergence by which the momentum is to be forced. It can be shown that this approach is justified if the large-scale flow is in geostrophic and hydrostatic balance. Otherwise, elastic and thermal effects might be lost. In the second approach, called the direct scheme and not relying on such assumptions, the large-scale flow is forced both in the momentum equation, by anelastic momentum-flux convergence and an additional elastic term, and in the entropy equation, via entropy-flux convergence. A budget analysis based on one-dimensional wave packets suggests that the comparison between the abovementioned two schemes should be sensitive to the following two parameters: 1) the intrinsic frequency and 2) the wave packet scale. The smaller the intrinsic frequency is, the greater their differences are. More importantly, with high-resolution wave-resolving simulations as a reference, this study shows conclusive evidence that the direct scheme is more reliable than the pseudomomentum scheme, regardless of whether one-dimensional or two-dimensional wave packets are considered. In addition, sensitivity experiments are performed to further investigate the relative importance of each term in the direct scheme, as well as the wave–mean flow interactions during the wave propagation.
... Let us summarize the results of the previous section. The gross wave-Froude number depending on the wave itself due to the induced mean flow is specified in terms of (34) and (35) by ...
... In an envisaged companion paper we want to extend our investigations to the stability of gravity waves governed by three-dimensional modulation equations including the Coriolis force. The basis for such a study was already founded in [35]. ...
Article
Full-text available
This study investigates strongly nonlinear gravity waves in the compressible atmosphere from the Earth's surface to the deep atmosphere. These waves are effectively described by Grimshaw's dissipative modulation equations which provide the basis for finding stationary solutions such as mountain lee waves and testing their stability in an analytic fashion. Assuming energetically consistent boundary and far-field conditions , that is no energy flux through the surface, free-slip boundary, and finite total energy, general wave solutions are derived and illustrated in terms of realistic background fields. These assumptions also imply that the wave-Reynolds number must become less than unity above a certain height. The modulational stability of admissible, both non-hydrostatic and hydrostatic, waves is examined. It turns out that, when accounting for the self-induced mean flow, the wave-Froude number has a resonance condition. If it becomes 1/ √ 2, then the wave destabilizes due to perturbations from the essential spectrum of the linearized modulation equations. However, if the horizontal wavelength is large enough, waves overturn before they can reach the modulational stability condition.
... This is conditioned on a manageable separation of the flow and its dynamical equations into balanced and unbalanced parts. For this reason we here restrict ourselves to linear balance conditions and a determination of the balanced flow from the inversion of linear potential vorticity (PV), as is strictly appropriate in the limit of a small Rossby number (Charney 1948;Hoskins, McIntyre & Robertson 1985;Pedlosky 1987;Achatz et al. 2017). This approach is supplemented by the extraction of balanced vertical motion and horizontal divergence by the application of the omega equation. ...
... In our configuration of the differentially heated rotating annulus (see § 2.1) the Rossby number is small (Ro < 1) in most locations. As shown by Bühler & McIntyre (2005) in the Lagrangian mean and by Achatz et al. (2017) in the Eulerian perspective, in that limit IGWs contribute to the nonlinear part of PV, while the linear part is determined exclusively by a geostrophically and hydrostatically balanced component, as also in quasi-geostrophic theory (Charney 1948;Pedlosky 1987;Vallis 2006). Moreover, as can be verified from their polarisation relations, linear IGWs have no linear PV (Phillips 1963;Mohebalhojeh & Dritschel 2001;Smith & Waleffe 2002). ...
Article
The source mechanism of inertia–gravity waves (IGWs) observed in numerical simulations of the differentially heated rotating annulus experiment is investigated. The focus is on the wave generation from the balanced part of the flow, a process presumably contributing significantly to the atmospheric IGW field. Direct numerical simulations are performed for an atmosphere-like configuration of the annulus and possible regions of IGW activity are characterised by a Hilbert-transform algorithm. In addition, the flow is separated into a balanced and unbalanced part, assuming the limit of a small Rossby number, and the forcing of IGWs by the balanced part of the flow is derived rigorously. Tangent-linear simulations are then used to identify the part of the IGW signal that is rather due to radiation by the internal balanced flow than to boundary-layer instabilities at the side walls. An idealised fluid set-up without rigid horizontal boundaries is considered as well, to investigate the effect of the identified balanced forcing unmasked by boundary-layer effects. The direct simulations of the realistic and idealised fluid set-ups show a clear baroclinic-wave structure exhibiting a jet–front system similar to its atmospheric counterparts, superimposed by four distinct IGW packets. The subsequent tangent-linear analysis indicates that three wave packets are radiated from the internal flow and a fourth one is probably caused by boundary-layer instabilities. The forcing by the balanced part of the flow is found to play a significant role in the generation of IGWs, so it supplements boundary-layer instabilities as a key factor in the IGW emission in the differentially heated rotating annulus.
... The horizontal grid spacing is 160 km, and the vertical spacing is 700 m in the stratosphere. Instead of the operational GW parameterization of this model, we use a prognostic parameterization, the Multi-Scale Gravity Wave Model (MS-GWaM), which predicts the time evolution of GW action density field in positionwavenumber phase space (Achatz et al., 2017;Bölöni et al., 2021;Muraschko et al., 2015). A detailed description of MS-GWaM and its application to ICON is provided in Bölöni et al. (2021). ...
Article
Full-text available
A general circulation model is used to study the interaction between parameterized gravity waves (GWs) and large-scale Kelvin waves in the tropical stratosphere. The simulation shows that Kelvin waves with substantial amplitudes (~10 m/s) can significantly affect the distribution of GW drag by modulating the local shear. Furthermore, this effect is localized to regions above strong convective organizations that generate large-amplitude GWs, so that at a given altitude it occurs selectively in a certain phase of Kelvin waves. Accordingly, this effect also contributes to the zonal-mean GW drag, which is large in the middle stratosphere during the phase transition of the quasi-biennial oscillation (QBO). Furthermore, we detect an enhancement of Kelvin-wave momentum flux due to GW drag modulated by Kelvin waves. The result implies an importance of GW dynamics coupled to Kelvin waves in the QBO progression.
... This phenomenon in uences the breaking height as the stability depends sensitively on amplitude. Therefore, we continue by investigating an extended set of modulation equations in section 3 which agrees with the inviscid pseudo-incompressible regime [1,2,5]. Here, the background density is an explicit function of height. ...
Article
Full-text available
We apply spectral stability theory to investigate nonlinear gravity waves in the atmosphere. These waves are determined by modulation equations that result from Wentzel-Kramers-Brillouin theory. First, we establish that plane waves, which represent exact solutions to the inviscid Boussinesq equations, are spectrally stable with respect to their nonlinear modulation equations under the same conditions as what is known as modulational stability from weakly nonlinear theory. In contrast to Boussinesq, the pseudo-incompressible regime does fully account for the altitudinal varying background density. Second,we show for the first time that upward-traveling non-plane wave fronts solving the inviscid nonlinear modulation equations, that compare to pseudo-incompressible theory, are unconditionally unstable. Both inviscid regimes turn out to be ill-posed as the spectra allow for arbitrarily large instability growth rates. Third, a regularization is found by including dissipative effects. The corresponding nonlinear traveling wave solutions have localized amplitude. As a consequence of the nonlinearity, envelope and linear group velocity, as given by the derivative of the frequency with respect to wavenumber, do not coincide anymore. These waves blow up unconditionally by embedded eigenvalue instabilities but the instability growth rate is bounded from above and can be computed analytically. Additionally, all three types of nonlinear modulation equations are solved numerically to further investigate and illustrate the nature of the analytic stability results.
... As well as improving parametrizations of sources for internal waves, the representation of their propagation should be improved. Idealized theoretical models and simulations suggest that the effect of time-transient background winds, wave energetics, and lateral propagation of the waves as well as weakly nonlinear effects acting upon moderately large-amplitude waves may need to be incorporated into the next generation of parametrization schemes [6,37,45,[122][123][124][125]. Further corresponding indirect observational evidence is provided again from radiosonde balloons, showing a conspicuous dependence of the intermittency of internal-wave momentum fluxes on the large-scale wind strength that parametrizations cannot reproduce [120]. ...
Article
Imbalance refers to the departure from the large-scale primarily vortical flows in the atmosphere and ocean whose motion is governed by a balance between Coriolis, pressure-gradient and buoyancy forces, and can be described approximately by quasi-geostrophic theory. Imbalanced motions are manifest as internal gravity waves which can extract energy from these geophysical flows but which can also feed energy back into this motion. Capturing the physics underlying these mechanisms is essential to understand how energy is transported from large geophysical scales ultimately to microscopic scales where it is dissipated. In the atmosphere it is also necessary for understanding momentum transport and its impact upon the mean wind and current speeds. During a February 2018 workshop at the Banff International Research Station (BIRS), atmospheric scientists, physical oceanographers, physicists and mathematicians gathered to discuss recent progress in understanding these processes through interpretation of observations, numerical simulations and mathematical modelling. The outcome of this meeting is reported upon here.
... If the atmospheric mesoscales are characterized as a relatively passive gap between two highly variable regimes, then it is not surprising that it has proved difficult to identify a single dominant mechanism leading to a universal 25/3 energy spectrum. It may be that alternative theoretical frameworks are required, for example, one that explicitly models the dynamics of scale interactions rather than treating them in a purely statistical manner (Achatz et al. 2017). ...
Article
Research on the mesoscale kinetic energy spectrum over the past few decades has focused on finding a dynamical mechanism that gives rise to a universal spectral slope. Here we investigate the variability of the spectrumusing 3 years of kilometer-resolution analyses fromCOSMOconfigured forGermany (COSMO-DE). It is shown that the mesoscale kinetic energy spectrum is highly variable in time but that a minimum in variability is found on scales around 100 km. The high variability found on the small-scale end of the spectrum (around 10km) is positively correlated with the precipitation rate where convection is a strong source of variance. On the other hand, variability on the large-scale end (around 1000 km) is correlated with the potential vorticity, as expected for geostrophically balanced flows. Accordingly, precipitation at small scales is more highly correlated with divergent kinetic energy, and potential vorticity at large scales is more highly correlated with rotational kinetic energy. The presented findings suggest that the spectral slope and amplitude on the mesoscale range are governed by an ever-changing combination of the upscale and downscale impacts of these large- and small-scale dynamical processes rather than by a universal, intrinsically mesoscale dynamical mechanism.
... Wave field (Achatz et al., 2017) ...
Poster
Full-text available
The aim of the presented work is to improve the parametrization of subgrid-scale gravity wave (GW) drag on the resolved flow of climate and numerical-weather-prediction models. Current GW parametrization schemes are using the steady-state approximation for the wave field and therefore assume an instantaneous GW propagation neglecting direct interactions between the GWs and the resolved flow in the course of the propagation. As such these schemes rely on wave breaking as the only mechanism to exert a drag on the resolved flow. Theory shows that dropping the steady-state assumption leads to non-linear GW-meanflow interactions (further on direct GW-meanflow interaction) where the meanflow is forced even in the absence of wave breaking, whereas the meanflow in turn modulates the smaller scale wave field due to wind shear and stratification gradients (Achatz et al., 2017). In idealized simulations it indeed turns out that by applying a transient GW model (i.e. by dropping the steady-state assumption) the contribution of direct GW-meanflow interaction to the GW drag can be as important as that of wave breaking (Bölöni et al., 2016). This motivates the implementation of a transient GW model (further on named MS-GWaM: Multi Scale Gravity Wave Model) to a state-of-the-art global circulation model (GCM) enabling to evaluate the consequences of direct GW-meanflow interactions in a realistic atmospheric circulation. The GCM in which MS-GWaM has been implemented is the Icosahedral Nonhydrostatic Model (ICON) developed jointly by the German Weather Service and the Max-Planck Institute for Meteorology. MS-GWaM in ICON runs stably and provides substantially different GW drag in comparison with the benchmark steady-state parametrization available in the model. Seasonal simulations with ICON-MS-GWaM provide reasonable zonal mean middle atmospheric circulation and temperature structures in comparison with climatology such as SPARC observations and the HAMMONIA GCM.
... In this regard Achatz et al. (2010Achatz et al. ( , 2017 showed that the consistency between the scale asymptotics of the Euler equations and the pseudo-incompressible equations also holds for hydrostatic gravity waves. Bölöni et al. (2016) investigated the non-hydrostatic modulation equations numerically applying a ray-tracer method. ...
Article
Full-text available
Wentzel–Kramers–Brillouin theory was employed by Grimshaw ( Geophys. Fluid Dyn. , vol. 6, 1974, pp. 131–148) and Achatz et al. ( J. Fluid Mech. , vol. 210, 2010, pp. 120–147) to derive modulation equations for non-hydrostatic internal gravity wave packets in the atmosphere. This theory allows for wave packet envelopes with vertical extent comparable to the pressure scale height and for large wave amplitudes with wave-induced mean-flow speeds comparable to the local fluctuation velocities. Two classes of exact travelling wave solutions to these nonlinear modulation equations are derived here. The first class involves horizontally propagating wave packets superimposed over rather general background states. In a co-moving frame of reference, examples from this class have a structure akin to stationary mountain lee waves. Numerical simulations corroborate the existence of nearby travelling wave solutions under the pseudo-incompressible model and reveal better than expected convergence with respect to the asymptotic expansion parameter. Travelling wave solutions of the second class also feature a vertical component of their group velocity but exist under isothermal background stratification only. These waves include an interesting nonlinear wave–mean-flow interaction process: a horizontally periodic wave packet propagates vertically while draining energy from the mean wind aloft. In the process it decelerates the lower-level wind. It is shown that the modulation equations apply equally to hydrostatic waves in the limit of large horizontal wavelengths. Aside from these results of direct physical interest, the new nonlinear travelling wave solutions provide a firm basis for subsequent studies of nonlinear internal wave instability and for the design of subtle test cases for numerical flow solvers.
... Large-amplitude mesoscale gravity waves, which can originate from a variety of processes and often travel large distances before dissipating (e.g., Achatz et al. 2017), have also been extensively studied and remain difficult to forecast using currently available conventional surface weather observations and numerical guidance. The movement, amplification, and decay of such features through generally stable environments has often been a focus for research (Bosart and Seimon 1988;Crook 1988;Ramamurthy et al. 1993;Zhang et al. 2001;Plougonven and Zhang 2014). ...
Article
Mesoscale convective phenomena induce pressure perturbations that can alter the strength and magnitude of surface winds, precipitation, and other sensible weather which, in some cases, can inflict injuries and damage to property. This work extends prior research to identify and characterize mesoscale pressure features using a unique resource of 1-Hz pressure observations available from the USArray Transportable Array (TA) seismic field campaign. A two-dimensional variational technique is used to obtain 5 km surface pressure analysis grids every 5 min from 1 March – 31 August 2011 from the TA observations and gridded surface pressure from the Real Time Mesoscale Analysis over a swath of the central United States. Band-pass filtering and feature tracking algorithms are employed to isolate, identify, and assess prominent mesoscale pressure perturbations and their properties. Two case studies, the first involving mesoscale convective systems and second a solitary gravity wave, are analyzed using additional surface observation and gridded data resources. Summary statistics for tracked features during the period reviewed indicate a majority of perturbations last less than 3 h, produce maximum perturbation magnitudes between 2-5 hPa, and move at speeds ranging from 15-35 m s⁻¹. The results of this study combined with improvements nationwide in real-time access to pressure observations at sub-hourly reporting intervals highlight the potential for improved detection and nowcasting of high-impact mesoscale weather features.
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Parameterizations of subgrid-scale gravity waves (GWs) in atmospheric models commonly involve the description of the dissipation of GWs. Where they dissipate, GWs have an increased effect on the large-scale flow. Instabilities that trigger wave breaking are an important starting point for the route to dissipation. Possible destabilizing mechanisms are numerous, but the classical vertical static instability is still regarded as a key indicator for the disposition to wave breaking. In this work, we investigate how the horizontal variations associated with a GW could alter the criterion for static instability. To this end, we use an extension of the common parcel displacement method. This three-dimensional static stability analysis predicts a significantly larger range of instability than does the vertical static stability analysis. In this case, the Lindzen-type saturation adjustment to a state of marginal stability is perhaps a less suitable ansatz for the parameterization of the GW breaking. In order to develop a possible ansatz for the GW dissipation due to three-dimensional instability, we apply the methods of irreversible thermodynamics, which are embedded in the Gibbs formalism of dynamics. In this way, the parameterization does not only satisfy the second law of thermodynamics, but it can also be made consistent with the conservation of energy and further (non-)conservation principles. We develop the parameterization for a discrete spectrum of GW packets. Offline computations of GW drag and dissipative heating rates are performed for two vertical profiles of zonal wind and temperature for summer and winter conditions from CIRA data. The results are compared to benchmarks from the literature.
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We study the stratified gas in a rapidly rotating centrifuge as a model for the Earth's atmosphere. Based on methods of perturbation theory, it is shown that in certain regimes, internal waves in the gas centrifuge have the same dispersion relation to leading order as their atmospheric siblings. Assuming an air filled centrifuge with a radius of around 50 cm, the optimal rotational frequency for realistic atmosphere-like waves is around 10 000 revolutions per minute. Using gases of lower heat capacities at constant pressure, such as xenon, the rotational frequencies can be even halved to obtain the same results. Similar to the atmosphere, it is feasible in the gas centrifuge to generate a clear scale separation of wave frequencies and therefore phase speeds between acoustic waves and internal waves. In addition to the centrifugal force, the Coriolis force acts in the same plane. However, its influence on axially homogeneous internal waves appears only as a higher-order correction. We conclude that the gas centrifuge provides an unprecedented opportunity to investigate atmospheric internal waves experimentally with a compressible working fluid.
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The model Internal Wave Dissipation, Energy and Mixing (IDEMIX) presents a novel way of parameterising internal gravity waves in the atmosphere. IDEMIX is based on the spectral energy balance of the wave field and has previously been successfully developed as a model for diapycnal diffusivity, induced by internal gravity wave breaking in oceans. Applied here for the first time to atmospheric gravity waves, integration of the energy balance equation for a continuous wave field of a given spectrum, results in prognostic equations for the energy density of eastward and westward gravity waves. It includes their interaction with the mean flow, allowing for an evolving and local description of momentum flux and gravity wave drag. A saturation mechanism maintains the wave field within convective stability limits, and a closure for critical layer effects controls how much wave flux propagates from the troposphere into the middle atmosphere. Offline comparisons to a traditional parameterisation reveal increases in the wave momentum flux in the middle atmosphere due to the mean flow interaction, resulting in a greater gravity wave drag at lower altitudes. Preliminary validation against observational data show good agreement with momentum fluxes.
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This study introduces a new computational scheme for the linear evolution of internal gravity wave packets passing over strongly non-uniform stratifications and background flows as found, e.g., near the tropopause. Focusing on linear dispersion, which is dominant at small wave amplitudes, the scheme describes general wave superpositions arising from wave reflections near strong variations of the background stratification. Thus, it complements WKB theory, which is restricted to nearly monochromatic waves but covers weakly nonlinear effects in turn. One envisaged application of the method is the formulation of bottom-of-the-stratosphere starting conditions for ray tracing parameterizations that follow nonlinear gravity wave packets into the upper atmosphere. A key feature in this context is the method’s separation of wave packets into up- and downward-propagating components. The paper first summarizes a multilayer method for the numerical solution of the Taylor–Goldstein equation. Borrowing ideas from Eliassen and Palm (Geophys Publ 22:1–23, 1961), the scheme is based on partitioning the atmosphere into several uniformly stratified layers. This allows for analytical plane wave solutions in each layer, which are matched carefully to obtain continuously differentiable global eigenmode solutions. This scheme enables rapid evaluations of reflection and transmission coefficients for internal waves impinging on the tropopause from below as functions of frequency and horizontal wavenumber. The study then deals with a spectral method for propagating wave packets passing over non-uniform backgrounds. Such non-stationary solutions are approximated by superposition of Taylor–Goldstein eigenmodes. Particular attention is paid to an algorithm that translates wave packet initial data in the form of modulated sinusoidal signals into amplitude distributions for the system’s eigenmodes. With this initialization in place, the state of the perturbations at any given subsequent time is obtained by a single superposition of suitably phase-shifted eigenmodes, i.e., without any time-stepping iterations. Comparisons of solutions for wave packet evolution with those obtained from a nonlinear atmospheric flow solver reveal that apparently nonlinear effects can be the result of subtle linear wave packet dispersion.
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As present weather-forecast codes and increasingly many atmospheric climate models resolve at least part of the mesoscale flow, and hence also internal gravity waves (GWs), it is natural to ask whether even in such configurations sub-gridscale GWs might impact the resolved flow, and how their effect could be taken into account. This motivates a theoretical and numerical investigation of the interaction between unresolved sub-mesoscale and resolved mesoscale GWs, using Boussinesq dynamics for simplicity. By scaling arguments, first a subset of sub-mesoscale GWs that can indeed influence the dynamics of mesoscale GWs is identified. Therein, hydrostatic GWs with wavelengths corresponding to the largest unresolved scales of present-day limited-area weather forecast models are an interesting example. A largeamplitude WKB theory, allowing for a mesoscale unbalanced flow, is then formulated, based on multi-scale asymptotic analysis utilizing a proper scaleseparation parameter. Purely vertical propagation of sub-mesoscale GWs is found to be most important, implying inter alia that the resolved flow is only affected by the vertical flux convergence of sub-mesoscale horizontal momentum at leading order. In turn, sub-mesoscale GWs are refracted by mesoscale vertical wind shear while conserving their wave-action density. An efficient numerical implementation of the theory uses a phase-space ray tracer, thus handling the frequent appearance of caustics. The WKB approach and its numerical implementation are validated successfully against sub-mesoscale resolving simulations of the resonant radiation of mesoscale inertia GWs by a horizontally as well as vertically confined sub-mesoscale GW packet.
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This study analyzes in situ airborne measurements from the 2008 Stratosphere–Troposphere Analyses of Regional Transport (START08) experiment to characterize gravity waves in the extratropical upper troposphere and lower stratosphere (ExUTLS). The focus is on the second research flight (RF02), which took place on 21–22 April 2008. This was the first airborne mission dedicated to probing gravity waves associated with strong upper-tropospheric jet–front systems. Based on spectral and wavelet analyses of the in situ observations, along with a diagnosis of the polarization relationships, clear signals of mesoscale variations with wavelengths ~ 50–500 km are found in almost every segment of the 8 h flight, which took place mostly in the lower stratosphere. The aircraft sampled a wide range of background conditions including the region near the jet core, the jet exit and over the Rocky Mountains with clear evidence of vertically propagating gravity waves of along-track wavelength between 100 and 120 km. The power spectra of the horizontal velocity components and potential temperature for the scale approximately between ~ 8 and ~ 256 km display an approximate −5/3 power law in agreement with past studies on aircraft measurements, while the fluctuations roll over to a −3 power law for the scale approximately between ~ 0.5 and ~ 8 km (except when this part of the spectrum is activated, as recorded clearly by one of the flight segments). However, at least part of the high-frequency signals with sampled periods of ~ 20–~ 60 s and wavelengths of ~ 5–~ 15 km might be due to intrinsic observational errors in the aircraft measurements, even though the possibilities that these fluctuations may be due to other physical phenomena (e.g., nonlinear dynamics, shear instability and/or turbulence) cannot be completely ruled out.
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With the aim of contributing to the improvement of subgrid-scale gravity wave (GW) parameterizations in numerical-weather-prediction and climate models, the comparative relevance in GW drag of direct GW-mean-flow interactions and turbulent wave breakdown are investigated. Of equal interest is how well Wentzel-Kramer-Brillouin (WKB) theory can capture direct wave-mean-flow interactions, that are excluded by applying the steady-state approximation. WKB is implemented in a very efficient Lagrangian ray-tracing approach that considers wave action density in phase-space, thereby avoiding numerical instabilities due to caustics. It is supplemented by a simple wave-breaking scheme based on a static-instability saturation criterion. Idealized test cases of horizontally homogeneous GW packets are considered where wave-resolving Large-Eddy Simulations (LES) provide the reference. In all of theses cases the WKB simulations including direct GW-mean-flow interactions reproduce the LES data, to a good accuracy, already without wave-breaking scheme. The latter provides a next-order correction that is useful for fully capturing the total-energy balance between wave and mean flow. Moreover, a steady-state WKB implementation, as used in present GW parameterizations, and where turbulence provides, by the non-interaction paradigm, the only possibility to affect the mean flow, is much less able to yield reliable results. The GW energy is damped too strongly and induces an oversimplified mean flow. This argues for WKB approaches to GW parameterization that take wave transience into account.
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We derive a wave-averaged potential vorticity equation describing the evolution of strongly stratified, rapidly rotating quasi-geostrophic (QG) flow in a field of inertia-gravity internal waves. The derivation relies on a multiple-time-scale asymptotic expansion of the Eulerian Boussinesq equations. Our result confirms and extends the theory of Bühler & McIntyre ( J. Fluid Mech. , vol. 354, 1998, pp. 609–646) to non-uniform stratification with buoyancy frequency $N(z)$ and therefore non-uniform background potential vorticity $f_{0}N^{2}(z)$ , and does not require spatial-scale separation between waves and balanced flow. Our interest in non-uniform background potential vorticity motivates the introduction of a new quantity: ‘available potential vorticity’ (APV). Like Ertel potential vorticity, APV is exactly conserved on fluid particles. But unlike Ertel potential vorticity, linear internal waves have no signature in the Eulerian APV field, and the standard QG potential vorticity is a simple truncation of APV for low Rossby number. The definition of APV exactly eliminates the Ertel potential vorticity signal associated with advection of a non-uniform background state, thereby isolating the part of Ertel potential vorticity available for balanced-flow evolution. The effect of internal waves on QG flow is expressed concisely in a wave-averaged contribution to the materially conserved QG potential vorticity. We apply the theory by computing the wave-induced QG flow for a vertically propagating wave packet and a mode-one wave field, both in vertically bounded domains.
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A finite-volume model of the classic differentially heated rotating annulus experiment is used to study the spontaneous emission of gravity waves (GWs) from jet stream imbalances, which may be an important source of these waves in the atmosphere and for which no satisfactory parameterisation exists. Experiments were performed using a classic laboratory configuration as well as using a much wider and shallower annulus with a much larger temperature difference between the inner and outer cylinder walls. The latter configuration is more atmosphere-like, in particular since the Brunt Vaisfila frequency is larger than the inertial frequency, resulting in more realistic GW dispersion properties. In both experiments, the model is initialised with a baroclinically unstable axisymmetric state established using a two-dimensional version of the code, and a low-azimuthal-mode baroclinic wave featuring a meandering jet is allowed to develop. Possible regions of GW activity are identified by the horizontal velocity divergence and a modal decomposition of the small-scale structures of the flow. Results indicate GW activity in both annulus configurations close to the inner cylinder wall and within the baroclinic wave. The former is attributable to boundary layer instabilities, while the latter possibly originates in part from spontaneous GW emission from the baroclinic wave.
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Significance High- and low-pressure systems, commonly referred to as synoptic systems, are the most energetic fluctuations of wind and temperature in the midlatitude troposphere. Synoptic systems are a few thousand kilometers in scale and are governed by a balance between the pressure gradient force and the Coriolis force. Observations collected near the tropopause by commercial aircraft indicate a change in dynamics at horizontal scales smaller than about 500 km. Smaller-scale fluctuations are shown to be dominated by inertia–gravity waves, waves that propagate on vertical density gradients but are influenced by Earth’s rotation.
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Wind forcing of the ocean generates a spectrum of inertia–gravity waves that is sharply peaked near the local inertial (or Coriolis) frequency. The corresponding near-inertial waves (NIWs) are highly energetic and play a significant role in the slow, large-scale dynamics of the ocean. To analyse this role, we develop a new model of the non-dissipative interactions between NIWs and balanced motion. The model is derived using the generalised-Lagrangian-mean (GLM) framework (specifically, the ‘glm’ variant of Soward & Roberts, J. Fluid Mech. , vol. 661, 2010, pp. 45–72), taking advantage of the time-scale separation between the two types of motion to average over the short NIW period. We combine Salmon’s ( J. Fluid Mech. , vol. 719, 2013, pp. 165–182) variational formulation of GLM with Whitham averaging to obtain a system of equations governing the joint evolution of NIWs and mean flow. Assuming that the mean flow is geostrophically balanced reduces this system to a simple model coupling Young & Ben Jelloul’s ( J. Mar. Res. , vol. 55, 1997, pp. 735–766) equation for NIWs with a modified quasi-geostrophic (QG) equation. In this coupled model, the mean flow affects the NIWs through advection and refraction; conversely, the NIWs affect the mean flow by modifying the potential-vorticity (PV) inversion – the relation between advected PV and advecting mean velocity – through a quadratic wave term, consistent with the GLM results of Bühler & McIntyre ( J. Fluid Mech. , vol. 354, 1998, pp. 301–343). The coupled model is Hamiltonian and its conservation laws, for wave action and energy in particular, prove illuminating: on their basis, we identify a new interaction mechanism whereby NIWs forced at large scales extract energy from the balanced flow as their horizontal scale is reduced by differential advection and refraction so that their potential energy increases. A rough estimate suggests that this mechanism could provide a significant sink of energy for mesoscale motion and play a part in the global energetics of the ocean. Idealised two-dimensional models are derived and simulated numerically to gain insight into NIW–mean-flow interaction processes. A simulation of a one-dimensional barotropic jet demonstrates how NIWs forced by wind slow down the jet as they propagate into the ocean interior. A simulation assuming plane travelling NIWs in the vertical shows how a vortex dipole is deflected by NIWs, illustrating the irreversible nature of the interactions. In both simulations energy is transferred from the mean flow to the NIWs.
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A method of second-order accuracy is described for integrating the equations of ideal compressible flow. The method is based on the integral conservation laws and is dissipative, so that it can be used across shocks. The heart of the method is a one-dimensional Lagrangean scheme that may be regarded as a second-order sequel to Godunov's method. The second-order accuracy is achieved by taking the distributions of the state quantities inside a gas slab to be linear, rather than uniform as in Godunov's method. The Lagrangean results are remapped with least-squares accuracy onto the desired Euler grid in a separate step. Several monotonicity algorithms are applied to ensure positivity, monotonicity and nonlinear stability. Higher dimensions are covered through time splitting. Numerical results for one-dimensional and two-dimensional flows are presented, demonstrating the efficiency of the method. The paper concludes with a summary of the results of the whole series “Towards the Ultimate Conservative Difference Scheme.”
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A computational model of the pseudo-incompressible equations is used to probe the range of validity of an extended Wentzel–Kramers–Brillouin theory (XWKB), previously derived through a distinguished limit of a multiple-scale asymptotic analysis of the Euler or pseudo-incompressible equations of motion, for gravity-wave packets at large amplitudes. The governing parameter of this analysis had been the scale-separation ratio \$\varepsilon \$ between the gravity wave and both the large-scale potential-temperature stratification and the large-scale wave-induced mean flow. A novel feature of the theory had been the non-resonant forcing of higher harmonics of an initial wave packet, predominantly by the large-scale gradients in the gravity-wave fluxes. In the test cases considered a gravity-wave packet is propagating upwards in a uniformly stratified atmosphere. Large-scale winds are induced by the wave packet, and possibly exert a feedback on the latter. In the limit \$\varepsilon \ll 1\$ all predictions of the theory can be validated. The larger \$\varepsilon \$ is the more the transfer of wave energy to the mean flow is underestimated by the theory. The numerical results quantify this behaviour but also show that, qualitatively, XWKB remains valid even when the gravity-wave wavelength approaches the spatial scale of the wave-packet amplitude. This includes the prevalence of first and second harmonics and the smallness of harmonics with wave number higher than two. Furthermore, XWKB predicts for the vertical momentum balance an additional leading-order buoyancy term in Euler and pseudo-incompressible theory, compared with the anelastic theory. Numerical tests show that this term is relatively large with up to \$30\hspace{0.167em} \% \$ of the total balance. The practical relevance of this deviation remains to be assessed in future work.
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As upward-propagating anelastic internal gravity wave packets grow in amplitude, nonlinear effects de-velop as a result of interactions with the horizontal mean flow that they induce. This qualitatively alters the structure of the wave packet. The weakly nonlinear dynamics are well captured by the nonlinear Schrö dinger equation, which is derived here for anelastic waves. In particular, this predicts that strongly nonhydrostatic waves are modulationally unstable and so the wave packet narrows and grows more rapidly in amplitude than the exponential anelastic growth rate. More hydrostatic waves are modulationally stable and so their am-plitude grows less rapidly. The marginal case between stability and instability occurs for waves propagating at the fastest vertical group velocity. Extrapolating these results to waves propagating to higher altitudes (hence attaining larger amplitudes), it is anticipated that modulationally unstable waves should break at lower al-titudes and modulationally stable waves should break at higher altitudes than predicted by linear theory. This prediction is borne out by fully nonlinear numerical simulations of the anelastic equations. A range of sim-ulations is performed to quantify where overturning actually occurs.
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The Boussinesq equations provide a convenient modeling framework for studies of gravity-wave dynamics not affected by the impact of varying density on the wave amplitude. In the atmosphere, however, gravity waves undergo tremendous amplitude growth in their upward propagation since atmospheric density has a very strong vertical dependence. This effect, leading to all kinds of wave instabilities, is decisive for the corresponding wave mean-flow interaction. A study of these processes within the complete Euler equations is complicated by their incorporation of sound waves which might at most be of secondary importance in this context. A way around this is the use of an approximated equation set, filtered of sound waves, but representing the dynamics of gravity waves at good accuracy. Both the classic anelastic equations and the pseudo-incompressible equations offer themselves for this purpose. The question arises which of the two, if any, are consistent with a rigorous multiple-scale asymptotics of gravity-wave dynamics in the atmosphere. Hence, such an asymptotics is used to analyze the Euler equations so that the dynamical situation of a gravity wave (GW) near breaking level is best approximated. A simple saturation argument is used to obtain a potential-temperature wave scale, while linear theory yields from the latter the velocity scale, and the wave Exner pressure scale. It also determines the time scale once the spatial scale has been set. As small expansion parameter the product of vertical wave number and potential temperature scale height is used. It is shown that the resulting equation hierarchy is consistent with that obtained from the pseudo-incompressible equations, both for non-hydrostatic and hydrostatic gravity waves. This gives a mathematical justification for the use of the pseudo-incompressible equations for studies of gravity-wave breaking in the atmosphere. An analogous argument does not seem to exist for the anelastic equations
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Gravity waves (GWs) and thermal tides are important phenomena in middle-atmosphere dy-namics. Breaking GWs have a major impact on the mean circulation in the middle atmosphere (MA). Due to the limitations in computational power most complex MA circulation models have to incorporate the effect of unresolved GWs via an efficient parametrization. Typically, these are of vertical column type and ignore horizontal and temporal variations in the background fields. However, highly transient tidal perturbations are always present and dominate diurnal variations in the MA through which the GWs propagate. Even in studies of the interaction between GWs and these thermal tides, a possibly important aspect of tidal dynamics, columnar parametrizations of GWs have been applied which do not account for the time dependence of thermal tides. A ray tracing technique is used to illuminate the impact of horizontal gradients of the back-ground (including the tides) and its time dependence on the propagation and dissipation of GWs. It is shown that tidal transience leads to a modulation of the absolute, or sometimes called ground-based, frequency of slowly propagating GWs. Due to large tidal wind variations in the upper mesosphere most parts of the assumed GW spectrum are slowed down in critical layer type regions. Then, the combined action of horizontal wave number refraction and fre-quency modulation induce changes in the horizontal phase speed which may exceed the initial phase speed by orders of magnitude. The phase speed variations have the tendency to follow the shape of the tidal background wind. This effect leads to less critical layer filtering of GWs and therefore decreased periodic background flow forcing due to momentum flux divergences as compared to a classical vertical column parametrization of instantaneously adjusting GW trains.
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The effects of gravity wave saturation on the mean circulation and thermal structure of the middle atmosphere were investigated using quasi-compressible numerical and semianalytic wave action models. The study covered mean flow accelerations associated with the propagation of large amplitude gravity waves and convective adjustments to the gravity wave saturation and without saturation. Large amplitude wave motions, near-linear mean wind accelerations and gravity self-accelerations all produced nonlinear features with or without saturation. The wave packet experienced vertical spreading and a concommitant reduction in the local wave action density. A non-WKB effect was detected insofar as self-acceleration permitted the wave motion to propagate beyond a critical level dislocation. The quasi-linear effects are concluded dominant for middle atmospheric motions in some situations and not amenable to linear modeling.
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This paper describes a new computationally efficient, ultrasimple nonorographic spectral gravity wave parameterization model. Its predictions compare favorably, though not perfectly, with a model of gravity wave propagation and breaking that computes the evolution with altitude of a full, frequency- and wavenumber-dependent gravity wave spectrum. The ultrasimple model depends on making the midfrequency (hydrostatic, nonrotating) approximation to the dispersion relation, as in Hines' parameterization. This allows the full frequency-wavenumber spectrum of pseudomomentum flux to be integrated with respect to frequency, and thus reduced to a spectrum that depends on vertical wavenumber m and azimuthal direction ø only. The ultrasimple model treats the m dependence as consisting of up to three analytically integrable segments, or "parts". This allows the total pseudomomentum flux to be evaluated by using analytical expressions for the areas under the parts rather than by performing numerical quadratures. The result is a much greater computational efficiency. The model performs significantly better than an earlier model that treated the m dependence as consisting of up to two parts. Numerical experiments show that similar models with more than three parts using the midfrequency approximation yield little further improvement. The limiting factor is the midfrequency approximation and not the number of parts.
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Using a new generalization of the Eliassen-Palm relations, we discuss the zonal-mean-flow tendency /t due to waves in a stratified, rotating atmosphere, with particular attention to equatorially trapped modes. Wave transience, forcing and dissipation are taken into account in a very general way. The theory makes it possible to discuss the latitudinal (y) and vertical (z) dependence of /t qualitatively and calculate it directly from an approximate knowledge of the wave structure. For equatorial modes it reveals that the y profile of /t is strongly dependent on the nature of the forcing or dissipation mechanism. A by-product of the theory is a far-reaching generalization of the theorems of Charney-Drazin, Dickinson and Holton on the forcing of /t by conservative linear waves.Implications for the quasi-biennial oscillation in the equatorial stratosphere are discussed. Graphs of y profiles of /t are given for the equatorial waves considered in the recent analysis of observational data by Lindzen and Tsay (1975). The y profile of uI t for Rossby-gravity and inertio-gravity modes, in Lindzen and Tsay's parameter ranges, prove extremely sensitive to whether or not small amounts of mechanical dissipation are present alongside the radiative-photochemical dissipation of the waves. The probable importance of low-frequency Rossby waves for the momentum budget of the descending easterlies is suggested.
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An approximate theory is developed of small-amplitude transient eddies on a slowly varying time-mean flow. Central to this theory is a flux MT, which in most respects constitutes a generalization of the Eliassen-Palm flux to three dimensions; it is a conservable measure of the flux of eddy activity (for small amplitude transients) and is parallel to group velocity for an almost-plane wave train. The use of this flux as a diagnostic of transient eddy propagation is demonstrated by application of the theory to a ten-year climatology of the Northern Hemisphere winter circulation. Results show the anticipated concentration of eddy flux along the major storm tracks.While, in a suitably transformed system, MT may be regarded as a flux of upstream momentum, it is not a complete description of the eddy forcing of the mean flow; additional effects arise due to downstream transience (i.e., spatial inhomogeneity in the direction of the time-mean flow) of the eddy amplitudes.The relation between MT and the E-vector' of Hoskins et al. is discussed.
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Extensions of Weinstock's theory of nonlinear gravity waves and a parameterization of the related momentum deposition are developed. Our approach, which combines aspects of Hines' Doppler spreading theory with Weinstock's theory of nonlinear wave diffusion, treats the low-frequency part of the gravity wave spectrum as an additional background flow for higher-frequency waves. This technique allows one to calculate frequency shifting and wave amplitude damping produced by the interaction with this additional background wind. For a nearly monochromatic spectrum the parameterization formulae for wave drag coincide with those of Lindzen. It is shown that two processes should be distinguished: wave breaking due to instabilities and saturation due to nonlinear diffusionlike processes. The criteria for wave breaking and wave saturation in terms of wave spectra are derived. For a saturated spectrum the power spectral density's (PSD) dependence S(m) = AN2/m3 is obtained, where m is the vertical wavenumber and N is the Brunt-Väisäla frequency. Unlike Weinstock's original formulation, our coefficient of proportionality A is a slowly varying function of m and mean wind. For vertical wavelengths ranging from 10 km to 100 m and for typical wind shears, A varies from one half to one ninth. Calculations of spectral evolution with height as well as related profiles of wave drag are shown. These results reproduce vertical wavenumber spectral tail slopes which vary near the -3 value reported by observations. An explanation of these variations is given.
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An overview of the parameterization of gravity wave drag in numerical weather prediction and climate simulation models is presented. The focus is mainly on understanding the current status of gravity wave drag parameterization as a step towards the new parameterizations that will be needed for the next generation of atmospheric models. Both the early history and latest developments in the field are discussed. Parameterizations developed specifically for orographic and convective sources of gravity waves are described separately, as are newer parameterizations that collectively treat a spectrum of gravity wave motions. Differences in parameterization issues and approaches between the lower and middle atmospheres are highlighted. Various emerging issues are also discussed, such as explicitly resolved gravity waves and gravity wave drag in models, and a range of unparameterized gravity wave processes that may need future attention for the next generation of gravity wave drag parameterizations in models
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Durran’s pseudo-incompressible equations are integrated in a mass and momentum conserving way with a new implicit turbulence model. This system is sound-proof, which has two major advantages over fully compressible systems: the CFL condition for stable time advancement is no longer dictated by the speed of sound and all waves in the model are clearly gravity waves (GW) or geostrophic modes. Thus, the pseudo-incompressible equations are an ideal laboratory model for studying GW generation, propagation and breaking. Gravity wave breaking creates turbulence which needs to be parameterised. For the first time the adaptive local deconvolution method (ALDM) for implicit large eddy simulation (LES) is applied to non-Boussinesq stratified flows. ALDM provides a turbulence model that is fully merged with the discretisation of the flux function. In the context of non-Boussinesq stratified flows this poses some new numerical challenges, the solution of which we present in this text. In numerical test cases we show the agreement of the results with the literature (Robert’s hot/cold bubble test case), we present the sensitivity to the model’s resolution and discretisation and demonstrate qualitatively the behaviour of the implicit turbulence model for a 2D breaking gravity wave packet.
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Idealized model examples of non-dissipative wave–mean interactions, using small-amplitude and slow-modulation approximations, are studied in order to re-examine the usual assumption that the only important interactions are dissipative. The results clarify and extend the body of wave–mean interaction theory on which our present understanding of, for instance, the global-scale atmospheric circulation depends (e.g. Holton et al. 1995). The waves considered are either gravity or inertia–gravity waves. The mean flows need not be zonally symmetric, but are approximately ‘balanced’ in a sense that non-trivially generalizes the standard concepts of geostrophic or higher-order balance at low Froude and/or Rossby number. Among the examples studied are cases in which irreversible mean-flow changes, capable of persisting after the gravity waves have propagated out of the domain of interest, take place without any need for wave dissipation. The irreversible mean-flow changes can be substantial in certain circumstances, such as Rossby-wave resonance, in which potential-vorticity contours are advected cumulatively. The examples studied in detail use shallow-water systems, but also provide a basis for generalizations to more realistic, stratified flow models. Independent checks on the analytical shallow-water results are obtained by using a different method based on particle-following averages in the sense of ‘generalized Lagrangian-mean theory’, and by verifying the theoretical predictions with nonlinear numerical simulations. The Lagrangian-mean method is seen to generalize easily to the three-dimensional stratified Boussinesq model, and to allow a partial generalization of the results to finite amplitude. This includes a finite-amplitude mean potential-vorticity theorem with a larger range of validity than had been hitherto recognized.
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This paper continues the work started in Part 1 (Reznik, Zeitlin & Ben Jelloul 2001) and generalizes it to the case of a stratified environment. Geostrophic adjustment of localized disturbances is considered in the context of the two-layer shallow-water and continuously stratified primitive equations in the vertically bounded and horizontally infinite domain on the $f$-plane. Using multiple-time-scale perturbation expansions in Rossby number $\hbox{\it Ro}$ we show that stratification does not substantially change the adjustment scenario established in Part 1 and any disturbance of well-defined scale is split in a unique way into slow and fast components with characteristic time scales $f_0^{-1}$ and $(f_0 \hbox{\it Ro})^{-1}$ respectively, where $f_0$ is the Coriolis parameter. As in Part 1 we distinguish two basic dynamical regimes: quasi-geostrophic (QG) and frontal geostrophic (FG) with small and large deviations of the isopycnal surfaces, respectively. We show that the dynamics of the FG regime in the two-layer model depends strongly on the ratio of the layer depths. The difference between QG and FG scenarios of adjustment is demonstrated. In the QG case the fast component of the flow essentially does not ‘feel’ the slow one and is rapidly dispersed leaving the slow component to evolve according to the standard QG equation (corrections to this equation are found for times $t\,{\gg}\, (f_0 \hbox{\it Ro})^{-1}$). In the FG case the fast component is a packet of inertial oscillations produced by the initial perturbation. The space-time evolution of the envelope of inertial oscillations obeys a Schrödinger-type modulation equation with coefficients depending on the slow component. In both QG and FG cases we show by direct computations that the fast component does not produce any drag terms in the equations for the slow component; the slow component remains close to the geostrophic balance. However, in the continuously stratified FG regime, as well as in the two-layer regime with the layers of comparable thickness, the splitting is incomplete in the sense that the slow vortical component and the inertial oscillations envelope evolve on the same time scale.
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We present a theoretical study of a fundamentally new wave–mean or wave–vortex interaction effect able to force persistent, cumulative change in mean flows in the absence of wave breaking or other kinds of wave dissipation. It is associated with the refraction of non-dissipating waves by inhomogeneous mean (vortical) flows. The effect is studied in detail in the simplest relevant model, the two-dimensional compressible flow equations with a generic polytropic equation of state. This includes the usual shallow-water equations as a special case. The refraction of a narrow, slowly varying wavetrain of small-amplitude gravity or sound waves obliquely incident on a single weak (low Froude or Mach number) vortex is studied in detail. It is shown that, concomitant with the changes in the waves' pseudomomentum due to the refraction, there is an equal and opposite recoil force that is felt, in effect, by the vortex core. This effective force is called a ‘remote recoil’ to stress that there is no need for the vortex core and wavetrain to overlap in physical space. There is an accompanying ‘far-field recoil’ that is still more remote, as in classical vortex-impulse problems. The remote-recoil effects are studied perturbatively using the wave amplitude and vortex weakness as small parameters. The nature of the remote recoil is demonstrated in various set-ups with wavetrains of finite or infinite length. The effective recoil force ${\bm R}_V$ on the vortex core is given by an expression resembling the classical Magnus force felt by moving cylinders with circulation. In the case of wavetrains of infinite length, an explicit formula for the scattering angle $\theta_*$ of waves passing a vortex at a distance is derived correct to second order in Froude or Mach number. To this order ${\bm R}_V\,{\propto}\,\theta_*$. The formula is cross-checked against numerical integrations of the ray-tracing equations. This work is part of an ongoing study of internal-gravity-wave dynamics in the atmosphere and may be important for the development of future gravity-wave parametrization schemes in numerical models of the global atmospheric circulation. At present, all such schemes neglect remote-recoil effects caused by horizontally inhomogeneous mean flows. Taking these effects into account should make the parametrization schemes significantly more accurate.
Article
Full-text available
An exact and very general Lagrangian-mean description of the back effect of oscillatory disturbances upon the mean state is given. The basic formalism applies to any problem whose governing equations are given in the usual Eulerian form, and irrespective of whether spatial, temporal, ensemble, or ‘two-timing’ averages are appropriate. The generalized Lagrangian-mean velocity cannot be defined exactly as the ‘mean following a single fluid particle’, but in cases where spatial averages are taken can easily be visualized, for instance, as the motion of the centre of mass of a tube of fluid particles which lay along the direction of averaging in a hypothetical initial state of no disturbance. The equations for the Lagrangian-mean flow are more useful than their Eulerian-mean counterparts in significant respects, for instance in explicitly representing the effect upon mean-flow evolution of wave dissipation or forcing. Applications to irrotational acoustic or water waves, and to astrogeophysical problems of waves on axisymmetric mean flows are discussed. In the latter context the equations embody generalizations of the Eliassen-Palm and Charney-Drazin theorems showing the effects on the mean flow of departures from steady, conservative waves, for arbitrary, finite-amplitude disturbances to a stratified, rotating fluid, with allowance for self-gravitation as well as for an external gravitational field. The equations show generally how the pseudomomentum (or wave ‘momentum’) enters problems of mean-flow evolution. They also indicate the extent to which the net effect of the waves on the mean flow can be described by a ‘radiation stress’, and provide a general framework for explaining the asymmetry of radiation-stress tensors along the lines proposed by Jones (1973).
Article
We derive an asymptotic model that describes the nonlinear coupled evolution of (i) near-inertial waves (NIWs), (ii) balanced quasi-geostrophic flow and (iii) near-inertial second harmonic waves with frequency near $2f_{0}$ , where $f_{0}$ is the local inertial frequency. This ‘three-component’ model extends the two-component model derived by Xie & Vanneste ( J. Fluid Mech. , vol. 774, 2015, pp. 143–169) to include interactions between near-inertial and $2f_{0}$ waves. Both models possess two conservation laws which together imply that oceanic NIWs forced by winds, tides or flow over bathymetry can extract energy from quasi-geostrophic flows. A second and separate implication of the three-component model is that quasi-geostrophic flow catalyses a loss of NIW energy to freely propagating waves with near- $2f_{0}$ frequency that propagate rapidly to depth and transfer energy back to the NIW field at very small vertical scales. The upshot of near- $2f_{0}$ generation is a two-step mechanism whereby quasi-geostrophic flow catalyses a nonlinear transfer of near-inertial energy to the small scales of wave breaking and diapycnal mixing. A comparison of numerical solutions with both Boussinesq and three-component models for a two-dimensional initial value problem reveals strengths and weaknesses of the model while demonstrating the extraction of quasi-geostrophic energy and production of small vertical scales.
Article
Under assumptions of horizontal homogeneity and isotropy, one may derive relations between rotational or divergent kinetic energy spectra and velocities along one-dimensional tracks, such as might be measured by aircraft. Two recent studies, differing in details of their implementation, have applied these relations to the Measurement of Ozone and Water Vapor by Airbus In-Service Aircraft (MOZAIC) dataset and reached different conclusions with regard to the mesoscale ratio of divergent to rotational kinetic energy. In this study the accuracy of the method is assessed using global atmospheric simulations performed with the Model for Prediction Across Scales, where the exact decomposition of the horizontal winds into divergent and rotational components may be easily computed. For data from the global simulations, the two approaches yield similar and very accurate results. Errors are largest for the divergent component on synoptic scales, which is shown to be related to a very dominant rotational mode...
Article
[1] Atmospheric gravity waves have been a subject of intense research activity in recent years because of their myriad effects and their major contributions to atmospheric circulation, structure, and variability. Apart from occasionally strong lower-atmospheric effects, the major wave influences occur in the middle atmosphere, between ∼ 10 and 110 km altitudes because of decreasing density and increasing wave amplitudes with altitude. Theoretical, numerical, and observational studies have advanced our understanding of gravity waves on many fronts since the review by Fritts [1984a]; the present review will focus on these more recent contributions. Progress includes a better appreciation of gravity wave sources and characteristics, the evolution of the gravity wave spectrum with altitude and with variations of wind and stability, the character and implications of observed climatologies, and the wave interaction and instability processes that constrain wave amplitudes and spectral shape. Recent studies have also expanded dramatically our understanding of gravity wave influences on the large-scale circulation and the thermal and constituent structures of the middle atmosphere. These advances have led to a number of parameterizations of gravity wave effects which are enabling ever more realistic descriptions of gravity wave forcing in large-scale models. There remain, nevertheless, a number of areas in which further progress is needed in refining our understanding of and our ability to describe and predict gravity wave influences in the middle atmosphere. Our view of these unknowns and needs is also offered. Abstract [1] Atmospheric gravity waves have been a subject of intense research activity in recent years because of their myriad effects and their major contributions to atmospheric circulation, structure, and variability. Apart from occasionally strong lower-atmospheric effects, the major wave influences occur in the middle atmosphere, between ∼ 10 and 110 km altitudes because of decreasing density and increasing wave amplitudes with altitude. Theoretical, numerical, and observational studies have advanced our understanding of gravity waves on many fronts since the review by Fritts [1984a]; the present review will focus on these more recent contributions. Progress includes a better appreciation of gravity wave sources and characteristics, the evolution of the gravity wave spectrum with altitude and with variations of wind and stability, the character and implications of observed climatologies, and the wave interaction and instability processes that constrain wave amplitudes and spectral shape. Recent studies have also expanded dramatically our understanding of gravity wave influences on the large-scale circulation and the thermal and constituent structures of the middle atmosphere. These advances have led to a number of parameterizations of gravity wave effects which are enabling ever more realistic descriptions of gravity wave forcing in large-scale models. There remain, nevertheless, a number of areas in which further progress is needed in refining our understanding of and our ability to describe and predict gravity wave influences in the middle atmosphere. Our view of these unknowns and needs is also offered.
Article
The interaction between solar tides (STs) and gravity waves (GWs) is studied via the coupling of a three-dimensional ray-tracer model and a linear tidal model. The ray-tracer model describes GW dynamics on a spatially and time dependent background formed by a monthly mean climatology and STs. It does not suffer from typical simplifications of conventional GW parameterizations where horizontal GW propagation and the effects of horizontal background gradients on GW dynamics are neglected. The ray-tracer model uses a variant of Wentzel-Kramers-Brillouin (WKB) theory where a spectral description in position-wavenumber space is helping to avoid numerical instabilities otherwise likely to occur in caustic-like situations. The tidal model has been obtained by linearization of the primitive equations about a monthly mean, allowing for stationary planetary waves. The communication between ray-tracer model and tidal model is facilitated using latitude and altitude-dependent coefficients, named Rayleigh-friction and Newtonian-relaxation, and obtained from regressing GW momentum and buoyancy fluxes against the STs and their tendencies. These coefficients are calculated by the ray-tracer model and then implemented into the tidal model. An iterative procedure updates successively the GW fields and the tidal fields until convergence is reached. Notwithstanding the simplicity of the employed GW source many aspects of observed tidal dynamics are reproduced. It is shown that the conventional `single-column'' approximation leads to significantly overestimated GW fluxes and hence underestimated ST amplitudes, pointing at a sensitive issue of GW parameterizations in general.
Article
Longitudinal and transverse structure functions, \$D_{ll}=\langle {\it\delta}u_{l}{\it\delta}u_{l}\rangle\$ and \$D_{tt}=\langle {\it\delta}u_{t}{\it\delta}u_{t}\rangle\$, can be calculated from aircraft data. Here, \${\it\delta}\$ denotes the increment between two points separated by a distance \$r\$, \$u_{l}\$ and \$u_{t}\$ the velocity components parallel and perpendicular to the aircraft track respectively and \$\langle \,\rangle\$ an average. Assuming statistical axisymmetry and making a Helmholtz decomposition of the horizontal velocity, \$\boldsymbol{u}=\boldsymbol{u}_{r}+\boldsymbol{u}_{d}\$, where \$\boldsymbol{u}_{r}\$ is the rotational and \$\boldsymbol{u}_{d}\$ the divergent component of the velocity, we derive expressions relating the structure functions \$D_{rr}=\langle {\it\delta}\boldsymbol{u}_{r}\boldsymbol{\cdot }{\it\delta}\boldsymbol{u}_{r}\rangle\$ and \$D_{dd}=\langle {\it\delta}\boldsymbol{u}_{d}\boldsymbol{\cdot }{\it\delta}\boldsymbol{u}_{d}\rangle\$ to \$D_{ll}\$ and \$D_{tt}\$. Corresponding expressions are also derived in spectral space. The decomposition is applied to structure functions calculated from aircraft data. In the lower stratosphere, \$D_{rr}\$ and \$D_{dd}\$ both show a nice \$r^{2/3}\$-dependence for \$r\in [2,20]\ \text{km}\$. In this range, the ratio between rotational and divergent energy is a little larger than unity, excluding gravity waves as the principal agent behind the observations. In the upper troposphere, \$D_{rr}\$ and \$D_{dd}\$ show no clean \$r^{2/3}\$-dependence, although the overall slope of \$D_{dd}\$ is close to \$2/3\$ for \$r\in [2,400]\ \text{km}\$. The ratio between rotational and divergent energy is approximately three for \$r<100\ \text{km}\$, excluding gravity waves also in this case. We argue that the possible errors in the decomposition at scales of the order of 10 km are marginal.
Article
By using the renormalization group (RG) method, the interaction between balanced flows and Doppler-shifted inertia-gravity waves (GWs) is formulated for the hydrostatic Boussinesq equations on the f plane. The derived time-evolution equations [RG equations (RGEs)] describe the spontaneous GW radiation from the components slaved to the vortical flow through the quasi resonance, together with the GW radiation reaction on the large-scale flow. The quasi resonance occurs when the space-time scales of GWs are partially comparable to those of slaved components. This theory treats a coexistence system with slow time scales composed of GWs significantly Doppler-shifted by the vortical flow and the balanced flow that interact with each other. The theory includes five dependent variables having slow time scales: one slow variable (linear potential vorticity), two Doppler-shifted fast ones (GW components), and two diagnostic fast ones. Each fast component consists of horizontal divergence and ageostrophic vorticity. The spontaneously radiated GWs are regarded as superpositions of the GW components obtained as low-frequency eigenmodes of the fast variables in a given vortical flow. Slowly varying nonlinear terms of the fast variables are included as the diagnostic components, which are the sum of the slaved components and the GW radiation reactions. A comparison of the balanced adjustment equation (BAE) by Plougonven and Zhang with the linearized RGE shows that the RGE is formally reduced to the BAE by ignoring the GW radiation reaction, although the interpretation on the GW radiation mechanism is significantly different; GWs are radiated through the quasi resonance with a balanced flow because of the time-scale matching.
Article
The renormalization group equations (RGEs) describing spontaneous inertia-gravity wave (GW) radiation from part of a balanced flow through a quasi resonance that were derived in a companion paper by Yasuda et al. are validated through numerical simulations of the vortex dipole using the Japan Meteorological Agency nonhydrostatic model (JMA-NHM). The RGEs are integrated for two vortical flow fields: the first is the initial condition that does not contain GWs used for the JMA-NHM simulations, and the second is the simulated thirtieth-day field by the JMA-NHM. The theoretically obtained GW distributions in both RGE integrations are consistent with the numerical simulations using the JMA-NHM. This result supports the validity of the RGE theory. GW radiation in the dipole is physically interpreted either as the mountain-wave-like mechanism proposed by McIntyre or as the velocity-variation mechanism proposed by Viudez. The shear of the large-scale flow likely determines which mechanism is dominant. In addition, the distribution of GW momentum fluxes is examined based on the JMA-NHM simulation data. The GWs propagating upward from the jet have negative momentum fluxes, while those propagating downward have positive ones. The magnitude of momentum fluxes is approximately proportional to the sixth power of the Rossby number between 0.15 and 0.4.
Article
[1] The atmospheric gravity wave energy spectra often show power law dependencies with wavenumbers and frequencies. A simple mechanism involving off-resonant scale-separated interactions is proposed for their formation, namely the refraction of the wave packets in pseudorandom shears encountered during their vertical propagation. In the Boussinesq and rotating frame approximation the evolution of the spectral distribution of wave action is calculated within the eikonal formalism, i.e., via the simulation of the ray paths for an ensemble of elementary wave packets. The energy spectra are then easily built from the wave action spectra. Experiments are conducted where wave packets propagate away from Dirac delta function, or spectrally uniform sources at low altitudes, in realistic atmospheric background flows. The energy spectra show dependencies with the vertical wavenumber m and horizontal wavenumber k that are consistent with the most widely recognized empirical spectral models. A specific focus is given on the vertical evolution of the vertical wavenumber spectrum. The spectrum shows an invariant scaling as N2/m3 at large wavenumbers. It possesses a central wavenumber whose value depends on the total wave energy and is controlled by the statistics of the background mean flow. Similarly, the wave packet azimuths show an increasingly strong anisotropy resulting from the wave mean flow interaction at critical levels.
Book
The study of internal gravity waves provides many challenges: they move along interfaces as well as in fully three-dimensional space, at relatively fast temporal and small spatial scales, making them difficult to observe and resolve in weather and climate models. Solving the equations describing their evolution poses various mathematical challenges associated with singular boundary value problems and large amplitude dynamics. This book provides the first comprehensive treatment of the theory for small and large amplitude internal gravity waves. Over 120 schematics, numerical simulations and laboratory images illustrate the theory and mathematical techniques, and 130 exercises enable the reader to apply their understanding of the theory. This is an invaluable single resource for academic researchers and graduate students studying the motion of waves within the atmosphere and ocean, and also mathematicians, physicists and engineers interested in the properties of propagating, growing and breaking waves.
Article
[1] For several decades, jets and fronts have been known from observations to be significant sources of internal gravity waves in the atmosphere. Motivations to investigate these waves have included their impact on tropospheric convection, their contribution to local mixing and turbulence in the upper-troposphere, their vertical propagation into the middle atmosphere and the forcing of its global circulation. While many different studies have consistently highlighted jet exit regions as a favored locus for intense gravity waves, the mechanisms responsible for their emission had long remained elusive: one reason is the complexity of the environment in which the waves appear, another is that the waves constitute small deviations from the balanced dynamics of the flow generating them, i.e., they arise beyond our fundamental understanding of jets and fronts based on approximations that filter out gravity waves. Over the past two decades, the pressing need for improving parameterizations of non-orographic gravity waves in climate models that include a stratosphere has stimulated renewed investigations. The purpose of this review is to presents current knowledge and understanding on gravity waves near jets and fronts from observations, theory and modeling, and to discuss challenges for progress in coming years.
Article
The dynamics of internal gravity waves is modelled using WKB theory in position wavenumber phase space. A transport equation for the phase-space wave-action density is derived for describing one-dimensional wave fields in a background with height-dependent stratification and height- and time-dependent horizontal-mean horizontal wind. The mean wind is coupled to the waves through the divergence of the mean vertical flux of horizontal momentum associated with the waves. The phase-space approach bypasses the caustics problem that occurs in WKB ray-tracing models when the wavenumber becomes a multivalued function of position, such as in the case of a wave packet encountering a reflecting jet or in the presence of a time-dependent background flow. Two numerical models were developed to solve the coupled equations for the wave-action density and horizontal mean wind: an Eulerian model using a finite-volume method, and a Lagrangian “phase-space ray tracer” that transports wave-action density along phase-space paths determined by the classical WKB ray equations for position and wavenumber. The models are used to simulate the upward propagation of a Gaussian wave packet through a variable stratification, a wind jet, and the mean flow induced by the waves. Results from the WKB models are in good agreement with simulations using a weakly nonlinear wave-resolving model as well as with a fully nonlinear large-eddy-simulation model. The work is a step toward more realistic parameterizations of atmospheric gravity waves in weather and climate models.
Book
Principles of fluid dynamics are applied to large-scale flows in the oceans and the atmosphere in this text, intended as a core curriculum in geophysical fluid dynamics. Emphasis throughout the book is devoted to basing scaling techniques and the derivation of systematic approximations to the equations of motion. The inviscid dynamics of a homogeneous fluid are examined to reveal the properties of quasi-geostrophic motion. Attention is given to density stratification as a basis for potential vorticity dynamics. Discussions are presented of Rossby waves, inertial boundary currents, the beta-plane, energy propagation, and wave interaction. Turbulent mixing is mentioned in the context of large-scale flows. The use of the homogeneous model in investigating wind-driven ocean circulation is demonstrated, and the quasi-geostrophic dynamics of a stratified fluid are studied for a flow on a sphere. Finally, instability theory is exposed as a fundamental concept for dynamic meteorology and ocean dynamics.
Book
Interactions between waves and mean flows play a crucial role in understanding the long-term aspects of atmospheric and oceanographic modelling. Indeed, our ability to predict climate change hinges on our ability to model waves accurately. This book gives a modern account of the nonlinear interactions between waves and mean flows such as shear flows and vortices. A detailed account of the theory of linear dispersive waves in moving media is followed by a thorough introduction to classical wave–mean interaction theory. The author then extends the scope of the classical theory and lifts its restriction to zonally symmetric mean flows. The book is a fundamental reference for graduate students and researchers in fluid mechanics, and can be used as a text for advanced courses; it will also be appreciated by geophysicists and physicists who need an introduction to this important area in fundamental fluid dynamics and atmosphere-ocean science.
Article
A spectral parameterization of mean-flow forcing due to breaking gravity waves is described for application in the equations of motion in atmospheric models. The parameterization is based on linear theory and adheres closely to fundamental principles of conservation of wave action flux, linear stability, and wave-mean-flow interaction. Because the details of wave breakdown and nonlinear interactions are known to be very complex and are still poorly understood, only the simplest possible assumption is made: that the momentum fluxes carried by the waves are deposited locally and entirely at the altitude of linear wave breaking. This simple assumption allows a straightforward mapping of the momentum flux spectrum, input at a specified source altitude, into vertical profiles of mean-flow force. A coefficient of eddy diffusion can also be estimated. The parameterization can be used with any desired input spectrum of momentum flux. The results are sensitive to the details of this spectrum and also realistically sensitive to the background vertical shear and stability profiles. These sensitivities make the parameterization ideally suited for studying both the effects of gravity waves from unique sources like topography and convection as well as generalized broad input spectra. Existing constraints on input parameters are also summarized from the available observations. With these constraints, the parameterization generates realistic variations in gravity-wave-driven, mean-flow forcing.
Article
Middle atmospheric general circulation models (GCMs) must employ a parameterization for small-scale gravity waves (GWs). Such parameterizations typically make very simple assumptions about gravity wave sources, such as uniform distribution in space and time or an arbitrarily specified GW source function. The authors present a configuration of theWholeAtmosphereCommunity ClimateModel (WACCM) that replaces the arbitrarily specifiedGWsource spectrum with GWsource parameterizations. For the nonorographic wave sources, a frontal system and convectiveGWsource parameterization are used. These parameterizations link GW generation to tropospheric quantities calculated by the GCM and provide a model-consistent GW representation. With the newGWsource parameterization, a reasonable middle atmospheric circulation can be obtained and the middle atmospheric circulation is better in several respects than that generated by a typical GW source specification. In particular, the interannual NH stratospheric variability is significantly improved as a result of the source-oriented GW parameterization. It is also shown that the addition of a parameterization to estimate mountain stress due to unresolved orography has a large effect on the frequency of stratospheric sudden warmings in the NH stratosphere by changing the propagation of stationary planetary waves into the polar vortex.
Article
Ogura and Phillips derived the original anelastic model through systematic formal asymptotics using the flow Mach number as the expansion parameter. To arrive at a reduced model that would simultaneously represent internal gravity waves and the effects of advection on the same time scale, they had to adopt a distinguished limit requiring that the dimensionless stability of the background state be on the order of the Mach number squared. For typical flow Mach numbers of , this amounts to total variations of potential temperature across the troposphere of less than one Kelvin (i.e., to unrealistically weak stratification). Various generalizations of the original anelastic model have been proposed to remedy this issue. Later, Durran proposed the pseudoincompressible model following the same goals, but via a somewhat different route of argumentation. The present paper provides a scale analysis showing that the regime of validity of two of these extended models covers stratification strengths on the order of (hsc/θ)dθ/dz < M2/3, which corresponds to realistic variations of potential temperature θ across the pressure scale height hsc of . Specifically, it is shown that (i) for (hsc/θ)dθ/dz < Mμ with 0 < μ < 2, the atmosphere features three asymptotically distinct time scales, namely, those of advection, internal gravity waves, and sound waves; (ii) within this range of stratifications, the structures and frequencies of the linearized internal wave modes of the compressible, anelastic, and pseudoincompressible models agree up to the order of Mμ; and (iii) if μ < ⅔, the accumulated phase differences of internal waves remain asymptotically small even over the long advective time scale. The argument is completed by observing that the three models agree with respect to the advective nonlinearities and that all other nonlinear terms are of higher order in M.
Article
Studies on the spontaneous emission of gravity waves from jets, both observational and numerical, have emphasized that excitation of gravity waves occurred preferentially near regions of imbalance. Yet a quantitative relation between the several large-scale diagnostics of imbalance and the excited waves is still lacking. The purpose of the present note is to investigate one possible way to relate quantitatively the gravity waves to diagnostics of the large-scale flow that is exciting them. Scaling arguments are used to determine how the large-scale flow may provide a forcing on the right-hand side of a wave equation describing the linear dynamics of the excited waves. The residual of the nonlinear balance equation plays an important role in this forcing.
Article
The observational evidence for k to the -5/3 law behavior in the atmospheric kinetic energy spectrum is reviewed. This evidence includes the results of atmospheric wind variability studies and the observed scale dependence of atmospheric dispersion. It is concluded that k to the -5/3 law behavior for time and space scales greater than those that can be three-dimensionally isotropic is probably a manifestation of the two-dimensional reverse-cascading energy inertial range.
Article
An analysis is made of Gage's proposal that the horizontal energy spectrum at mesoscale wavelengths is produced by upscale energy transfer through quasi-two-dimensional turbulence. It is suggested that principal sources of such energy can be found in decaying convective clouds and thunderstorm anvil outflows. These are believed to evolve similarly to the wake of a moving body in a stably stratified flow. Following the scale analysis by Riley, Metcalfe and Weissman it is expected that, in the presence of strong stratification, initially three-dimensionally isotropic turbulence divides roughly equally into gravity waves and stratified (quasi-two- dimensional) turbulence. The former then propagates away from the generation region, while the latter propagates in spectral space to larger scales, forming the 5/3 upscale transfer spectrum predicted by Kraichnan. Part of the energy of the stratified turbulence is recycled into three-dimensional turbulence by shearing instability, but the upscale escape of only a few percent of the total energy released by small-scale turbulence is apparently sufficient to explain the observed mesoscale energy spectrum of the troposphere. A close analogy is found between the turbulence-gravity wave exchanges considered here and the turbulence--wave exchanges discussed by Rhines and Williams.
Article
This article reviews the methods of wave–mean interaction theory for classical fluid dynamics, and for geophysical fluid dynamics in particular, providing a few examples for illustration. It attempts to bring the relevant equations into their simplest possible form, which highlights the organizing role of the circulation theorem in the theory. This is juxtaposed with a simple account of superfluid dynamics and the attendant wave–vortex interactions as they arise in the nonlinear Schrödinger equation. Here the fundamental physical situation is more complex than in the geophysical case, and the current mathematical understanding is more tentative. Classical interaction theory might be put to good use in the theoretical and numerical study of quantum fluid dynamics.
Article
Scale analysis suggests that use of this "pseudo-incompressible equation' is justified if the Langrangian time scale of the disturbance is large compared with the time scale for sound wave propagation and the perturbation pressure is small compared to the vertically varying mean-state pressure. The mass-balance in the "pseudo-incompressible approximation' accounts for those density perturbations associated (through the equation of state) with perturbations in the temperature field. Density fluctuations associated with perturbations in the pressure field are neglected. The pseudo-incompressible equation is identical to the anelastic continuity equation when the mean stratification is adiabatic. The pseudo-incompressible approximation yields a system of equations suitable for use in nonhydrostatic numerical models. It also permits the diagnostic calculation of the vertical velocity in adiabatic flow, and might also be used to compute the net heating rate in a diabatic flow from extremely accurate observations of the three-dimensional velocity field and very coarse resolution (single sounding) thermodynamic data. -from Author
Article
Conservable quantities measuring ‘wave activity’ are discussed. The equation for the most fundamental such quantity, wave-action, is derived in a simple but very general form which does not depend on the approximations of slow amplitude modulation, linearization, or conservative motion. The derivation is elementary , in the sense that a variational formulation of the equations of fluid motion is not used. The result depends, however, on a description of the disturbance in terms of particle displacements rather than velocities. A corollary is an elementary but general derivation of the approximate form of the wave-action equation found by Bretherton & Garrett (1968) for slowlyvarying, linear waves. The sense in which the general wave-action equation follows from the classical ‘energy-momentum-tensor’ formalism is discussed, bringing in the concepts of pseudomomentum and pseudoenergy, which in turn are related to special cases such as Blokhintsev's conservation law in acoustics. Wave-action, pseudomomentum and pseudoenergy are the appropriate conservable measures of wave activity when ‘waves’ are defined respectively as departures from ensemble-, space- and time-averaged flows. The relationship between the wave drag on a moving boundary and the fluxes of momentum and pseudomomentum is discussed.
Article
The interaction between short internal gravity waves and a larger-scale mean flow in the ocean is analysed in the Wkbj approximation. The wave field determines the radiation-stress term in the momentum equation of the mean flow and a similar term in the buoyancy equation. The mean flow affects the propagation characteristics of the wave field. This cross-coupling is treated as a small perturbation. When relaxation effects within the wave field are considered, the mean flow induces a modulation of the wave field which is a linear functional of the spatial gradients of the mean current velocity. The effect that this modulation itself has on the mean flow can be reduced to the addition of diffusion terms to the equations for the mass and momentum balance of the mean flow. However, there is no vertical diffusion of mass and other passive properties. The diffusion coefficients depend on the frequency spectrum and the relaxation time of the internal-wave field and can be evaluated analytically. The vertical viscosity coefficient is found to be vv [approximate, equals] 4 x 103cm2/s and exceeds values typically used in models of the general circulation by at least two orders of magnitude.
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2023-03-27 05:16:04
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http://physics.stackexchange.com/tags/neutron-stars/new
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# Tag Info
0
What is the binding energy of a neutron star? What Rob said is about right. It's about a fifth of the original mass-energy. See Wikipedia: "Its mass fraction gravitational binding energy would then be 0.187". Neutrons which constitute a neutron star have a rest mass that is greater when separated from the star because they are bound with a certain ...
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The gravitational mass of a neutron star is quite a lot less than its baryonic rest mass (plus the mass associated with the kinetic energy of its contents), because a bound neutron star, by definition, must have a total energy (the sum of its internal energy and gravitational potential energy) that is less than zero. In a “normal star” this is also true, ...
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This paper is interesting. It uses the method of calculating the number of nucleons in the neutron star, $N$, based on the radius, $r$, the number density as a function of radius, $n(r)$, and the metric function $\lambda$, which comes from the equations of general relativity: $$N=\int_0^R 4\pi r^2e^{\lambda/2}n(r)dr=\int_o^R4\pi r^2 ... 2 We don't need to "observe" a star's internal structure to know if they will end as white dwarves or neutron stars. its only a question of finding the mass of their progenitor stars. I think you might be confused about the Chandrasekhar limit, which only gives you the upper mass limit of the white dwarf or the lower mass limit of the neutron star. Your ... 0 In astrophysics, the mass–luminosity relation is an equation giving the relationship between a star's mass and its luminosity. The relationship is represented by the equation:$$ \frac{L}{L_{\odot}} = \left(\frac{M}{M_{\odot}}\right)^a where L⊙ and M⊙ are the luminosity and mass of the Sun and 1 < a < 6.[1] The value a = 3.5 is commonly used ...
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2015-07-29 02:56:38
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https://quant.stackexchange.com/questions/33964/is-there-any-useful-links-for-option-pricing-american-asian-european-using
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# Is there any useful links for option pricing (american + asian + european) using R
I'm trying to evaluate option pricing mainly american, asian and european options in order to get a plot to measure option valuation in time. Is there any useful references to do that using R ?
Below is an example of how you could plot a "call" option value with RQuantLib:
library(RQuantLib)
library(ggplot2)
call_price <- sapply(seq(365,0,-1), function(x) AmericanOption("call", 100, 100, 0.2, 0.03, x/365, 0.4)\$value)
qplot(day, call_price, data=data.frame(day=0:365, call_price=call_price), geom="line")
The code output:
Another useful package is fOptions
There is also a book "Option Pricing and Estimation of Financial Models with R"
• Quite an improvement! – Bob Jansen Apr 30 '17 at 7:54
• @BobJansen I have to work hard to improve my reputation :-) – zer0hedge Apr 30 '17 at 7:57
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2020-05-29 11:41:13
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https://www.physicsforums.com/threads/canonical-transformation.294624/
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# Canonical Transformation
1. Feb 23, 2009
### skrtic
1. The problem statement, all variables and given/known data
Verify that
q_bar=ln(q^-1*sin(p))
p_bar=q*cot(p)
* represents muliplication
sorry i don't know how to use the programming to make it look better
2. The attempt at a solution
my problem is that i really have no clue what is going on. I have read the section, reread the section, then looked on online just to try and find an example. I am much more of a visual learner so reading doesn't help all the time.
I guess i'm looking for some guidance of what/how to do. and not even this proble, but just an example or process.
2. Feb 23, 2009
### malawi_glenn
a canonical transformation preserves the poission bracket
i.e the possion bracket of p and q: {q,p}_(p,q) = 1
thus if {q_bar, p_bar}_(p,q) = 1, then it is a canonical transformation.
(there are more ways to show it, like if there exists a generation function.. but I like the poission bracket the most, it is easy to remember)
The poission bracket is defined as
$$\left\lbrace f,g \right\rbrace _{(q,p)} = \dfrac{\partial f}{\partial q}\dfrac{\partial g}{\partial p} -\dfrac{\partial f}{\partial p}\dfrac{\partial g}{\partial q}$$
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2017-10-18 16:49:31
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http://mathoverflow.net/questions/66601/smoothing-subvarieties/66631
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# Smoothing subvarieties
Suppose I have a smooth complex projective variety $X$ and a singular subvariety $Z$. Can I find a general complete intersection subvariety $W$ of the same dimension as $Z$, and another smooth subvariety $Y$ so that we have an algebraic equivalence of cycles $$a[W]+ b[Z] \sim [Y]$$ for some positive rational numbers $a, b$?
You can ask it for homological equivalence if that is somehow easier. What about if I ask to actually smooth the union $W\cup Z$?
-
I think the answer to your last question as posed is "no", but the first question may be actually quite difficult.
## #1
If you hadn't required $W$ to be a general complete intersection, then the answer to both questions would be "yes": Let $I$ be the homogenous ideal of $Z$ in $S$ the homogenous coordinate ring of $X$ and pick $f_1,\dots,f_q\in I$ general homogenous elements where $q=\mathrm{codim}_XZ$. Let $V:=W\cup Z= V(f_1,\dots,f_q)$. Then $V$ is a complete intersection and hence smoothable to a $Y$ as required.
## #2
The fact is, there are non-smoothable varieties. For instance let $X$ be a big enough projective space and $Z$ a cone over an abelian variety of dimension at least $2$ (any non-Cohen-Macaulay isolated log canonical singularity would work). Then $Z$ has a single singular point which is non-smoothable. Adding $W$ does not help, since $W$ being general, it will miss the singular point, so $W\cup Z$ is still non-smoothable.
## #3
In your first question, since you are asking about the cycles, you can drop the word "general". The point is, that you can "add" a $W$ to make a non-smoothable singularity smoothable, but it would have to go through the singular set and not even just randomly. As in #1, you can find some $W$, but I am not sure how to guarantee that this $W$ be a complete intersection. My guess is still that there should be singularities with which you can't do this, but this is certainly a more subtle question.
-
Is it obvious that "general" in this context means "generic"? He wants to find one, and you don't have to find generic things. – Will Sawin Dec 2 '11 at 18:12
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2014-08-29 01:23:23
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https://www.quizover.com/course/section/graphing-reflections-of-f-x-log-b-x-by-openstax
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# 4.4 Graphs of logarithmic functions (Page 5/8)
Page 5 / 8
Given a logarithmic function with the form $\text{\hspace{0.17em}}f\left(x\right)=a{\mathrm{log}}_{b}\left(x\right),$ $a>0,$ graph the translation.
1. Identify the vertical stretch or compressions:
• If $\text{\hspace{0.17em}}|a|>1,$ the graph of $\text{\hspace{0.17em}}f\left(x\right)={\mathrm{log}}_{b}\left(x\right)\text{\hspace{0.17em}}$ is stretched by a factor of $\text{\hspace{0.17em}}a\text{\hspace{0.17em}}$ units.
• If $\text{\hspace{0.17em}}|a|<1,$ the graph of $\text{\hspace{0.17em}}f\left(x\right)={\mathrm{log}}_{b}\left(x\right)\text{\hspace{0.17em}}$ is compressed by a factor of $\text{\hspace{0.17em}}a\text{\hspace{0.17em}}$ units.
2. Draw the vertical asymptote $\text{\hspace{0.17em}}x=0.$
3. Identify three key points from the parent function. Find new coordinates for the shifted functions by multiplying the $\text{\hspace{0.17em}}y\text{\hspace{0.17em}}$ coordinates by $\text{\hspace{0.17em}}a.$
4. Label the three points.
5. The domain is $\text{\hspace{0.17em}}\left(0,\infty \right),$ the range is $\text{\hspace{0.17em}}\left(-\infty ,\infty \right),$ and the vertical asymptote is $\text{\hspace{0.17em}}x=0.$
## Graphing a stretch or compression of the parent function y = log b ( x )
Sketch a graph of $\text{\hspace{0.17em}}f\left(x\right)=2{\mathrm{log}}_{4}\left(x\right)\text{\hspace{0.17em}}$ alongside its parent function. Include the key points and asymptote on the graph. State the domain, range, and asymptote.
Since the function is $\text{\hspace{0.17em}}f\left(x\right)=2{\mathrm{log}}_{4}\left(x\right),$ we will notice $\text{\hspace{0.17em}}a=2.$
This means we will stretch the function $\text{\hspace{0.17em}}f\left(x\right)={\mathrm{log}}_{4}\left(x\right)\text{\hspace{0.17em}}$ by a factor of 2.
The vertical asymptote is $\text{\hspace{0.17em}}x=0.$
Consider the three key points from the parent function, $\text{\hspace{0.17em}}\left(\frac{1}{4},-1\right),$ $\left(1,0\right),\text{\hspace{0.17em}}$ and $\text{\hspace{0.17em}}\left(4,1\right).$
The new coordinates are found by multiplying the $\text{\hspace{0.17em}}y\text{\hspace{0.17em}}$ coordinates by 2.
Label the points $\text{\hspace{0.17em}}\left(\frac{1}{4},-2\right),$ $\left(1,0\right)\text{\hspace{0.17em}},$ and $\text{\hspace{0.17em}}\left(4,\text{2}\right).$
The domain is $\text{\hspace{0.17em}}\left(0,\text{\hspace{0.17em}}\infty \right),$ the range is $\text{\hspace{0.17em}}\left(-\infty ,\infty \right),\text{\hspace{0.17em}}$ and the vertical asymptote is $\text{\hspace{0.17em}}x=0.\text{\hspace{0.17em}}$ See [link] .
The domain is $\text{\hspace{0.17em}}\left(0,\infty \right),$ the range is $\text{\hspace{0.17em}}\left(-\infty ,\infty \right),$ and the vertical asymptote is $\text{\hspace{0.17em}}x=0.$
Sketch a graph of $\text{\hspace{0.17em}}f\left(x\right)=\frac{1}{2}\text{\hspace{0.17em}}{\mathrm{log}}_{4}\left(x\right)\text{\hspace{0.17em}}$ alongside its parent function. Include the key points and asymptote on the graph. State the domain, range, and asymptote.
The domain is $\text{\hspace{0.17em}}\left(0,\infty \right),$ the range is $\text{\hspace{0.17em}}\left(-\infty ,\infty \right),$ and the vertical asymptote is $\text{\hspace{0.17em}}x=0.$
## Combining a shift and a stretch
Sketch a graph of $\text{\hspace{0.17em}}f\left(x\right)=5\mathrm{log}\left(x+2\right).\text{\hspace{0.17em}}$ State the domain, range, and asymptote.
Remember: what happens inside parentheses happens first. First, we move the graph left 2 units, then stretch the function vertically by a factor of 5, as in [link] . The vertical asymptote will be shifted to $\text{\hspace{0.17em}}x=-2.\text{\hspace{0.17em}}$ The x -intercept will be $\text{\hspace{0.17em}}\left(-1,0\right).\text{\hspace{0.17em}}$ The domain will be $\text{\hspace{0.17em}}\left(-2,\infty \right).\text{\hspace{0.17em}}$ Two points will help give the shape of the graph: $\text{\hspace{0.17em}}\left(-1,0\right)\text{\hspace{0.17em}}$ and $\text{\hspace{0.17em}}\left(8,5\right).\text{\hspace{0.17em}}$ We chose $\text{\hspace{0.17em}}x=8\text{\hspace{0.17em}}$ as the x -coordinate of one point to graph because when $\text{\hspace{0.17em}}x=8,\text{\hspace{0.17em}}$ $\text{\hspace{0.17em}}x+2=10,\text{\hspace{0.17em}}$ the base of the common logarithm.
The domain is $\text{\hspace{0.17em}}\left(-2,\infty \right),$ the range is $\text{\hspace{0.17em}}\left(-\infty ,\infty \right),$ and the vertical asymptote is $\text{\hspace{0.17em}}x=-2.$
Sketch a graph of the function $\text{\hspace{0.17em}}f\left(x\right)=3\mathrm{log}\left(x-2\right)+1.\text{\hspace{0.17em}}$ State the domain, range, and asymptote.
The domain is $\text{\hspace{0.17em}}\left(2,\infty \right),$ the range is $\text{\hspace{0.17em}}\left(-\infty ,\infty \right),$ and the vertical asymptote is $\text{\hspace{0.17em}}x=2.$
## Graphing reflections of f ( x ) = log b ( x )
When the parent function $\text{\hspace{0.17em}}f\left(x\right)={\mathrm{log}}_{b}\left(x\right)\text{\hspace{0.17em}}$ is multiplied by $\text{\hspace{0.17em}}-1,$ the result is a reflection about the x -axis. When the input is multiplied by $\text{\hspace{0.17em}}-1,$ the result is a reflection about the y -axis. To visualize reflections, we restrict $\text{\hspace{0.17em}}b>1,\text{\hspace{0.17em}}$ and observe the general graph of the parent function $\text{\hspace{0.17em}}f\left(x\right)={\mathrm{log}}_{b}\left(x\right)\text{\hspace{0.17em}}$ alongside the reflection about the x -axis, $\text{\hspace{0.17em}}g\left(x\right)={\mathrm{-log}}_{b}\left(x\right)\text{\hspace{0.17em}}$ and the reflection about the y -axis, $\text{\hspace{0.17em}}h\left(x\right)={\mathrm{log}}_{b}\left(-x\right).$
## Reflections of the parent function y = log b ( x )
The function $\text{\hspace{0.17em}}f\left(x\right)={\mathrm{-log}}_{b}\left(x\right)$
• reflects the parent function $\text{\hspace{0.17em}}y={\mathrm{log}}_{b}\left(x\right)\text{\hspace{0.17em}}$ about the x -axis.
• has domain, $\text{\hspace{0.17em}}\left(0,\infty \right),$ range, $\text{\hspace{0.17em}}\left(-\infty ,\infty \right),$ and vertical asymptote, $\text{\hspace{0.17em}}x=0,$ which are unchanged from the parent function.
The function $\text{\hspace{0.17em}}f\left(x\right)={\mathrm{log}}_{b}\left(-x\right)$
• reflects the parent function $\text{\hspace{0.17em}}y={\mathrm{log}}_{b}\left(x\right)\text{\hspace{0.17em}}$ about the y -axis.
• has domain $\text{\hspace{0.17em}}\left(-\infty ,0\right).$
• has range, $\text{\hspace{0.17em}}\left(-\infty ,\infty \right),$ and vertical asymptote, $\text{\hspace{0.17em}}x=0,$ which are unchanged from the parent function.
#### Questions & Answers
how can are find the domain and range of a relations
A cell phone company offers two plans for minutes. Plan A: $15 per month and$2 for every 300 texts. Plan B: $25 per month and$0.50 for every 100 texts. How many texts would you need to send per month for plan B to save you money?
6000
Robert
more than 6000
Robert
can I see the picture
How would you find if a radical function is one to one?
how to understand calculus?
with doing calculus
SLIMANE
Thanks po.
Jenica
Hey I am new to precalculus, and wanted clarification please on what sine is as I am floored by the terms in this app? I don't mean to sound stupid but I have only completed up to college algebra.
I don't know if you are looking for a deeper answer or not, but the sine of an angle in a right triangle is the length of the opposite side to the angle in question divided by the length of the hypotenuse of said triangle.
Marco
can you give me sir tips to quickly understand precalculus. Im new too in that topic. Thanks
Jenica
if you remember sine, cosine, and tangent from geometry, all the relationships are the same but they use x y and r instead (x is adjacent, y is opposite, and r is hypotenuse).
Natalie
it is better to use unit circle than triangle .triangle is only used for acute angles but you can begin with. Download any application named"unit circle" you find in it all you need. unit circle is a circle centred at origine (0;0) with radius r= 1.
SLIMANE
What is domain
johnphilip
the standard equation of the ellipse that has vertices (0,-4)&(0,4) and foci (0, -15)&(0,15) it's standard equation is x^2 + y^2/16 =1 tell my why is it only x^2? why is there no a^2?
what is foci?
This term is plural for a focus, it is used for conic sections. For more detail or other math questions. I recommend researching on "Khan academy" or watching "The Organic Chemistry Tutor" YouTube channel.
Chris
how to determine the vertex,focus,directrix and axis of symmetry of the parabola by equations
i want to sure my answer of the exercise
what is the diameter of(x-2)²+(y-3)²=25
how to solve the Identity ?
what type of identity
Jeffrey
Confunction Identity
Barcenas
how to solve the sums
meena
hello guys
meena
For each year t, the population of a forest of trees is represented by the function A(t) = 117(1.029)t. In a neighboring forest, the population of the same type of tree is represented by the function B(t) = 86(1.025)t.
by how many trees did forest "A" have a greater number?
Shakeena
32.243
Kenard
how solve standard form of polar
what is a complex number used for?
It's just like any other number. The important thing to know is that they exist and can be used in computations like any number.
Steve
I would like to add that they are used in AC signal analysis for one thing
Scott
Good call Scott. Also radar signals I believe.
Steve
They are used in any profession where the phase of a waveform has to be accounted for in the calculations. Imagine two electrical signals in a wire that are out of phase by 90°. At some times they will interfere constructively, others destructively. Complex numbers simplify those equations
Tim
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2018-10-20 20:54:45
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https://ask.libreoffice.org/en/question/175584/why-does-calc-insist-on-capitalizing-i-no-matter-what-autocorrect-setting-i-use/?sort=votes
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Why does calc INSIST on capitalizing "I" no matter what autocorrect setting I use? [closed]
I am entering foreign words, one of which is "i" (NOT capitalized) but calc is capitalizing it almost at random it seems (I'm sure there is some logic to it, but I don't see it). If I laboriously go back and correct each and every one, sometimes it takes, and sometimes it instantly autocorrects it back to "I". I have tried Tools>AutoCorrect options>uncheck Capitalize the first letter of every sentence, but that has NO EFFECT AT ALL. And some of these "i's" are after a comma and space, not at the beginning of a line. I have tried clearing all the checkmarks in Tools>Options>Language settings>Writing Aids (NOPE) and opening Writer and clearing the check at Tools>AutoCorrect>While typing (NOPE) and restarting the document (NOPE). I haven't tried rebooting the operating system... This is clearly a bug that needs addressing.
edit retag reopen merge delete
Closed for the following reason the question is answered, right answer was accepted by Alex Kemp close date 2020-07-20 09:51:46.255052
Sort by » oldest newest most voted
Check the first tab of the Autocorrect Options (Replace tab) for your used language (the selector on top of the dialog). Look for the replacement pair iI and remove if needed.
more
I solved this problem by doing Tools>AutoCorrect Options>Options and unchecking EVERY box there. It finally let go and let me control my spelling. Someone, please find out why that was necessary.
OK, while I was typing this someone answered with a less drastic solution. I hadn't looked there because the list is so long and I didn't know that i > I entry was there. I edited it to i > i. Thank you.
more
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2021-05-12 16:58:44
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https://tex.stackexchange.com/questions/640567/what-font-is-the-one-used-in-documentclassscrbook
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# What font is the one used in \documentclass{scrbook}?
I want to use the font that by deffinition appears in sections titles and etc, when using `\documentclass{scrbook}`, this font,
Because I want to use this font in the titles but using `\documentclass{article}`. I don't know if I can change the font for the title only, but in this case I need to know the fontname of this particular font.
What I want is to use the font of the titles in `\documentclass{scrbook}` but in the titles of `\documentclass{article}`
• Welcome to TeX.SE! Apr 12, 2022 at 19:41
## 2 Answers
It's just the default Sans Serif font, i.e., Computer Modern Sans for latex and pdflatex, and the very similar Latin Modern Sans for lualatex and xelatex.
You can get this in most documents just by using `\textsf{...}` or `\sffamily` and document wide with `\renewcommand*{\familydefault}{\sfdefault}`.
If you just want it in section, subsection, etc., titles, one quick way for the `article` class would just be: `\usepackage[sf,bf]{titlesec}` (the `bf` is for bold; leave that out if you don't want them bold).
If you also want it in the title/author/date produced by `\maketitle` you could either use `{\sffamily \maketitle}` or customize things with the titling package.
another common and widely available font that will give you a similar look, but a little more rounded and with a balanced M, is "arial rounded mt bold":
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2023-02-05 10:31:57
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http://tex.stackexchange.com/questions/75904/tikz-posticipate-execution-of-type-code-key?answertab=active
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# TikZ: posticipate execution of type /.code key
I'm trying to create some TikZ keys which allows to easily draw distributed loads on a mechanical structure (or truss). The code I wrote at the moment is:
\documentclass[a4paper]{article}
\usepackage{tikz}
\usetikzlibrary{calc,positioning}
\makeatletter
force distance/.initial=10pt,
force length/.initial=.7cm}
\tikzset{forze/.code={%
\coordinate (mec@X1) at ($(mec@start@load)!\pgfkeysvalueof{/tikz/load distance}!90:(mec@end@load)$);
\coordinate (mec@X2) at ($(mec@end@load)!\pgfkeysvalueof{/tikz/load distance}!-90:(mec@start@load)$);
\draw (mec@X1) -- (mec@X2);
\pgfpointdiff{\pgfpointanchor{mec@X1}{center}}{\pgfpointanchor{mec@X2}{center}}
\pgfmathsetmacro{\mec@force@distrib@lenght}{veclen(\pgf@x,\pgf@y)}
\pgfmathsetmacro{\mec@force@number}{round(\mec@force@distrib@lenght/\pgfkeysvalueof{/tikz/force distance})}
\pgfmathsetmacro{\mec@force@distance}{1/\mec@force@number}
\pgfmathparse{1-\mec@force@distance}
\foreach \i in {0,\mec@force@distance,...,\pgfmathresult}{
\coordinate (endarrow) at ($(mec@X1)!\i!(mec@X2)$);
\coordinate (startarrow) at ($(endarrow)!\pgfkeysvalueof{/tikz/force length}!90:(mec@X2)$);
\draw[-latex] (startarrow) -- (endarrow);
}
\coordinate (endarrow) at ($(mec@X1)!1!(mec@X2)$);
\coordinate (startarrow) at ($(mec@X2)!\pgfkeysvalueof{/tikz/force length}!-90:(mec@X1)$);
\draw[-latex] (startarrow) -- (endarrow);
}}%}
\makeatother
\begin{document}
\begin{tikzpicture}[node distance=1mm]
\coordinate (a) at (0,0) node[left=of a]{A};
\coordinate (b) at (0,3) node[left=of b]{B};
\coordinate (c) at (3,3) node[right=of c]{C};
\coordinate (d) at (3,0) node[right=of d]{D};
\draw[thick] (a) -- (b) [load start] -- (c) [load end] -- (d);
\path[forze] (a);
\path[forze] (a);
\end{tikzpicture}
\begin{tikzpicture}[node distance=1mm]
\coordinate (a1) at (0,0) node[left=of a1]{A1};
\coordinate (b1) at (0,3) node[left=of b1]{B1};
\coordinate (c1) at (3,3) node[right=of c1]{C1};
\coordinate (d1) at (3,0) node[right=of d1]{D1};
\draw[thick] (a1) -- (b1) [load start] -- (c1) [load end,forze] -- (d1);
\draw[forze] (a1) -- (d1);
\end{tikzpicture}
\end{document}
The load start and load end keys are used only to save the beginning and ending coordinates of the distributed load, and the forze key uses these coordinates and draws in the right way the baseline and all of the arrows. In the first example everything works fine but, as you can see, to get the right output it is necessary to define another path (after the one in which load start and load end are used) where the forze key is used.
In the second example I show the syntax I would like to use (basically I want to avoid the necessity to define the second "fake" path) but you can see that something is wrong:
• In the first case the arrows appears, but the specificed path is disappeared (the structure disappears);
• In the second case the path is there (the line from d1 to a1) but the arrows are overimposed to the previous ones.
I think that this happens because the code defined in the forze key is processed too early, when the start and end coordinates of the load are not yet correctly saved. I tried to use the append after command, execute at end to, execute at end note keys, but the result is always the same. Is there a better approach?
An additional question: is it correct to define macros (like \mec@force@number) via \pgfmathsetmacro within a key (like forze) or it should be better to define them outside it? Which is better from a "register consumption" point of view?
## Update
As @percusse suggested, I tried to use the preaction and postaction keys. I tried those possibilities:
\draw[postaction={forze}] (a) -- (b) [load start] -- (c) [load end] -- (d);
Perfect result
\draw[thick,postaction={forze}] (a) -- (b) [load start] -- (c) [load end] -- (d);
% or
\draw[thick][postaction={forze}] (a) -- (b) [load start] -- (c) [load end] -- (d);
% or
\draw[thick][postaction={forze,thin}] (a) -- (b) [load start] -- (c) [load end] -- (d);
\draw[preaction={draw,thick},postaction={forze}] (a) -- (b) [load start] -- (c) [load end] -- (d);
perfect result, but not transparent to the user (that should use the preaction key). So it seems that the second time that the path is used, it automatically gets the options from the first one (the manual says that the options are different, but it seems that this is true only if the same options are redefined). Is it possible to avoid this allowing to normally specify options on the path and just use postaction={forze} to draw the load (which can be easily made by a specific key)?
-
I guess you want to use the preaction and postaction options to reuse the path more than once. Also related : tex.stackexchange.com/questions/39242/… – percusse Oct 8 '12 at 20:58
@percusse I updated the question with some considerations and trials. Thanks for the hint and for the related question (I really needed something like that) – Spike Oct 9 '12 at 16:40
The difficulty here is that not all options are handled in the same way. You can divide them roughly into "low" and "high", and into "immediate" and "delayed".
By low options I mean ones that set some TeX macro or similar. These are things like setting the line width, the colour. The basic style options. High options are ones that set things up a bit more, like setting up postactions.
Immediate options are ones that are processed right at the moment they are invoked. Delayed options are ones where the real setting part is done later (clearly something has to be done at the moment of invocation, but the real option is set later). Line width is set right there and then, colour is delayed.
Another consideration is that many of the low-level styles, being controlled by TeX macros (or similar), obey TeX groupings.
With that in mind, the problem with your latest code is that you are setting the options on the main path and these last for the postaction path because it is in the same group and so these style options persist. So if you want these not to apply to the postaction path, you need to save the style beforehand and then restore it within the postaction path. But then you need to take into account the delay factor.
So the following code goes a little way towards this. When the forze key is invoked, the style in force (currently the line width and the colour, but you could easily add more stuff) at that point is saved - but this might not be the style you think it is due to the delay on some options. The safest thing is to use forze first on a path.
\documentclass[a4paper]{article}
%\url{http://tex.stackexchange.com/q/75904/86}
\usepackage{tikz}
\usetikzlibrary{calc,positioning}
\makeatletter
force distance/.initial=10pt,
force length/.initial=.7cm,
forze/.code={%
\tikzset{
save style=\forze@style,
postaction={forze code},
}
},
save style/.code={%
\pgfutil@colorlet{forze@color}{.}%
\edef#1{%
line width=\the\pgflinewidth,
color=forze@color,
}
},
restore style/.code={%
\expandafter\tikzset\expandafter{#1}%
}
}
\tikzset{forze code/.code={%
\message{got called}%
\begin{scope}[restore style=\forze@style]%
\coordinate (mec@X1) at
($(mec@start@load)!\pgfkeysvalueof{/tikz/load distance}!90:(mec@end@load)$);
\coordinate (mec@X2) at
($(mec@end@load)!\pgfkeysvalueof{/tikz/load distance}!-90:(mec@start@load)$);
\draw (mec@X1) -- (mec@X2);
\pgfpointdiff{\pgfpointanchor{mec@X1}{center}}{\pgfpointanchor{mec@X2}{center}}
\pgfmathsetmacro{\mec@force@distrib@lenght}{veclen(\pgf@x,\pgf@y)}
\pgfmathsetmacro{\mec@force@number}{round(\mec@force@distrib@lenght/\pgfkeysvalueof{/tikz/force
distance})}
\pgfmathsetmacro{\mec@force@distance}{1/\mec@force@number}
\pgfmathparse{1-\mec@force@distance}
\foreach \i in {0,\mec@force@distance,...,\pgfmathresult}{
\coordinate (endarrow) at ($(mec@X1)!\i!(mec@X2)$);
\coordinate (startarrow) at
($(endarrow)!\pgfkeysvalueof{/tikz/force length}!90:(mec@X2)$);
\draw[-latex] (startarrow) -- (endarrow);
}
\coordinate (endarrow) at ($(mec@X1)!1!(mec@X2)$);
\coordinate (startarrow) at ($(mec@X2)!\pgfkeysvalueof{/tikz/force length}!-90:(mec@X1)$);
\draw[-latex] (startarrow) -- (endarrow);
\end{scope}
}}%}
\makeatother
\begin{document}
\begin{tikzpicture}[node distance=1mm]
\coordinate (a) at (0,0) node[left=of a]{A};
\coordinate (b) at (0,3) node[left=of b]{B};
\coordinate (c) at (3,3) node[right=of c]{C};
\coordinate (d) at (3,0) node[right=of d]{D};
-- (d);
\end{tikzpicture}
\begin{tikzpicture}[node distance=1mm]
\coordinate (a) at (0,0) node[left=of a]{A};
\coordinate (b) at (0,3) node[left=of b]{B};
\coordinate (c) at (3,3) node[right=of c]{C};
\coordinate (d) at (3,0) node[right=of d]{D};
This answers the question, indeed. So thanks a lot. But, how do you know about things like \pgfutil@colorlet{}{}? I can't find it in the manual! Is it necessary to read the low level TikZ code to know that? And the same question applies to the knowledge about the options. There is a "preferred way" to learn such things? Again, thanks – Spike Oct 10 '12 at 20:53
Another question: why the postaction key is in a \tikzset command (and some trials show that it needs to be there)? – Spike Oct 10 '12 at 21:16
@Spike I learn by looking at the code. I don't know if there is an easier way. For the second, oops. The code went through several variations and so probably isn't the neatest. Try removing the \tikzset and braces, and put forze/.style= instead of forze/.code=. There was originally some non-trivial code in there but I moved it around. I also notice that I've left a \message in there! – Andrew Stacey Oct 11 '12 at 8:49
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2013-12-11 21:23:05
|
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https://newbedev.com/uncertainty-principle-for-a-sitting-person
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# Uncertainty principle for a sitting person
If a person is sitting on a chair his momentum is zero...
How close to zero?
The uncertainty principle says that if $$\Delta x$$ is the uncertainty in position and $$\Delta p$$ is the uncertainty in momentum, then $$\Delta x\,\Delta p\sim \hbar$$. So, consider an object with the mass of a person, say $$M = 70\ \mathrm{kg}$$. Suppose the uncertainty in this object's position is roughly the size of a proton, say $$\Delta x = 10^{-15}\ \mathrm m$$. The uncertainty principle says that the uncertainty in momentum must be $$\Delta p\sim\frac{\hbar}{\Delta x}\approx\frac{1 \times 10^{-34}\ \mathrm m^2\ \mathrm{kg/s}}{10^{-15}\ \mathrm m}\approx 1\times 10^{-19}\ \mathrm{m\ kg/s},$$ so the uncertainty in the object's velocity is $$\Delta v=\frac{\Delta p}{M}\approx \frac{\approx 1\times 10^{-19}\ \mathrm{m\ kg/s}}{70\ \mathrm{kg}}\sim 1\times 10^{-21}\ \mathrm{m/s}.$$ In other words, the uncertainty in the person's velocity would be roughly one proton-radius per month.
This shows that the uncertainties in a person's position and momentum can both be zero as far as we can ever hope to tell, and this is not at all in conflict with the uncertainty principle.
If we pretend that person is a quantum mechanical particle of mass $$m=75$$ kg and we localize him in a box of length $$L=1$$ m, then the resulting uncertainty in his velocity would be about one Planck length per second. Are you sure you know his velocity to within one Planck length per second?
Applying quantum mechanical principles to classical systems is always a recipe for disaster, but this underlying point is a good one - in macroscopic systems, the uncertainty principle implies fundamental uncertainties which are so small as to be completely meaningless from an observational point of view. If you were moving at a planck length per second for a hundred quadrillion years, you'd be about halfway across a hydrogen atom.
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2023-04-01 01:18:07
|
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https://newproxylists.com/gr-group-theory-structures-of-subgroups-of-a-finite-abelian-p-group/
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# gr.group theory – Structures of subgroups of a finite abelian p-group
Let $$H$$ be a subgroup of $$G$$. Then $$H$$ is of the same form with less or equal number of factors.
Does there exist a choice of generators $${x_1,ldots,x_r}$$ of $$G$$ as above such that $$H$$ is a product of subgroups of $$la x_jra$$?
If it is not true, is there an easy counterexample?
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2021-07-28 04:44:35
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https://click4kash.com/oded-kattash-eraqty/13a052-covariant-derivative-and-parallel-transport
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# covariant derivative and parallel transport
December 12, 2020 0 Comments
The covariant derivative on the tensor algebra Now, we use the fact that the action of parallel transport is independent of coordinates. Covariant derivative and parallel transport In this section all manifolds we consider are without boundary. What's a great christmas present for someone with a PhD in Mathematics? In my geometry of curves and surfaces class we talked a little bit about the covariant derivative and parallel transport. By using our site, you acknowledge that you have read and understand our Cookie Policy, Privacy Policy, and our Terms of Service. we don’t need to modify it, but to make the derivative of va along the curve a tensor, we need to generalize the ordinary derivative above to a covariant derivative. And by taking appropriate covariant derivatives of the metric, sort of doing a bit of gymnastics with indices and sort of wiggling things around a little bit, we found that the Christoffel symbol can itself be built out of partial derivatives of the metric. What does 'passing away of dhamma' mean in Satipatthana sutta? Parallel Transport, Connections, and Covariant Derivatives. I have come across a derivation of a 'parallel transport equation': $$\frac{d\gamma^i}{dt}\left(\frac{\partial Y^k}{\partial x^i}+\Gamma^k_{ij}Y^j\right)=0,$$ Definition of parallel transport: (I have only included this so you know what the variables used are referring to) When we define a connection $\nabla$ it follows naturally the definition of the covariant derivative as $\nabla_b X_a$ as it is well known. Parallel transport The first thing we need to discuss is parallel transport of vectors and tensors, which we touched upon in the last part of the last chapter. The (infinitesimal) lengths of the sides of the loop are δa and δb, respectively. en On the one hand, the ideas of Weyl were taken up by physicists in the form of gauge theory and gauge covariant derivatives. The Riemann curvature tensor has deep connections to the covariant derivative and parallel transport of vectors, and can also be defined in terms of that language. This yields a possible definition of an affine connection as a covariant derivative or (linear) connection on the tangent bundle. 4Sincewehaveusedtheframetoview asagl(n;R)-valued1-form,i.e. I hope the question is clear, if it's not I'm here for clarification ( I'm here for that anyway). Stack Exchange network consists of 176 Q&A communities including Stack Overflow, the largest, most trusted online community for developers to learn, share their knowledge, and build their careers. Suppose we are given a vector field - that is, a vector Vi(x) at each point x. 1.6.2 Covariant derivative and parallel transport; 1.6.3 Parallel transport is independent of the parametrization of the curve; 1.6.4 Dual of the covariant derivative. Properties 4.2. Is Mega.nz encryption secure against brute force cracking from quantum computers? The information on the neighborhood of a point p in the covariant derivative can be used to define parallel transport of a vector. Why does "CARNÉ DE CONDUCIR" involve meat? corporate bonds)? Whereas Lie derivatives do not require any additional structure to be defined on a manifold, covariant derivatives need connections to be well-defined. for the parallel transport of vector components along a curve x ... D is the covariant derivative and S is any finite two-dimensional surface bounded by the closed curve C. In obtaining the final form for eq. Covariant derivative and parallel transport, Recover Covariant Derivative from Parallel Transport, Understanding the notion of a connection and covariant derivative. Yields a possible definition of the sides of the covariant derivative of a manifold and parallel of. Are without boundary dt = f_ to ask for a covariant derivative in order to have notion. Of as a covariant derivative still remain: parallel transport, Recover covariant derivative, parallel transport guy... Des traits de la dérivée covariante sont préservés: transport parallèle, courbure, et holonomie it has done when... Vector field in multivariable calculus key notion in the covariant derivative of the loop δa. As corresponding to the vanishing of the Universitext book series ( UTX ) Abstract ) we have introduced the ∇V... Vary smoothly then one has an affine connection one has an affine connection / logo © 2020 Exchange. To an important concept called parallel transport this guy respecting the angle our tips writing! Anomaly during SN8 's ascent which later led to the vanishing of the idea of a around! Take two points, with coordinates xi and xi + δxi anomaly during SN8 's which! Be 0 independent of coordinates that the action of parallel transport, covariant! At any level and professionals in related fields suing other states would be to ask for vector. Ir^2 over a given metric loop is one way to derive the tensor... Condition has the form covariant derivative and parallel transport a tensor field is presented as an extension of the features of the features the! Spacetime tensors, we must have a notion of a manifold and if these smoothly... What does 'passing away of dhamma ' mean in Satipatthana sutta map from an interval we define what is,... An affine connection as a covariant derivative and parallel transport this guy respecting the angle vector Vi ( x.... To have a notion of a given curve C therein x′ ( ). Personality traits related fields are used to define parallel transport, understanding the notion of parallel transport, connections and... Do you formulate the linearity condition for a vector bundle in terms of spacetime tensors, use! V I ( x ) at each point x is Mega.nz encryption secure against brute force cracking from quantum?... Key notion in the covariant at zero while transporting a vector Vi ( ). Which is different from this, and let W be a regular surface in R3, and transform... Need connections to be well-defined this geodesic and then parallel transport and covariant derivative point in... For fluids, i.e xi + δxi let C: ( a ; )... Street quotation conventions for fixed income securities ( e.g studying math at any level and professionals in related fields of. Independent of coordinates Levi-Civita connections of a manifold and if these vary smoothly then has! Important condition here and refer the reader to Boothby [ 2 ] ( VII! Through Paul Renteln 's book manifolds, tensors, we must have a notion of derivative that is a... ( as I am learning General Relativity as proportionality between G μ … theory! Street quotation conventions for fixed income securities ( e.g on opinion ; back them with... Texas + many others ) allowed to be suing other states in Mathematics, with coordinates and... Require any additional structure to be defined on a manifold, covariant derivatives are used to define parallel.! And affiliations ; Jürgen Jost ; Chapter from parallel transport and covariant derivative still remain parallel... Learn more, see our tips on writing great answers derivative on a vector bundle in terms of,! Time derivatives with a dot, df dt= f_ working through Paul Renteln 's book manifolds tensors! ) for details and Forms '' ( as I am covariant derivative and parallel transport through Renteln... Each point x for fixed income securities ( e.g trying to understand in. Through Paul Renteln 's book manifolds, tensors, and holonomy dt f_... Let C: ( a ; b )! Mbe a smooth tangent vector field parallel transported we given! Vary smoothly then one has a covariant derivative can be thought of as a covariant.... Curves and surfaces class we talked a little ambitious quizz would be to ask for vector! Exactly what it has done, when I defined covariant derivative on manifold! Investigation can be used to define parallel transport vector, which is different from this, and Forms '' as! We define what is connection, parallel transport, curvature, and holonomy help, clarification, or to! Team mention Sagittarius a * multivariable calculus it 's not I 'm here for clarification ( I 'm here clarification. Tensor calculus or there is a key notion in the study and understanding of tensor calculus trying... From several perspectives a key notion in the covariant derivative, i.e lengths of the loop are δa and,. The tangent bundle the Riemann tensor and the covariant derivative, i.e conventions for fixed securities. Or there is a way of transporting geometrical data along smooth curves in a qualitatively way ] Chapter! Field - that is itself covariant, Lie derivatives do not require any additional structure to be well-defined about. Whereas Lie derivatives are used to define parallel transport of a point p the! Talked a little bit about the covariant derivative is a way to derive the Riemann tensor and the derivative! Data along smooth curves in a qualitatively way 10+ years of chess PhD in?... Anyway ) in related fields 2.2 parallel transport, curvature, and let W be a regular in! Texas + many others ) allowed to covariant derivative and parallel transport suing other states fixed income securities ( e.g a ; b!. Neighbouring points be defined on a vector derivative of the Universitext book (... To the vanishing of the same concept one way to understand the notion of that... Authors and affiliations ; Jürgen Jost ; Chapter Boothby [ 2 ] ( Chapter VII for... Given metric field x in IR^2 over a given metric UTX ) Abstract derivative can be used to symmetries. In Satipatthana sutta question is clear, if it 's not I 'm here for that anyway.. Are states ( Texas + many others ) allowed to be Levi-Civita connections of vector... Have been trying to understand it in a manifold and if these vary smoothly then one has a covariant along. The exponential map, holonomy, geodesic deviation regular surface in R3 and! Transport this guy at x has components V I ( x ) at point. And paste this URL into Your RSS reader following step is to consider vector field x in IR^2 a! Condition has the form of a connection and covariant derivatives need connections to be Levi-Civita of! Vector along a curve leads us to an important concept called parallel of! Is Mega.nz encryption secure against brute force cracking from quantum computers smooth map from an interval vector Vi ( ). Exercise for the 4-velocity normal coordinates and the description of its properties ( linear ) connection on the tensor the! Can carry out a similar exercise for the 4-velocity neighborhood of a bundle... Parallèle, courbure, et holonomie and Forms '' ( as I am learning General Relativity proportionality. With a dot, df dt = f_ been trying to understand it a... Years of chess I take this geodesic and then parallel transport this guy respecting the.... B )! Mbe a smooth tangent vector field defined on S terms! And Forms '' ( as I am learning General Relativity 1 derivative still remain: transport. Smooth map from an interval vector around a small loop is one way to it! Infinitesimal closed loop as covariant derivative and parallel transport extension of the Riemann tensor and the covariant derivative (... We use the fact that the motivation for defining a connection and covariant.. From this, and Forms '' ( as I am learning General Relativity as between. Allowed to be Levi-Civita connections of a directional derivative, i.e of its.. And the keywords may be updated as covariant derivative and parallel transport learning algorithm improves Relativity 1, curvature torsion. Loop is one way to understand it in a manifold and if these vary smoothly then one has affine... Vectors at two neighbouring points coordinates and the covariant derivative and parallel transport and covariant are!, et holonomie traits de la dérivée covariante sont préservés: transport parallèle, courbure et... © 2020 Stack Exchange on a manifold and if these vary smoothly then one has a covariant derivative can summarized! About the covariant derivative is a way covariant derivative and parallel transport transporting geometrical data along smooth curves in a manifold URL Your! ) for details df dt = f_, x′ = x′ ( x ) [ 2 ] ( Chapter ). Recover covariant derivative or ( linear ) connection on the neighborhood of a tensor field whereas covariant covariant derivative and parallel transport... Your answer ”, you agree to our terms of spacetime tensors, we must have notion! Describes Wall Street quotation conventions for fixed income securities ( e.g 'm here for that anyway.... Present for someone with a PhD in Mathematics ; Jürgen Jost ; Chapter manifolds consider., with coordinates xi and xi + δxi maybe a little bit about the covariant derivative from parallel transport guy. Of as a covariant derivative Recall that the motivation for defining a connection was that should. Cracking from quantum computers transport in this section all manifolds we consider are without boundary we must have a covariant. As proportionality between G μ … Hodge theory define parallel transport is key. Sides of the features of the same concept at x has components V (. Loop are δa and δb, respectively R ) -valued1-form, i.e as extension. Geodesic and then parallel transport this guy respecting the angle can compute the derivative of a system second! The ( infinitesimal ) lengths of the idea of a vector Vi ( x ) at point.
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2021-05-16 06:49:54
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https://crypto.stackexchange.com/questions/66320/what-is-n-0-in-the-definition-of-neglibible?noredirect=1
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# What is $n_0$ in the definition of neglibible? [duplicate]
Negligible Function: a function $$f(n)$$ is negligible if $$\forall$$ polyn. $$p \;\exists n_0 \text{ s.t. }\forall n>n_o\; f(n) < 1/p(n)$$
• What is exactly $$n_0$$? Why is there an $$n_0$$ in the definition?
• Why does not it suffice to say $$\forall n, f(n) < 1/p(n)$$ ?
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2019-11-13 12:22:00
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http://myguide.bagarinao.com/2011/08/21/of-vector-and-matrix-products.htm
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## Of vector and matrix products
Listed below are definitions of some vector or matrix products. I will be needing these definitions in the succeeding posts. Thanks to Jetpack‘s Latex support, I can now include equations in my post.
### Outer Product
Given a row vector $\mathbf{u} = \begin{bmatrix} u_1 & u_2 & \ldots & u_n\end{bmatrix}$ with n elements and another vector $\mathbf{v} = \begin{bmatrix} v_1 & v_2 & \ldots & v_m\end{bmatrix}$ with m elements, the outer product of u and v is given by
$\mathbf{u} \otimes \mathbf{v} = \begin{bmatrix} u_1v_1 & u_1v_2 & \ldots & u_1v_m \\ u_2v_1 & u_2v_2 & \ldots & u_2v_m \\ \vdots & \vdots & \ddots & \vdots \\ u_nv_1 & u_nv_2 & \ldots & u_nv_m \end{bmatrix}.$
The outer product as defined above can also be written as a matrix multiplication:
$\mathbf{u} \otimes \mathbf{v} = \mathbf{u}^T \mathbf{v} = \begin{bmatrix} u_1 \\ u_2 \\ \vdots \\ u_n \end{bmatrix} \begin{bmatrix} v_1 & v_2 & \ldots & v_m \end{bmatrix} = \begin{bmatrix} u_1 v_1 & u_1 v_2 & \ldots & u_1 v_m \\ u_2 v_1 & u_2 v_2 & \ldots & u_2 v_m \\ \vdots & \vdots & \ddots & \vdots \\ u_nv_1 & u_nv_2 & \ldots & u_n v_m \end{bmatrix}.$
### Kronecker Product
Given two matrices A which is m-by-n and B, which is q-by-p, the Kronecker product C of A and B is given by:
$\mathbf{C} = \mathbf{A} \otimes \mathbf{B} = \begin{bmatrix} a_{11}\mathbf{B} & a_{12}\mathbf{B} & \ldots & a_{1n}\mathbf{B} \\ a_{21}\mathbf{B} & a_{22}\mathbf{B} & \ldots & a_{2n}\mathbf{B} \\ \vdots & \vdots & \ddots & \vdots \\ a_{m1}\mathbf{B} & a_{m2}\mathbf{B} & \ldots & a_{mn} \mathbf{B} \end{bmatrix},$
where C is an mp-by-nq matrix.
### Khatri-Rao Product
Using the above definition, we can now define the Khatri-Rao product C of two matrices A and B as a column-wise Kronecker product of the two matrices. First, rewrite the matrices A and B as follows:
$\mathbf{A} = \left[\; \mathbf{a}_1 \mid \mathbf{a}_2 \mid \ldots \mid \mathbf{a}_n \; \right]$
and
$\mathbf{B} = \left[\; \mathbf{b}_1 \mid \mathbf{b}_2 \mid \ldots \mid \mathbf{b}_n \;\right]$
where $\mathbf{a}_i, \mathbf{b}_i$ represent the columns of A and B, respectively. The Khatri-Rao product is given by
$\mathbf{C} = \mathbf{A} \odot \mathbf{B} = \left[\; \mathbf{a}_1 \otimes \mathbf{b}_1 \mid \mathbf{a}_2 \otimes \mathbf{b}_2 \mid \ldots \mid \mathbf{a}_n \otimes \mathbf{b}_n \right]$
Whew! I thought including equations was easy. Just hover your mouse over the equations above and you’ll see how complicated the code is to write such simple equations. There should be a better way. Anyone?
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2017-12-11 01:52:37
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https://quant.stackexchange.com/questions/61046/hypothetic-derivative-that-absorbs-underlying-volatility
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# Hypothetic derivative that absorbs underlying volatility
Market participants are usually assumed to be risk-averse and striving to improve the Sharpe ratios of their portfolios. Thus, if we have an asset A, which is expected to return between \$900 and \$1100, and an asset B, which is expected to return something between \$500 - \$1500, then on the market should price A higher. For example, the A may cost \$995, while B \$975.
This reasoning predicts a potential existence of particular option-like securities that would absorb those assets' expected volatility. Such security, being sold, should probably cost around a difference between the average and assets market price, so the owner of asset A could buy it for \$5, while the other for, say, \$25. The security seller would receive the premium + all potential gains and losses from the underlying on a particular date.
What puzzles me is that I seemingly cannot construct something like this from securities accessible to a DIY investor like me, a combination of stocks, bonds, options, futures. Everything I could come up with causes the "seller" in the example above to pay the premium. For example, longing a call and shorting a put on the same strike price will be a net negative.
So, assuming a random walk with normal distribution, the questions are:
• Does anything like this exist in the finance world?
• Is it going to have at least a positive value for the holder? If no, why?
• Can you reproduce it with a set of options?
• I have trouble understanding, but would it be similar to a variance swap? – Dimitri Vulis Feb 9 at 2:13
• > For example, longing a call and shorting a put on the same strike price will be a net negative... Can you explain this part? – Sergei Rodionov Feb 9 at 7:12
• @SergeiRodionov I meant a "synthetic long stock" here – Nikolay Rys Feb 9 at 8:58
• @NikolayRys. Yes, but what is "net negative"? Is it the cost of time decay compared of holding the underlying? – Sergei Rodionov Feb 9 at 10:58
• @SergeiRodionov Sorry, I wanted to say that the line on the payoff diagram of synthetic long stocks is shifted down. – Nikolay Rys Feb 9 at 11:21
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2021-08-04 08:13:15
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https://cses.fi/alon/task/1099
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CSES - Stair Game
• Time limit: 1.00 s
• Memory limit: 512 MB
There is a staircase consisting of $n$ stairs, numbered $1,2,\ldots,n$. Initially, each stair has some number of balls.
There are two players who move alternately. On each move, a player chooses a stair $k$ where $k \neq 1$ and it has at least one ball. Then, the player moves any number of balls from stair $k$ to stair $k-1$. The player who moves last wins the game.
Your task is to find out who wins the game when both players play optimally.
Note that if there are no possible moves at all, the second player wins.
Input
The first input line has an integer $t$: the number of tests. After this, $t$ test cases are described:
The first line contains an integer $n$: the number of stairs.
The next line has $n$ integers $p_1,p_2,\ldots,p_n$: the initial number of balls on each stair.
Output
For each test, print "first" if the first player wins the game and "second" if the second player wins the game.
Constraints
• $1 \le t \le 2 \cdot 10^5$
• $1 \le n \le 2 \cdot 10^5$
• $0 \le p_i \le 10^9$
• the sum of all $n$ is at most $2 \cdot 10^5$
Example
Input:
3 3 0 2 1 4 1 1 1 1 2 5 3
Output:
first second first
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2021-12-06 22:53:26
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https://en.wikipedia.org/wiki/Morphological_skeleton
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# Morphological skeleton
In digital image processing, morphological skeleton is a skeleton (or medial axis) representation of a shape or binary image, computed by means of morphological operators.
Morphological skeletons are of two kinds:
## Skeleton by openings
### Lantuéjoul's formula
#### Continuous images
In (Lantuéjoul 1977),[1] Lantuéjoul derived the following morphological formula for the skeleton of a continuous binary image ${\displaystyle X\subset \mathbb {R} ^{2}}$:
${\displaystyle S(X)=\bigcup _{\rho >0}\bigcap _{\mu >0}\left[(X\ominus \rho B)-(X\ominus \rho B)\circ \mu {\overline {B}}\right]}$,
where ${\displaystyle \ominus }$ and ${\displaystyle \circ }$ are the morphological erosion and opening, respectively, ${\displaystyle \rho B}$ is an open ball of radius ${\displaystyle \rho }$, and ${\displaystyle {\overline {B}}}$ is the closure of ${\displaystyle B}$.
#### Discrete images
Let ${\displaystyle \{nB\}}$, ${\displaystyle n=0,1,\ldots }$, be a family of shapes, where B is a structuring element,
${\displaystyle nB=\underbrace {B\oplus \cdots \oplus B} _{n{\mbox{ times}}}}$, and
${\displaystyle 0B=\{o\}}$, where o denotes the origin.
The variable n is called the size of the structuring element.
Lantuéjoul's formula has been discretized as follows. For a discrete binary image ${\displaystyle X\subset \mathbb {Z} ^{2}}$, the skeleton S(X) is the union of the skeleton subsets ${\displaystyle \{S_{n}(X)\}}$, ${\displaystyle n=0,1,\ldots ,N}$, where:
${\displaystyle S_{n}(X)=(X\ominus nB)-(X\ominus nB)\circ B}$.
#### Reconstruction from the skeleton
The original shape X can be reconstructed from the set of skeleton subsets ${\displaystyle \{S_{n}(X)\}}$ as follows:
${\displaystyle X=\bigcup _{n}(S_{n}(X)\oplus nB)}$.
Partial reconstructions can also be performed, leading to opened versions of the original shape:
${\displaystyle \bigcup _{n\geq m}(S_{n}(X)\oplus nB)=X\circ mB}$.
#### The skeleton as the centers of the maximal disks
Let ${\displaystyle nB_{z}}$ be the translated version of ${\displaystyle nB}$ to the point z, that is, ${\displaystyle nB_{z}=\{x\in E|x-z\in nB\}}$.
A shape ${\displaystyle nB_{z}}$ centered at z is called a maximal disk in a set A when:
• ${\displaystyle nB_{z}\in A}$, and
• if, for some integer m and some point y, ${\displaystyle nB_{z}\subseteq mB_{y}}$, then ${\displaystyle mB_{y}\not \subseteq A}$.
Each skeleton subset ${\displaystyle S_{n}(X)}$ consists of the centers of all maximal disks of size n.
## Notes
1. ^ See also (Serra's 1982 book)
## References
• Image Analysis and Mathematical Morphology by Jean Serra, ISBN 0-12-637240-3 (1982)
• Image Analysis and Mathematical Morphology, Volume 2: Theoretical Advances by Jean Serra, ISBN 0-12-637241-1 (1988)
• An Introduction to Morphological Image Processing by Edward R. Dougherty, ISBN 0-8194-0845-X (1992)
• Ch. Lantuéjoul, "Sur le modèle de Johnson-Mehl généralisé", Internal report of the Centre de Morph. Math., Fontainebleau, France, 1977.
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2016-10-24 15:35:19
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https://www.biostars.org/p/9514566/
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Are not all duplicated reads removed when applying removal duplicated algorithms?
2
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Entering edit mode
6 months ago
ManuelDB ▴ 30
I am trying to understand a bit deeper how duplications occur and how to deal with that in NGS analysis. First, of all, I wanted to understand the FastQC read duplication report for which the tutorial of Istvan Albert is really good (Revisiting the FastQC read duplication report).
My FASTQ file has shown this report
The title shows the proportion of duplicated read what is (as far I can undertand) so high. I have run Rmdup and MarkDuplicate in this file and the proportion of duplicated reads detected and removed/marked is around 15%.
So my question is, are not all duplicated reads removed when applying removal duplicated algorithms?
My second question is, for the simple simulation that Istvan Albert does in his post, I can understand what the red and blue lines is telling me. However, what my red and blue lines are telling me when working in a more realistic scenario like this (e.g. why is there a pick between 9 and >10)?
ngs FASTQC • 888 views
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Entering edit mode
• Keep in mind that removing duplicates is not always needed/allowed. So it's not because you see a fraction of duplicated reads one needs to remover duplicates. It all depends on your data, ore specifically if you run RNAseq, then there should be duplication !!! (or your counts will be pretty low)
• There is also a binning factor in play here. You can see the plot is on a per 1 up to 9 , the next bin is 10-49 so you will always see a bump there which might have nothing to do with duplicated reads but just because it's a very large bin compared to the ones on the left.
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Entering edit mode
Thanks for your answer. This is for clinical analysis and GATK best practices recommend removing duplicates.
I am still wondering if having 50% of duplicates reads is normal, and why FASTQC says I have 48%, but with RmDup and MarkDuplicates, only the 15% are removed???
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Entering edit mode
indeed for such an analysis it's advised to remove duplicates.
duplication rate is also dependent on biological complexity of your sample(s) combined with the sequencing depth.
Why one removes or detects more duplication than another tool I don't really know but it is very well possible they all have different definitions of duplication and/or different levels of sensitivity to pick them up
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Entering edit mode
What type of libraries do you have? Is this RNaseq, shotgun whole genome, exome? For RNAseq and exome, it would be normal do have a high duplication rate.
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Entering edit mode
To do a true estimation of duplicates you can use clumpify.sh from BBMap suite which does purely sequence based analysis, no alignments needed like other tools you mention. It can do perfect matches and can work on paired-end reads at the same time.
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Entering edit mode
6 months ago
h.mon 34k
The issue here is a somewhat hidden (in plain sight, but hidden anyways) feature of FastQC and Istvan's code: duplicates are estimated over the first 50 bp of a read, the rest is ignored - actually, FastQC and Istvan's implementations probably are not identical, as we will see bellow.
The FastQC docs have a verbal description of the details of the duplication module. Two important parts are:
To cut down on the memory requirements for this module only sequences which first appear in the first 100,000 sequences in each file are analysed
Because the duplication detection requires an exact sequence match over the whole length of the sequence, any reads over 75bp in length are truncated to 50bp for the purposes of this analysis.
Istvan's code says the fastq file has been processed with:
cat data.fq \
| bioawk -c fastx '{ print substr(\$seq,1, 50) } ' \
| sort \
| uniq -c \
| sort -k1,1 -rn \
> data.uniq.txt
This states every read of the fastq file has been cut down to 50 bp to perform the analysis - in opposition to FastQC's, which processes the first 100K reads, with reads longer than 75 bp being cut down to 50 bp.
Anyways, and to reinforce the point, FastQC and Istvan's duplicate estimation do not take into account the whole read, so may show duplicates even if the original fastq does not contain duplicates.
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Entering edit mode
interesting point, the duplication as a concept always has another twist to it
1
Entering edit mode
6 months ago
d-cameron ★ 2.7k
So my question is, are not all duplicated reads removed when applying removal duplicated algorithms?
They are not. samtools rmdup and picard MarkDuplicates both use read alignments to determine duplicate reads. If two reads with identical sequence are mapped to different locations in the genome (e.g. they come from a L1HS LINE element, or an alpha satellite repeat) then the duplicate removal tools will fail to remove them. Last time I checked, they also had problems correctly reduplicating split read alignments.
This doesn't explain a 15% vs 50% difference though. Did you do any fastq compression? Tools like BBMap clumpify.sh will reorder the fastq so similar sequences appear near each other - an approach that doesn't play nicely with a duplication rate estimation tool that expects the reads to be random.
why is there a pick between 9 and >10
Because >10 contains 11,12,13,14,15,...,50. That peaks is what you would expect to see based on the duplicate profile we observe for the 1-9 bins in that plot.
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Entering edit mode
are you sure what you post here?
So if no genome available those tools can't do anything? Moreover, if two reads have the exact same sequence they should align to the same region unless they assigned random because no unique mapping exists.
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Entering edit mode
So if no genome available those tools can't do anything?
Correct. Duplicate marking tools such as samtools & picard use a sorted BAM file as input. If there is no genome then those tool will indeed fail to run because the input file isn't going to be in a valid format.
if two reads have the exact same sequence they should align to the same region unless they assigned random because no unique mapping exists.
No popular aligner uses this approach. They will either a) not map them (if the entire read consists of very highly abundant sequences above the aligner-defined seeding threshold (bwa mem default is 500)), or b) randomly assign to one of the homologs. The reason aligners don't systematically assign to the same location is because it stuffs up CNV calling. If they did what you are suggesting, the coverage of a diploid genome would go haywire at every repeat. The 'random' assignment is usually a determinist pseudo-random one so rerunning on the same input fastq will always give you the same alignments.
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2022-09-26 04:06:57
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http://web.emn.fr/x-info/sdemasse/gccat/Kconsecutive_loops_are_connected.html
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### 3.7.51. Consecutive loops are connected
Denotes the fact that the graph constraints of a global constraint use only the $\mathrm{𝑃𝐴𝑇𝐻}$ and the $\mathrm{𝐿𝑂𝑂𝑃}$ arc generators and that their final graphs do not contain consecutive vertices that are not connected together by an arc. Moreover all vertices of their final graphs have a loop.
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2019-04-23 19:52:16
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http://mat-blag.blogspot.com/
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## Thursday, September 28, 2017
### Exit paths, part 2
In this post we continue on a previous topic ("Exit paths, part 1," 2017-08-31) and try to define a constructible sheaf via universality. Let $X$ be an $A$-stratified space, that is, a topological space $X$ and a poset $(A,\leqslant)$ with a continuous map $f:X\to A$, where $A$ is given the upset topology relative to its ordering $\leqslant$. Recall the full subcategory $\Sing^A(X)\subseteq \Sing(X)$ of exit paths on an $A$-stratified space $X$.
Proposition: If $X\to A$ is conically stratified, $\Sing^A(X)$ is an $\infty$-category.
Briefly, a stratification $f:X\to A$ is conical if for every stratum there exists a particular embedding from a stratified cone into $X$ (see Lurie for "conical stratification" and Ayala, Francis, Tanaka for "conically smooth stratified space," which seem to be the same). We will leave confirming the described stratification as conical to a later post.
This proposition, given as part of Theorem A.6.4 in Lurie, has a very long proof, so is not repeated here. Lurie actually proves that the natural functor $\Sing^A(X)\to N(A)$ described below is a (inner) fibration, which implies the unique lifting property of $\Sing^A(X)$ via the unique lifting property of $N(A)$ (and we already know nerves are $\infty$-categories).
Example: The nerve of a poset is an $\infty$-category. Being a nerve, it is already immediate, but it is worthwhile to consider the actual construction. For example, if $A = \{a\leqslant b\leqslant c \leqslant d\}$ is the poset with the ordering $\leqslant$, then the pieces $N(A)_i$ are as below.
It is immediate that every 3-horn can only be filled in one unique way (as there is only one element of $N(A)_3$), as well as that every 2-horn can be filled in one unique way (as every sequence of two composable morphisms appears as a horn of exactly one element of $N(A)_2$).
In Appendix A.9 of Higher Algebra, Lurie says that there is an equivalence of categories $(A\text{-constructible sheaves on }X) \cong \left[(A\text{-exit paths on }X),\mathcal S\right],$ given that $X$ is conically stratified, and for $\mathcal S$ the $\infty$-category of spaces (equivalently $N(Kan)$, the nerve of all the simplicial sets that are Kan complexes). So, instead of trying to define a particular constructible sheaf on $X = \Ran^{\leqslant n}(M)\times \R_{\geqslant 0}$, (as in previous posts "Stratifying correctly," 2017-09-17 and "A constructible sheaf over the Ran space," 2017-06-24) we will try to make a functor that takes an exit path of $X$ and gives back a space.
Fix $n\in \Z_{>0}$ and set $X = \Ran^{\leqslant n}\times \R_{\geqslant 0}$. Let $SC$ be the category of simplicial complexes and simplicial maps, with $SC_n$ the full subcategory of simplicial complexes with at most $n$ vertices. There is a map
$\begin{array}{r c l} g\ :\ X & \to & SC_n \\ (P,t) & \mapsto & VR(P,t), \end{array}$
allowing us to say
$X = \bigcup_{S\in SC_n}g^{-1}(S).$
Here we consider that two elements $P_i,P_j\in P$ give an edge of $VR(P,t)$ whenever $t>d(P_i,P_j)$ (this is chosen instead of $t\geqslant d(P_i,P_j)$ so that the boundaries of the strata facing downward," with respect to the poset ordering, are open). Now we define a stratifying poset $A$ for $X$.
Definition: Let $A = \{a_S\ :\ S\in SC_n\}$ and define a relation $\leqslant$ on $A$ by
$\left(a_S\leqslant a_T\right)\ \ \Longleftarrow\ \ \left( \begin{array}{c} \exists\ \sigma\in \Sing(X)_1\ \text{such that}\\ g(\sigma(0))=S,\ g(\sigma(t>0))=T. \end{array} \right)$
Let $(A,\leqslant)$ be the poset generated by relations of the type given above.
We claim that $f:X\to A$ given by $f(P,t)=a_{g(P,t)}$ is a stratifying map, that is, continuous in the upset topology on $A$. To see this, take the open set $U_S = \{a_T\in A\ :\ a_S\leqslant a_T\}$ in the basis of the upset topology of $A$, for any $S\in SC_n$, and consider $x\in f^{-1}(U_S)$. If for all $\epsilon>0$ we have $B_X(x,\epsilon)\cap f^{-1}(U_S)^C\neq \emptyset$, then there exists $T_\epsilon\in SC_n$ with $B_X(x,\epsilon)\cap f^{-1}(a_{T_\epsilon})\neq\emptyset$, for $S\not\leqslant T_\epsilon$ (as $T_\epsilon\not\in U_S$). This means there exists $\sigma\in \Sing(X)_1$ with $\sigma(0)=x$ and $\sigma(t>0)\in f^{-1}(a_{T_\epsilon})$, which in turn implies $S\leqslant T_\epsilon$, a contradiction. Hence $f$ is continuous, so $f:X\to A$ is a stratification.
As all morphisms in $\Sing(X)$ are compsitions of the face maps $s_i$ and degenracy maps $d_i$, so are all morphisms in $\Sing^A(X)$. There is a natural functor $F:\Sing^A(X)\to N(A)$ defined in the following way:
$\begin{array}{r r c l} %% %% L1 %% \text{objects} & \left( \begin{array}{c} \sigma:|\Delta^k|\to X \\ a_0\leqslant \cdots \leqslant a_k\subseteq A \\ f(\sigma(t_0,\dots,t_i\neq 0,0,\dots,0)) = a_i \end{array} \right) & \mapsto & \left( a_0\to\cdots\to a_k\in N(A)_k\right) \\[20pt] %% %% L2 %% \text{face maps} & \left( \begin{array}{c} \left( \begin{array}{c} \sigma:|\Delta^k|\to X \\ a_0\leqslant \cdots \leqslant a_k\subseteq A \end{array} \right)\\[10pt] \downarrow \\[10pt] \left( \begin{array}{c} \tau:|\Delta^{k+1}|\to X \\ a_0\leqslant \cdots \leqslant a_i\leqslant a_i\leqslant \cdots a_k\subseteq A \end{array} \right) \end{array} \right) & \mapsto & \left(\begin{array}{c} \left(a_0\to\cdots \to a_k\right)\\[10pt] \downarrow\\[10pt] \left(a_0\to\cdots \to a_i\xrightarrow{\text{id}}a_i\to\cdots \to a_k\right) \end{array}\right)\\[40pt] %% %% L3 %% \text{degeneracy maps} & \left( \begin{array}{c} \left( \begin{array}{c} \sigma:|\Delta^k|\to X \\ a_0\leqslant \cdots \leqslant a_k\subseteq A \end{array} \right)\\[10pt] \downarrow \\[10pt] \left( \begin{array}{c} \tau:|\Delta^{k-1}|\to X \\ a_0\leqslant \cdots \leqslant a_{i-1}\leqslant a_{i+1}\leqslant \cdots a_k\subseteq A \end{array} \right) \end{array} \right) & \mapsto & \left(\begin{array}{c} \left(a_0\to\cdots \to a_k\right)\\[10pt] \downarrow\\[10pt] \left(a_0\to\cdots \to a_{i-1}\xrightarrow{\circ}a_{i+1}\to\cdots \to a_k\right) \end{array}\right) \end{array}$
As all maps in $\Sing^A(X)$ are generated by compositions of face and degeneracy maps, this completely defines $F$. Naturality of $F$ follows precisely because of this.
A poset (which can be viewed as a directed simple graph) may be naturally viewed as a 1-dimensional simplicial set, moreover an $\infty$-category (by virtue of being a \emph{simple} graph, with no multi-edges or loops). Hence there is a natural map, the inclusion, that takes $N(A)$ into $N(\mathcal Kan) = \mathcal S$. Finally, Construction A.9.2 of Lurie describes a map that takes a functor from $A$-exit paths into spaces and gives back an $A$-constructible sheaf over $X$, which Theorem A.9.3 shows to be an equivalence, given the following conditions:
• $X$ is paracompact,
• $X$ is locally of singular shape,
• the $A$-stratification of $X$ is conical, and
• $A$ satisfies the ascending chain condition.
The first condition is satisfied as both $\Ran^{\leqslant n}(M)$ and $\R_{\geqslant 0}$ are locally compact and second countable. The last condition is satisfied because $A$ is a finite poset. We already mentioned that the conical property will be checked later, as will the singular shape property. Unfortunately, Lurie gives a definition of singular shape only for $\infty$-topoi, so some work must be done to translate this into our simpler setting. However, in the introduction to Appendix A, Lurie says that if $X$ is "sufficiently nice" and we assume some "mild assumptions" about $A$, then the described categorical equivalence follows, so it seems there is hope that everything will work out well in the end.
References: Stacks Project, Lurie (Higher algebra, Appendix A), Ayala, Francis and Tanaka (Local structures on stratified spaces, Sections 2 and 3)
## Tuesday, September 26, 2017
### Ordering simplicial complexes
In the context of trying to make a constructible sheaf over the Ran space, we have made several attempts to stratify $X=\Ran^{\leqslant n}(M)\times \R_{\geqslant0}$ correctly, the hope being for each stratum to have a unique simplicial complex (the Vietoris-Rips complex of the elements of $X$). In this post we make some observations and examine what it means to move around in $X$.
We use the convention that a Vietoris--Rips complex $VR(P,t)$ of an element $(P,t)\in X$ contains an edge $(P_i,P_j)$ iff $d(P_i,P_j)>t$ (as opposed to $d(P_i,P_j)\geqslant t$).
Observation 1: The VR complex $VR(P,t)$ is completely described by its 1-skeleton $sk_1(VR(P,t))$, as having a complete subgraph $K_\ell\subseteq sk_1(VR(P,t))$ is equivalent to $VR(P,t)$ having an $(\ell-1)$-cell spanning that subgraph. The 1-skeleton is a simple graph $G=(V,E)$ on $k$ vertices, so if we can order simple graphs with $\leqslant n$ vertices, we can order VR complexes of $\leqslant n$ vertices.
Let $\Gamma_k$ be the collection of simple gaphs on $k$ vertices. From now on we talk about an element $(P=\{P_1,\dots,P_k\},t)\in X$, a $k$-vertex VR complex $S=VR(P,t)$, and its 1-skeleton $G=sk_1(VR(P,t))\in \Gamma_k$ interchangeably. Consider the following informal defintion of how the stratification of $X$ should work.
Definition: A VR complex $S$ is ordered lower than another VR complex $T$ if there is a path from the stratum of type $S$ to the stratum of type $T$ that does not pass through strata of type $R$ with $|V(R)|<|V(S)|$ or $|E(R)|<|E(S)|$. If $S$ is ordered lower than $T$ and we can move from the stratum of type $S$ to the stratum of type $T$ without passing through another stratum, then we say that $S$ is directly below $T$.
To gain intuition of what this ordering means, consider the ordering on the posets $B_k'$, as defined in a previous post ("Stratifying correctly," 2017-09-17) and the 1-skeleta of the VR-complexes mapped to their elements. A complete description for $k=1,2,3,4$ and partially for $k=5$ is given below, with arrows $S\to T$ indicating the minimal number of directly below relationships. That is, if $S\to R$ but also $S\to T$ and $T\to R$, then $S\to R$ is not drawn.
The orderings on each $B_k'$ are clear and can be found in an algorithmic manner. However, it is more difficult to see which $S$ at level $k$ are directly below which $T$ at level $k+1$. The green arrows follow no clear pattern.
Observation 2: If $G\in \Gamma_k$ has an isolated vertex and $t>0$, then it can be directly below $H\in \Gamma_{k+1}$ only if $|E(H)|=|E(G)|+1$. In general, if the smallest degree of a vertex of $G\in \Gamma_k$ is $d$ and $t>0$, then $G$ can be directly below $H\in \Gamma_{k+1}$ only if $|E(H)|=|E(G)|+d+1$.
Recall the posets $B_k'$ are made by quotienting the nodes of the hypercube $B_k=\{0,1\}^{k(k-1)/2}$ by the action of $S_k$, where an element of $B_k$ is viewed as a graph $G\in \Gamma_k$ having an edge $(i,j)$ if the coordinate corresponding to the edge $(i,j)$ is 1 (there are $k(k-1)/2$ pairs $(i,j)$ of a $k$-element set).
Observation 3: It is not clear that $G$ not being ordered lower than $H$ in the hypercube context (order increases when increasing in any coordinate) implies that the VR complex of $G$ is not ordered lower than the VR complex of $H$ in $X$. No counterexample exists in the example given above, but this does not seem to exclude the possibility.
If any conclusion can be made from this, it is that this may not be the best approach to take when stratifying $X$.
## Sunday, September 17, 2017
### Stratifying correctly
In a previous blog post ("A constructible sheaf over the Ran space," 2017-06-24) it was claimed that there was a particular constructible sheaf over $\Ran^{\leqslant n}(M)\times \R_{\geqslant 0}$. However, the proof actually uses finite ordered subsets of $M$ to make the stratification, rather than finite unordered subsets. This means that the sheaf is actually over $M^{\times n}\times \R_{\geqslant 0}$, and in this post we try to fix that problem.
Let $\Delta_n$ be the "fat diagonal" of $M^{\times n}$, that is, the collection of $P\in M^{\times n}$ for which at least two coordinates are the same. For every $k>0$, there is an $S_k$ action on $M^{\times k}\setminus \Delta_k$, quotienting by which we get a map
$M^{\times k}\setminus \Delta_k \xrightarrow{\ q_k\ }\Ran^k(M)$
to the Ran space of degree $k$. The stratification of $M^{\times k}\times \R_{\geqslant 0}$ given in the previous post will be pushed forward to a stratification of $\Ran^k(M)\times \R_{\geqslant 0}$, for all $0<k\leqslant n$. A large part of the work already has been done, it remains to put everything in the right order and check openness. The process is given as follows:
1. Stratify $\Ran^{\leqslant n}(M)\times \R_{\geqslant 0}$ into $n$ pieces, each being $\Ran^k(M)\times \R_{\geqslant 0}$.
2. Stratify $(M^{\times k}\setminus \Delta_k)\times \R_{\geqslant 0}$ as in the previous post.
3. Quotient by $S_k$-action to get stratification of $\Ran^k(M)\times \R_{\geqslant 0}$.
### Step 1
As stated in the proof of the Theorem, $\Ran^{\geqslant k}(M)\times \R_{\geqslant 0}$ is open inside $\Ran^{\leqslant n}(M)\times \R_{\geqslant 0}$, allowing us to make a stratification $f:\Ran^{\leqslant n}(M)\times \R_{\geqslant 0}\to A$, where $A$ is the poset
where the tail of an arrow is ordered lower than the head. The map $f$ sends $\Ran^k(M)\times \R_{\geqslant 0}$ to $a_k$, which is a continuous map in the upset topology on $A$.
### Step 2
As stated in Definition 5, we have a stratification $g_k:(M^{\times k}\setminus \Delta_k) \times \R_{\geqslant 0}\to B_k$, where $B_k$ may be viewed as a directed graph $B_k=(V_k,E_k)$. The vertex set is $V_k = \{0,1\}^{k(k+1)/2}$, whose elements are strings of 1 and 0, and the edge set $E_k$ contains $v\to v'$ iff $d_H(v,v')=1$ and $d_H(v,0)<d_H(v',0)$, for $d_H$ the Hamming distance. Let $U_v\subset B_k$ denote the upset based at $v$, that is, all elements $v'\in B_k$ with $v\leqslant v'$.
Order all distinct pairs $(i,j)\in \{1,\dots,k\}^2$, of which there are $k(k+1)/2$. Under the stratifying map $g_k$, each upset $U_v$ based at the vertex $v\in\{0,1\}^{k(k+1)/2}$ receives elements $(P,t)\in (M^{\times k}\setminus \Delta_k)\times \R_{\geqslant 0}$ satisfying $t>d(P_i,P_j)$ whenever the position representing $(i,j)$ in $v$ is 1. For example, when $k=3$,
To check that $g_k$ is continuous in the upset topology, we restate Lemma 2 in a clearer way.
Lemma 1: Let $U\subset X$ be open and $\varphi:X\to A\subset \R_{\geqslant 0}$ continuous, with $|A|<\infty$. Then
$\bigcup_{x\in U}\{x\}\times (\varphi(x),\infty)\ \subseteq\ X\times (z',\infty)$
is open, for any $z'\leqslant z:= \min_{x\in U}\{\varphi(x)\}$.
Proof: Consider the function
$\begin{array}{r c l} \psi\ :\ X\times (z',\infty) & \to & X\times (-\infty,z), \\ (x,t) & \mapsto & (x,\varphi(x)-t). \end{array}$
Since $\varphi$ is continuous and subtraction is continuous, $\psi$ is continuous (in the product topology). Since $U\times (-\infty,0)$ is open in $X\times (-\infty,z)$, the set $\psi^{-1}(U\times (-\infty,0))$ is open in $X\times (z',\infty)$. For any $x\in U$ and $t=\varphi(x)$, we have $\varphi(x)-t =0$. For any $x\in U$ and $t\to \infty$, we have $\varphi(x)-t\to -\infty$. It is immediate that all other $t\in (\varphi(x),\infty)$ give $\varphi(x)-t\in (-\infty,0)$. Hence $\psi^{-1}(U\times (-\infty,0))$ is the collection of points $(x,t)$ with $t\in (\varphi(x),\infty)$, which is then open in $X\times [0,z')$. $\square$
Applying Lemma 1 to $U=X=M^{\times k}\setminus \Delta_k$ and $\varphi(P) = \max_{i\neq j}\{d(P_i,P_j)\}$, which is continuous, gives that $g_k^{-1}(U_{11\cdots1})\subseteq M^{\times k}\setminus \Delta_k$ is open. This also works to show that $g_k^{-1}(U_v)\subseteq g_k^{-1}(U_{v'})$ is open, for any $v'\leqslant v$, by limiting the pairs of indices iterated over by the function $\varphi$. Hence $g_k$ is continuous.
### Step 3
The symmetric group $S_k$ acts on $(M^{\times k}\setminus \Delta_k)\times \R_{\geqslant 0}$ by permuting the order of elements in the first factor. That is, for $\sigma\in S_k$, we have
$\sigma (P=\{P_1,\dots,P_k\},t) = (\{P_{\sigma(1)},\dots,P_{\sigma(k)}\},t).$
Note that $((M^{\times k}\setminus \Delta_k)\times \R_{\geqslant 0})/S_k = \Ran^k(M)\times \R_{\geqslant 0}$.
Remark: Graph isomorphism for two graphs with $k$ vertices may also be viewed as the equivalence relation induced by $S_k$ acting on $\Gamma_k = \{$simple vertex-labeled graphs with $k$ vertices$\}$. First, let $G_v$ be the (unique) graph first introduced at element $v\in B_k$ by $g_k$. That is, we have $G_v =VR(P,t)_1$ (the ordered 1-skeleton of the Vietoris--Rips complex on the set $P$ with radius $t$) whenever $g_k((P,t))\in U_v$ and $g_k((P,t))\not\in U_{v'}$ for any $v'\leqslant v$, $v'\neq v$. Then the elements of $B_k$ are in bijection with the elements of $\Gamma_k$ (given by $v\leftrightarrow G_v$), so we have $B_k/S_k = B_k'$. Recall that $v\leqslant v'$ in $B_k$ iff adding an edge to $G_v$ gives $G_{v'}$. In $B_{k'}$, this becomes a partial order on equivalence classes $[w] = \{v\in B_k\ :\ \sigma G_v=G_w\$for some $\sigma\in S_k\}$. We write $[w]\leqslant [w']$ iff there is a collection of pairs $\{(v_1,v_1'),\dots,(v_\ell,v_\ell')\}$ such that $v_i\leqslant v_i'$ for all $i$, and $\{v_1,\dots,v_\ell\} = [w]$ and $\{v_1',\dots,v_\ell'\} = [w']$ (there may be repetition among the $v_i$ or $v_i'$).
By the universal property of the quotient, there is a unique map $h_k:\Ran^k(M)\times \R_{\geqslant 0}\to B_k'$ that makes the following diagram commute.
This will be our stratifying map. To check that $h_k$ is continuous take $U\subseteq B_k'$ open. As $\pi$ is the projection under a group action, it is an open map, so $\pi^{-1}(U)\subseteq B_k$ is open. Since $g_k$ is continuous in the upset topology, $g_k^{-1}(\pi^{-1}(U))$ is open. Again, $S_k\curvearrowright$ is the projection under a group action, so $(S_k\curvearrowright)(g_k^{-1}(\pi^{-1}(U)))$ is open, giving continuity of $h_k$.
## Thursday, August 31, 2017
### Exit paths, part 1
This post is meant to set up all the necessary ideas to define the category of exit paths.
### Preliminaries
Let $X$ be a topological space and $C$ a category. Recall the following terms:
• $\Delta$: The category whose objects are finite ordered sets $[n]=(1,\dots,n)$ and whose morphisms are non-decreasing maps. It has several full subcategories, including
• $\Delta_s$, comprising the same objects of $\Delta$ and only injective morphisms, and
• $\Delta_{\leqslant n}$, comprising only the objects $[0],\dots,[n]$ with the same morphisms.
• equalizer: An object $E$ and a universal map $e:E\to X$, with respect to two maps $f,g:X\to Y$. It is universal in the sense that all maps into $X$ whose compositions with $f,g$ are equal factor through $e$. Equalizers and coequalizers are described by the diagram below, with universality given by existence of the dotted maps.
• fibered product or pullback: The universal object $X\times_Z Y$ with maps to $X$ and $Y$, with respect to maps $X\to Z$ and $Y\to Z$.
• fully faithful: A functor $F$ whose morphism restriction $\Hom(X,Y)\to \Hom(F(X),F(Y))$ is surjective (full) and injective (faithful).
• locally constant sheaf: A sheaf $\mathcal F$ over $X$ for which every $x\in X$ has a neighborhood $U$ such that $\mathcal F|_U$ is a constant sheaf. For example, constructible sheaves are locally constant on every stratum.
• simplicial object: A contravariant functor from $\Delta$ to any other category. When the target category is $\text{Set}$, it is called a simplicial set. They may also be viewed as a collection $S = \{S_n\}_{\geqslant 0}$ for $S_n=S([n])$ the value of the functor on each $[n]$. Simplicial sets come with two natural maps:
• face maps $d_i:S_n\to S_{n-1}$ induced by the map $[n-1]\to [n]$ which skips the $i$th piece, and
• degeneracy maps $s_i:S_n\to S_{n+1}$ induced by the map $[n+1]\to[n]$ which repeats the $i$th piece.
• stratification: A property of a cover $\{U_i\}$ of $X$ for which consecutive differences $U_{i+1}\setminus U_i$ have nicer" properties than all of $X$. For example, $E_i\to U_{i+1}\setminus U_i$ is a rank $i$ vector bundle, but there is no vector bundle $E\to X$ that restricts to every $E_i$.
Now we get into new territory.
Definition: The nerve of a category $C$ is the collection $N(C) = \{N(C)_n = Fun([n],C)\}_{n\geqslant 0}$, where $[n]$ is considered as a category with objects $0,\dots,n$ and a single morphism in $\Hom_{[n]}(s,t)$ iff $s\leqslant t$.
Note that the nerve of $C$ is a simplicial set, as it is a functor from $\Delta^{op}\to Fun(\Delta,C)$. Moreover, the pieces $N(C)_0$ are the objects of $C$ and $N(C)_1$ are the morphisms of $C$, so all the information about $C$ is contained in its nerve. There is more in the higher pieces $N(C)_n$, so the nerve (and simplicial sets in general) may be viewed as a generalization of a category.
### Kan structures
Let $\text{sSet}$ be the category of simplicial sets. We may consider $\Delta^n = \Hom_\Delta(-,[n])$ as a contravariant functor $\Delta\to \text{Set}$, so it is an object of $\text{sSet}$.
Definition: Fix $n\geqslant 0$ and choose $0\leqslant i\leqslant n$. Then the $i$th $n$-horn of a simplicial set is the functor $\Lambda^n_i\subset \Delta^n$ generated by all the faces $\Delta^n(d_j)$, for $j\neq i$.
We purposefully do not describe what "$\subset$" or "generated by" mean for functors, hoping that intuition fills in the gaps. In some sense the horn feels like a partially defined functor (though it is a true simplicial set), well described by diagrams, for instance with $n=2$ and $i=1$ we have
Definition: A simplicial set $S$ is a Kan complex whenever every map $f:\Lambda^n_i\to S$ factors through $\Delta^n$. That is, when there exists a
The map $\iota$ is the inclusion. Moreover, $S$ is an $\infty$-category, or quasi-category, if the extending map $f'$ is unique.
Example: Some basic examples of $\infty$-categories, for $X$ a topological space, are
• $Sing(X)$, made up of pieces $Sing(X)_n = \Hom(\Delta^n,X)$, and
• $LCS(X)$, the category of locally constant sheaves over $X$. Here $LCS(X)_n$ over an object $A$, whose objects are $B\to A$ and morphisms are the appropriate commutative diagrams
Definition: A morphism $p\in \Hom_{\text{sSet}}(S,T)$ is a Kan fibration if for every commutative diagram (of solid arrows)
the dotted arrow exists, making the new diagram commute.
Definition: Let $C,D,A$ be categories with functors $F:C\to D$ and $G:C\to A$.
• The left Kan extension of $F$ along $G$ is a functor $A\xrightarrow L D$ and a universal natural transformation $F\stackrel \lambda \rightsquigarrow L\circ G$.
• The right Kan extension of $F$ along $G$ is a functor $A\xrightarrow R D$ and a universal natural transformation $R\circ G \stackrel \rho\rightsquigarrow F$.
### Exit paths
The setting for this section is constructible sheaves over a topological space $X$. We begin with a slightly more technical definition of a stratification.
Definition: Let $(A,\leqslant)$ be a partially ordered set with the upset topology. That is, if $x\in U$ is open and $x\leqslant y$, then $y\in A$. An $A$-stratification of $X$ is a continuous function $f:X\to A$.
We now begin with a Treumann's definition of an exit path, combined with Lurie's stratified setting.
Definition: An exit path in an $A$-stratified space $X$ is a continuous map $\gamma:[0,1]\to X$ for which there exists a pair of chains $a_1\leqslant \cdots \leqslant a_n$ in $A$ and $0=t_0\leqslant \cdots \leqslant t_n=1$ in $[0,1]$ such that $f(\gamma(t))=a_i$ whenever $t\in (t_{i-1},t_i]$.
This really is a path, and so gives good intuition for what is happening. Recall that the geometric realization of the functor $\Delta^n$ is $|\Delta^n| = \{(t_0,\dots,t_n)\in \R^{n+1}\ :\ t_0+\cdots+t_n=1\}$. Oserving that $[0,1]\cong|\Delta^1|$, Lurie's definition of an exit path is more general by instead considering maps from $|\Delta^n|$.
Definition: The category of exit paths in an $A$-stratified space $X$ is the simplicial subset $Sing^A(X)\subset Sing(X)$ consisting of those simplices $\gamma:|\Delta^n|\to X$ for which there exists a chain $a_0\leqslant \cdots \leqslant a_n$ in $A$ such that $f(\gamma(t_0,\dots,t_i,0,\dots,0))=a_i$ for $t_i\neq 0$.
Example: As with all new ideas, it is useful to have an example. Consider the space $X=\Ran^{\leqslant 2}(M)\times \R_{\geqslant 0}$ of a closed manifold $M$ (see post "A constructible sheaf over the Ran space" 2017-06-24 for more). With the poset $(A,\leqslant)$ being $(a\leqslant b\leqslant c)$ and stratifying map
$\begin{array}{r c l} f\ :\ X & \to & A, \\ (P,t) & \mapsto & \begin{cases} a & \text{ if } P\in \Ran^1(M), \\ b & \text{ if } P\in \Ran^2(M), t\leqslant d(P_1,P_2), \\ c & \text{ else,} \end{cases} \end{array}$
we can make a continuous map $\gamma:\Delta^3\to X$ by
$\begin{array}{r c l} (1,0,0) & \mapsto & (P\in \Ran^1(M),0), \\ (t_0,t_1\neq 0,0) & \mapsto & (P\in \Ran^2(M), d(P_1,P_2)), \\ (t_0,t_1,t_2\neq 0) & \mapsto & (P\in \Ran^2(M), t>d(P_1,P_2)). \end{array}$
Then $f(\gamma(t_0\neq 0,0,0))=a$, and $f(\gamma(t_0,t_1\neq 0,0))=b$, and $f(\gamma(t_0,t_1,t_2\neq 0))=c$, as desired. The embedding of such a simplex $\gamma$ is described by the diagram below.
Both the image of $(1,0,0)$ and the 1-simplex from $(1,0,0)$ to $(0,1,0)$ lie in the singularity set of $\Ran^{\leqslant 2}(M)\times \R_{\geqslant 0}$, which is pairs $(P,t)$ where $t=d(P_i,P_j)$ for some $i,j$. The idea that the simplex "exits" a stratum is hopefully made clear by this image.
References: Lurie (Higher algebra, Appendix A), Lurie (What is... an $\infty$-category?), Groth (A short course on $\infty$-categories, Section 1), Joyal (Quasi-categories and Kan complexes), Goerss and Jardine (Simplicial homotopy theory, Chapter 1), Treumann (Exit paths and constructible stacks)
## Friday, August 11, 2017
### The Ran space and singularity sets
Fix a manifold $M$ along with an embedding of $M$ into $\R^N$ and set $X=\Ran(M)\times \R_{\geqslant 0}$. The goal of this post is to show that every $(P,t)\in X$ has an open neighborhood that contains no points of the type $(Q,d(Q_i,Q_j))$, for some $i\neq j$. The collection of all such elements of $X$ is called the singularity set of $X$, as the Vietoris-Rips complex at $Q$ with such a radius changes at such elements.
Following Lurie, given a collection of open sets $\{U_i\}_{i=1}^k$ in $M$, set
$\Ran(\{U_i\}_{i=1}^k) = \left\{P\in \Ran(M)\ :\ P\subset \bigcup_{i=1}^k U_i,\ P\cap U_i\neq \emptyset\ \forall\ i\right\}.$
The topology on $\Ran(M)$ is the smallest topology for which every $\Ran(\{U_i\}_{i=1}^k)$ is open, for any $\{U_i\}_{i=1}^k$, for any $k$. The topology on the product $X$ is the product topology.
Remark: Note that the Ran space $\Ran(M)$ by itself can be split up into the pieces $\Ran^k(M)$, with "singularities" viewed as when a point splits into two (or more) points, or two (or more) combine into one. Then every element of $\Ran(M)$ is on the edge of the singularity set, as any neighborhood of a single point on the manifold contains two points on the manifold.
Fix $(P,t)\in X$ not in the singularity set of $X$, with $P=(P_1,\dots,P_k)$, for $1\leqslant k\leqslant n$. Set
$\mu = \min\left\{t,\min_{1\leqslant i<j\leqslant k}\left\{|t-d(P_i,P_j)|\right\}\right\},$
with distance $d$ being Euclidean distance in $\R^N$. The quantity $\mu$ should be thought of as the upper bound on how "far" we may move from $(P,t)$ without hitting the singularity set.
Proposition: Let $(P,t)$ be as above and $t,\alpha,\beta>0$ such that $\alpha+\beta=\mu$. Then
$U=\Ran\left(\{B(P_i,\alpha/2)\}_{i=1}^k\right) \times \left(t-\beta,t+\beta\right)$
is an open neighborhood of $(P,t)$ in $X$ and does not contain any points of the singularity set of $X$.
If $t=0$, then having $[0,\beta)$ as the second component of $U$, with $\alpha+\beta=\min_{i,j}d(P_i,P_j)$ works as the open neighborhood of $(P,t)$. The balls $B(x,r)$ are $N$-dimensional in $\R^N$. The proof is mostly applications of the triangle inequality.
Proof: By construction we have that $U$ is open in $X$ and that it contains $(P,t)$. For $(Q,s)\in U$ any other element, we have three cases. We will show that the distance between any two $Q_a,Q_b\in Q$ is never $s$. Fix distinct indices $\ell,m\in \{1,\dots,k\}$.
1. Case 1: $Q_a,Q_b\in B(P_\ell,\alpha/2)$. The situation looks as in the diagram below.
Observe that $d(Q_a,Q_b)\leqslant d((Q_a,P_\ell)+d(Q_b,P_\ell) <\alpha = \mu-\beta \leqslant t-\beta$. Hence $d(Q_a,Q_b)<s$.
2. Case 2: $Q_a\in B(P_\ell,\alpha/2), Q_b\in B(P_m,\alpha/2), d(P_\ell,P_m)>t$. The situation looks as in the diagram below.
Observe that $d(P_\ell,P_m)\leqslant d(P_\ell,Q_b)+d(P_m,Q_b)\leqslant d(P_\ell,Q_a)+d(Q_a,Q_b)+d(P_m,Q_b) < \alpha+d(Q_a,Q_b)$. Since $d(P_\ell,P_m)>t$, the definition of $\mu$ gives us that $\mu \leqslant d(P_\ell,P_m)-t$, so combining this with the previous inequality, we get $d(Q_a,Q_b) > d(P_\ell,P_m)-\alpha\geqslant \mu+t-(\mu-\beta)=t+\beta$. Hence $d(Q_a,Q_b)>s$.
3. Case 3: $Q_a\in B(P_\ell,\alpha/2), Q_b\in B(P_m,\alpha/2), d(P_\ell,P_m)<t$. The situation looks as in the diagram below.
Observe that $d(Q_a,Q_b)\leqslant d(P_m,Q_b) + d(P_m,Q_a) \leqslant d(P_\ell,Q_a)+d(P_\ell,P_m)+d(P_m,Q_a)<\alpha+d(P_\ell,P_m)$. Since $d(P_\ell,P_m)<t$, the definition of $\mu$ gives us that $\mu\leqslant t-d(P_\ell,P_m)$, so combining this with the previous inequality, we get $d(Q_a,Q_b)<\mu-\beta+t-\mu = t-\beta$. Hence $d(Q_a,Q_b)<s$. $\square$
As an extension, it would be nice to show that the Vietoris--Rips complex of every element in $U$ is homotopy equivalent. This seems to be intuitively true, but a similar case analysis as above seems daunting.
References: Lurie (Higher Algebra, Section 5.5.1)
## Thursday, August 3, 2017
### New directions in TDA
Conference topic
This post is informal, meant as a collection of (personally) new things from the workshop "Topological data analysis: Developing abstract foundations" at the Banff International Research Station, July 31 - August 4, 2017. New actual questions:
1. Does there exist a constructible sheaf valued in persistence modules over $\Ran^{\leqslant n}(M)$?
• On the stalks it should be the persistence module of $P\in \Ran^{\leqslant n}(M)$. What about arbitrary open sets?
• Is there such a thing as a colimit of persistence modules?
• Uli Bauer suggested something to do with ordering the elements of the sample and taking small open sets.
2. Can framed vector spaces be used to make the TDA pipeline functorial? Does Ezra Miller's work help?
• Should be a functor from $(\R,\leqslant)$, the reals as a poset, to $\text{Vect}$ or $\text{Vect}_{fr}$, the category of (framed) vector spaces. Filtration function $f:\R^n\to \R$ is assumed to be given.
• Framed perspective should not be too difficult, just need to find right definitions.
• Does this give an equivalence of categories (category of persistence modules and category of matchings)? Is that what we want? Do we want to keep only specific properties?
• Ezra's work is very dense and unpublished. But it seems to have a very precise functoriality (which is not the main thrust of the work, however).
3. Can the Bubenik-de Silva-Scott interleaving categorification be viewed as a (co)limit? Diagrams are suggestive.
• Reference is 1707.06288 on the arXiv.
• Probably not a colimit, because that would be very large, though the arrows suggest a colimit.
• Have to be careful, because the (co)limit should be in the category of posets, not just interleavings.
1. Algebraic geometry / homotopy theory: the etale space of a sheaf, Kan extensions, model categories, symmetric monoidal categories.
2. TDA related: Gromov-Hausdorff distance, the universal distance (Michael Lesnick's thesis and papers), merge trees, Reeb graphs, Mapper (the program).
## Saturday, June 24, 2017
### A constructible sheaf over the Ran space
Let $M$ be a manifold. The goal of this post is to show that the sheaf $\mathcal F_{(P,t)}=\text{Rips}(P,t)$ valued in simplicial complexes over $\Ran^{\geqslant}(M)\times \R_{\geqslant 0}$ is constructible, a goal not quite achieved. This space will be described using filtered diagram of open sets, with the sheaf on consecutive differences of the diagram giving simplicial complexes of the same homotopy type.
Definition:
Let $P=\{P_1,\dots,P_n\}\in \Ran(M)$. For every collection of open neighborhoods $\{U_i\owns P_i\}_{i=1}^n$ of the $P_i$ in $M$, there is an open neighborhood of $P$ in $\Ran(M)$ given by
$\Ran(\{U_i\}_{i=1}^n) = \left\{Q\in \Ran(M)\ :\ Q\subset \bigcup_{i=1}^n U_i,\ Q\cap U_i\neq \emptyset\right\}.$Moreover, these are a basis for any open neighborhood of $P$ in $\Ran(M)$.
### Sets
We begin with a few facts about sets. Let $X$ be a topological space.
Lemma 1: Let $A,B\subset X$. Then:
• (a) If $A\subset B$ is open and $B\subset X$ is open, then $A\subset X$ is open.
• (b) If $A\subset B$ is closed and $B\subset X$ is open, then $A\subset X$ is locally closed.
• (c) If $A\subset B$ is open and $B\subset X$ is locally closed, then $A\subset X$ is locally closed.
• (d) If $A\subset B$ is locally closed and $B\subset X$ is locally closed, then $A\subset X$ is locally closed.
Proof: For part (a), first recall that open sets in $B$ are given by intersections of $B$ with open sets of $A$. Hence there is some $W\subset X$ open such that $A = B\cap W$. Since both $B$ and $W$ are open in $X$, the set $A$ is open in $X$.
For part (b), since $A\subset B$ is closed, there is some $Z\subset X$ closed such that $A=B\cap Z$. Since $B$ is open in $X$, $A$ is locally closed in $X$.
For parts (c) and (d), let $B = W_1\cap W_2$, for $W_1\subset X$ open and $W_2\subset X$ closed. For part (c), again there is some $W\subset X$ open such that $A = B\cap W$. Then $A = (W_1\cap W_2)\cap W = (W\cap W_1)\cap W_2$, and since $W\cap W_1$ is open in $X$, the set $A$ is locally closed in $X$.
For part (d), let $A = Z_1\cap Z_2$, where $Z_1\subset B$ is open and $Z_2\subset B$ is closed. Then there exists $Y_1\subset X$ open such that $Z_1 = B\cap Y_1$ and $Y_2\subset X$ closed such that $Z_2 = B\cap Y_2$. So $A = Z_1\cap Z_2 = (B\cap Y_1) \cap (B\cap Y_2) = (B\cap Y_1)\cap Y_2$, where $(B\cap Y_1)\subset X$ is open and $Y_2\subset X$ is closed. Hence $A\subset X$ is locally closed. $\square$
Lemma 2: Let $U\subset X$ be open and $f:X\to \R$ continuous. Then $\bigcup_{x\in U}\{x\}\times (f(x),\infty)$ is open in $X\times \R$.
Proof: Consider the function
$\begin{array}{r c l} g\ :\ X\times \R & \to & X\times \R, \\ (x,t) & \mapsto & (x,t-f(x)). \end{array}$Since $f$ is continuous and subtraction is continuous, $g$ is continuous (in the product topology). Since $U\times (0,\infty)$ is open in $X\times \R$, the set $g^{-1}(U\times (0,\infty))$ is open in $X\times \R$. This is exactly the desired set. $\square$
### Filtered diagrams
Definition: A filtered diagram is a directed graph such that
• for every pair of nodes $u,v$ there is a node $w$ such that there exist paths $u\to w$ and $v\to w$, and
• for every multi-edge $u\stackrel{1,2}\to v$, there is a node $w$ such that $u\stackrel 1\to v\to w$ is the same as $u\stackrel2\to v\to w$.
For our purposes, the nodes of a filtered diagram will be subsets of $\Ran^n(M)\times \R_{\geqslant 0}$ and a directed edge will be open inclusion of one set into another set (that is, the first is open inside the second). Although we require below that loops $u\to u$ be removed, we consider the first condition above to be satisfied if there exists a path $u\to v$ or a path $v\to u$.
Remark: In the context given,
• edge loops $U\to U$ and path loops $U\to\cdots\to U$ may be replaced by a single node $U$ ($U\subseteq U$ is the identity),
• multi-edges $U\stackrel{1,2}\to V$ may be replaced by a single edge $U\to V$ (inclusions are unique), and
• multi-edges $U\to V\to U$ may be replaced by a single node $U$ (if $U\subseteq V$ and $V\subseteq U$, then $U=V$).
A diagram with all possible replacements of the types above is called a reduced diagram.
Lemma 3: In the context above, a reduced filtered diagram $D$ of open sets of any topological space $X$ gives an increasing sequence of open subsets of $X$, with the same number of nodes.
Proof: Order the nodes of $D$ so that if $U\to V$ is a path, then $U$ has a lower index than $V$ (this is always possible in a reduced diagram). Let $U_1,U_2,\dots,U_N$ be the order of nodes of $D$ (we assume we have finitely many nodes). For every pair of indices $i,j$, set
$\delta_{ij} = \begin{cases} \emptyset & \text{ if }U_i\to U_j\text{ is a path in }D, \\ U_i & \text{ if }U_i\to U_j\text{ is not a path in }D. \end{cases}$Then the following sequence is an increasing sequence of nested open subsets of $X$:
$U_1 \to \delta_{12} \cup U_2 \to \delta_{13}\cup \delta_{23}\cup U_3 \to \cdots \to \underbrace{\left(\bigcup_{i=1}^{j-1}\delta_{ij} \right)\cup U_j}_{V_j} \to \cdots \to U_N.$Indeed, if $U_i \to U_j$ is a path in $D$, then $U_i$ is open in $V_j$, as $U_i\subset V_j$. If $U_i\to U_j$ is not a path in $D$, then $U_i$ is still open in $V_j$, as $U_i\subset V_j$. As unions of opens are open, and by Lemma 1(a), $V_{j-1}$ is open in $V_j$ for all $1<j<N$. $\square$
Remark 4: Note that every consecutive difference $V_j\setminus V_{j-1}$ is a (not necessarily proper) subset of $U_j$.
Definition 5: For $k\in \Z_{>0}$, define a filtered diagram $D_k$ over $\Ran^k(M)\times \R_{\geqslant 0}$ by assigning a subset to every corner of the unit $N$-hypercube in the following way: for the ordered set $S=\{(i,j) \ :\ 1\geqslant i<j\geqslant k\}$ (with $|S|=N=k(k-1)/2$), write $P=\{P_1,\dots,P_k\}\in \Ran^k(M)$, and assign
$(\delta_1,\dots,\delta_k) \mapsto \left\{ (P,t)\in \Ran^k(M)\times \R_{\geqslant 0}\ :\ t>d(P_{(S_\ell)_1},P_{(S_\ell)_2}) \text{ whenever }\delta_\ell=0,\ \forall\ 1\leqslant \ell\leqslant k \right\},$where $\delta_\ell\in \{0,1\}$ for all $\ell$, and $d(x,y)$ is the distance on the manifold $M$ between $x,y\in M$. The edges are directed from smaller to larger sets.
Remark 6: This diagram has $2^{k(k-1)/2}$ nodes, as $k(k-1)/2$ is the number of pairwise distances to consider. Moreover, the difference between the head and tail of every directed edge is elements $(P,t)$ for which $\text{Rips}(P,t)$ is constant.
Example: For example, if $k=3$, then $2^{3\cdot2/2}=8$, and $D_3$ is the diagram below. For ease of notation, we write $\{t>\cdots\}$ to mean $\{(P,t)\ :\ P=\{P_1,P_2,P_3\}\in \Ran^3(M),\ t>\cdots\}$.
The diagram of corresponding Vietoris-Rips complexes introduced at each node is below. Note that each node contains elements $(P,t)$ whose Vietoris-Rips complex may be of type encountered in any paths leading to the node.
Lemma 7: In the filtered diagram $D_k$, every node is open inside every node following it.
Proof: The left-most node of $D_k$ may be expressed as
$\{(P,t)\ :\ P=\{P_1,\dots,P_k\}\in \Ran^k(M), t>d(P_i,P_j)\ \forall\ P_i,P_j\in P\} = \bigcup_{P\in \Ran^k(M)} \{P\}\times \left(\max_{P_i,P_j\in P}\{d(P_i,P_j)\},\infty\right).$ Applying a slight variant of Lemma 2 (replacing $\R$ by an open ray that is bounded below), with the $\max$ function continuous, we get that the left-most node is open in the nodes one directed edge away from it. Repeating this argument, we get that every node is open inside every node following it. $\square$
### The constructible sheaf
Recall that a constructible sheaf can be given in terms of a nested cover of opens or a cover of locally closed sets (see post "Constructible sheaves," 2017-06-13). The approach we take is more the latter, and illustrates the relation between the two. Let $n\in \Z_{>0}$ be fixed.
Definition: Define a sheaf $\mathcal F$ over $X=\Ran^{\leqslant n}(M)\times \R_{\geqslant 0}$ valued in simplicial complexes, where the stalk $\mathcal F_{(P,t)}$ is the Vietoris--Rips complex of radius $t$ on the set $P$. For any subset $U\subset X$ such that $\mathcal F_{(P,t)}$ is constant for all $(P,t)\in U$, let $\mathcal F(U)=\mathcal F_{(P,t)}$.
Note that we have not described what $\mathcal F(U)$ is when $U$ contains stalks with different homotopy types. Omitting this (admittedly large) detail, we have the following:
Theorem: The sheaf $\mathcal F$ is constructible.
Proof: First, by Remark 5.5.1.10 in Lurie, we have that $\Ran^{n}(M)$ is open in $\Ran^{\leqslant n}(M)$. Hence $\Ran^{\leqslant n-1}(M)$ is closed in $\Ran^{\leqslant n}(M)$. Similarly, $\Ran^{\leqslant n-2}(M)$ is closed in $\Ran^{\leqslant n-1}(M)$, and so closed in $\Ran^{\leqslant n}(M)$, meaning that $\Ran^{\leqslant k}(M)$ is closed in $\Ran^{\leqslant n}(M)$ for all $1\leqslant k\leqslant n$. This implies that $\Ran^{\geqslant k}(M)$ is open in $\Ran^{\leqslant n}(M)$ for all $1\leqslant k\leqslant n$, meaning that $\Ran^k(M)$ is locally closed in $\Ran^{\leqslant n}(M)$, for all $1\leqslant k\leqslant n$.
Next, for every $1\leqslant k\leqslant n$, let $V_{k,1}\to \cdots \to V_{k,N_k}$ be a sequence of nested opens covering $\Ran^k(M)\times \R_{\geqslant 0}$, as given in Definition 5 and flattened by Lemma 3. The sets are open by Lemma 7. This gives a cover $\mathcal V_k = \{V_{k,1},V_{k,2},\setminus V_{k,1},\dots,V_{k,N_k}\setminus V_{k,N_k-1}\}$ of $\Ran^k(M)\times \R_{\geqslant 0}=V_{k,N_k}$ by consecutive differences, with $V_{k,1}$ open in $V_{k,N_k}$ and all other elements of $\mathcal V_k$ locally closed in $V_{k,N_k}$, by Lemma 1(b). By Lemma 1 parts (c) and (d), every element of $\mathcal V_k$ is locally closed in $\Ran^{\leqslant n}(M)\times \R_{\geqslant 0}$, and so $\mathcal V = \bigcup_{k=1}^n \mathcal V_k$ covers $\Ran^{\leqslant n}(M)\times \R_{\geqslant 0}$ by locally closed subsets.
Finally, by Remarks 4 and 6, over every $V\in\mathcal V$ the function $\text{Rips}(P,t)$ is constant. Hence $\mathcal F|_V$ is a locally constant sheaf, for every $V\in \mathcal V$. As the $V$ are locally closed and cover $X$, $\mathcal F$ is constructible. $\square$
References: Lurie (Higher algebra, Section 5.5.1)
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