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## If x is a positive number, is x an even integer?

##### This topic has expert replies
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### If x is a positive number, is x an even integer?

by BTGmoderatorDC » Wed Sep 01, 2021 7:39 pm

00:00

A

B

C

D

E

## Global Stats

If x is a positive number, is x an even integer?

(1) 3x is an even integer.

(2) 5x is an even integer.

OA C

Source: Manhattan Prep

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Posts: 2035
Joined: 29 Oct 2017
Followed by:6 members

### Re: If x is a positive number, is x an even integer?

by swerve » Fri Sep 03, 2021 3:24 am

00:00

A

B

C

D

E

## Global Stats

BTGmoderatorDC wrote:
Wed Sep 01, 2021 7:39 pm
If x is a positive number, is x an even integer?

(1) 3x is an even integer.

(2) 5x is an even integer.

OA C

Source: Manhattan Prep
$$x$$ is positive Integer means we need to think only in positive fractions and integers.

1) $$3x$$ is an even integer.

Let $$x= 2 \Rightarrow 3x=6 \Rightarrow x$$ is even integer

Let $$x=\dfrac{2}{3} \Rightarrow 3x=2 \Rightarrow x$$ is NOT even integer

Insufficient $$\Large{\color{red}\chi}$$

2) $$5x$$ is an even integer.

Let $$x= 2 \Rightarrow 5x=10 \Rightarrow x$$ is even integer

Let $$x=\dfrac{2}{5} \Rightarrow 5x=2 \Rightarrow x$$ is NOT even integer

Insufficient $$\Large{\color{red}\chi}$$

Combining 1 and 2, we have

There is no fraction could be multiplied simultaneously to $$3 \& 5$$ and give EVEN INTEGER. We left with that $$x=$$EVEN INTEGER to give make both $$3x \& 5x$$ even integers.

As $$3 \& 5$$ are odd numbers, then $$x$$ needs to be EVEN Integer to make both statements Even integers. $$\Large{\color{green}\checkmark}$$

Therefore, C

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5 Answered Questions for the topic consumer theory

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#### 题目列表

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1 2 ... 6 7 8 9 10 11 12 ... 53 54

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# Tag Info

## Hot answers tagged newtonian-gravity

67

If you allow for non-Newtonian gravity (i.e., general relativity), then an extended body can "swim" through spacetime using cyclic deformations. See the 2003 paper "Swimming in Spacetime: Motion by Cyclic Changes in Body Shape" (Science, vol. 299, p. 1865) and the 2007 paper "Extended-body effects in cosmological spacetimes" (Classical and Quantum Gravity, ...

49

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16

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14

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7

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6

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6

Essentially, yes. The derivation of the escape velocity is based only on the energy balance and energy does not depend on the direction. This follows from the property of the gravitational field to be conservative, so work required to move between any 2 points is independent from the path you take. Vague intuition: the vertical speed you estimate is about $... 4 By conservation of momentum and energy, the only possible way to change a planet's trajectory is to eject some (large) mass at high velocity in specific direction, like rockets do. But you are also correct that by increasing the moment of inertia, the rotational speed can be changed. But this cannot influence the movement of center of mass. Edit2: Other ... 4 In newtonian mechanics the angle does not matter, but in relativity it does. For example: An object close to the speed of light launched horizontally will orbit circular at a distance of 3GM/c² from the center of mass (the so called photon sphere), but it will escape if launched vertically. When you launch it at a distance just above 2GM/c² (the so called ... 4 Yes, the sun's mass isn't constant causing disturbances of orbits, but should you care. Mass of sun now:$\approx 2 * 10^{30}$kg so$E \approx 1.8 *10^{47}$J. Radiation per year is about$10^{34}$J. You've got a lot bigger problems that affect your calculation of orbits. Especially like where Jupiter is relative to where you thought it was. Namely the ... 4 For point sources of a field or energy source, such as a charged particle, a gravitational body (which acts like a point source), or a loudspeaker on top of a tall column, the geometry of the problem controls how energy and fields distribute themselves in space. At all points that are an equal distance from a point source, the energy or field strength is ... 3 As the question is finite versus infinite, I guess we don't need the exact result for a finite disc (though it is not difficult to calculate). The simple intuitive answer is that even though the disk mass is infinite, most of the forces from the bits of disk going out to infinity will cancel out due to symmetry, so the answer is finite. Let us assume we ... 3 Half. The escape velocity for an object at a distance$D$from an object of mass$M$is$\sqrt{2GM/D}$. The circular orbital velocity (the Moon is on an orbit that's close enough to circular that I'll just assume this) at the same distance is$\sqrt{GM/D}$. Setting the escape velocity from the Earth with it's new reduced mass$M_{\rm new}$equal to the ... 2 That is the case for central gravity fields. You can derive Newtonian gravitation from a spacetime metric $$ds^2 = c^2d\tau^2 = c^2\left(1 - \frac{2GM}{rc^2}\right)dt^{2} - dr^2 - r^2d\Omega^2$$ where the only metric element different than unity is$g_{tt} = c^2 - GM/r$. The relevant Christoffel symbol is $$\Gamma^r_{tt} = \frac{1}{2}g^{rr}\frac{\partial ... 2 According to Newton's theory of gravity, the orbit would be an ellipse, which could take the form of a perfect circle. However, Einstein's theory of General Relativity tells us that this elliptical orbit would very gradually decay due to the emission of gravitational waves, and perhaps also precess. The Wikipedia page on the two-body problem in general ... 2 The stable orbits around a star are given by the Kepler's laws oft planetary motion. In general these are ellipses with the center star in one of the two foci. Circular orbits are the special case when there is only one focus. For a orbit with a given radius, there is only one speed which allows a circular orbit. If you have a starting condition where the ... 2 From the equations \dot q = p/m and \dot p=V'(q) and the definition Q=D(q) you can derive by differentiation m\dot Q=D'(q)p and \ddot m^2Q=D''(q)pp-mD'(q)V'(q), where m is a diagonal matrix of masses. The second equation produces an ODE for Q if you can express p in terms of \dot Q up to terms in the null space of D''(q) (i.e., ... 2 Gravitational field is a vector field and is determined by negative gradient of the gravitational potential.$$\vec g=-\vec\nabla \phi$$Frome equation above, it is obvious that$|\vec g|=|-\vec\nabla \phi|$(magnitude of$\vec g$is equal to magnitude of$-\vec \nabla \phi$) and we know that$|-\vec\nabla \phi|$is a non-negative quantity. You have made a ... 2 Negative mass and by corollary negative energy have some strange consequences. If you have a set of masses in a region of space with a volume$V$the density of energy is$\rho = \sum_im_ic^2/V$This defines$T^{00} = \rho$component of the stress energy tensor. The Hawking-Penrose energy conditions are that$T^{00} \ge 0$. Violations of this creates various ... 2 The force straining a grain (which may soon be squashed) is equal and opposite to the weight (gravity force) supplied by the grain pile above. That's Newton's third law. If ALL the weight of the pile were held up by one single kernel, F_grain ~= Mass_of_pile * g and it would be easy for the ton of grain to smash the bottom kernel. It doesn't work ... 2 First, notice that this happens on earth too, the tidal effect of the moon is about 2.3 times larger than that of the sun. The fact that both orbits go in opposite directions will not be qualitatively important, only quantitatively in terms of the irregular variation in tidal periods and sizes. If the two moons have the same density and angular size in ... 2 When you jump from a height, you gather momentum. Absorbing this momentum at landing reduces the size of the maximum force, and thus the "pain". Let us assume that the distance over which a person can absorb the momentum of the fall is proportional to their height (proportional to the length of their legs). In that case, the taller person can absorb the ... 2 If we are at the equator on the surface of an atmosphere-free, non-rotating, perfectly spherical planet (no mountains or trees, etc) then the optimal angle should be zero (launch towards the horizon, tangent to the planet's surface). This assumes the initial velocity is the orbital speed at the surface of the planet, which is approximately the escape ... 2 Assuming that the gravity well acts as an attracting mass whose force follows Newton's law of gravitation, your well will suck the object from Earth's surface when the force from the well is larger than the force from the earth. That is,$GM_{earth}m/R_{earth}^2<GM_{well}m/r_{well}^2$, where$r_{well}\$ is the distance between object and well. Earth ...

Only top voted, non community-wiki answers of a minimum length are eligible


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# Fact Triangles Multiplication And Division Worksheets

Fact Triangles Multiplication And Division Worksheets. Web help children work on multiplication and division with these fact family. Web categories division tags fact families for multiplication and division, fact.

Web help children work on multiplication and division with these fact family. Web • before practicing facts, the student must first understand what multiplication and. Web categories division tags fact families for multiplication and division, fact.


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# Partitioning in groups - Combinatorial Proof

Define $G(n;k)$ as the number of possibilities to portion out $n$ distinguishable elements into $k$ groups. For each group we count the number of unique orders. Two orders are different if they produce different cycles. (for example ABC and ACB are different, ABC and BCA are equal).

Prove the following combinatorial: $G(n;k)=(n-1) \cdot G(n-1;k)+G(n-1;k-1)$ and $\sum\limits_{k=0}^n G(n;k) = n!$.

I am struggling with the combinatorial part of the proof. Can somebody help me with starting this proof in words? Thanx in advance!

• Can you restate the question more clearly? Your defintion of $G(n;k)$ has nothing to do with "regulations". – MarkG Feb 14 '15 at 14:02
• Is this more clear? – iJup Feb 14 '15 at 14:29
• No. What does the number of unique orders have to do with $G(n;k)$? A small example might help. – MarkG Feb 14 '15 at 14:33

It seems to me that $G(n;k)$ is the number of permutations $w \in S_n$ that have exactly $k$ cycles. Rather than using letters, I'll use numbers. Then an element of $G(5;3)$ could look like $(1)(5)(243)$ which would be the same as $(1)(5)(432)$. We can thus always put the smallest number at the beginning of each cycle.
In Combinatorics, these numbers $G(n;k)$ are referred to as Stirling Numbers of the First Kind. When viewing these as cycles, it's easier to see that $$\sum\limits_{k = 0}^n G(n;k) = \#\{\text{Permutations of } n \text{ elements with any number of cycles}\} = \# S_n = n!$$
Choose a permutation $w \in G(n-1;k)$. We can insert the symbol $n$ after any of the numbers $1,2,\ldots,n-1$ in the disjoint cycle decomposition of $w$ in $n-1$ ways, yielding a new permutation $w' \in G(n;k)$ where $n$ appears in a cycle of length at least $2$. On the other hand, given a cycle $\sigma \in G(n-1;k-1)$, we can add the number $n$ as a cycle by itself, yielding a new element $\sigma' \in G(n;k)$. We then see that every element of $G(n;k)$ can either be constructed by the above process from either a $w \in G(n-1,k)$ or a $\sigma \in G(n;k)$. Every $w \in G(n-1;k)$ gives us $n-1$ unique elements of $G(n;k)$, and every $\sigma \in G(n-1;k-1)$ gives us $1$ element of $G(n;k)$. Thus, we have the desire relation: $$G(n;k) = (n-1)G(n-1;k) + G(n-1;k-1).$$
If you'd like, we can also prove that $\sum_k G(n;k) = n!$ using this relation. We note that $G(n;0) = 0$ by definition and $G(n;k) = 0$ for $k > n$. Then clearly $\sum G(1,k) = G(1,1) = 1$. We proceed inductively. Suppose that $\sum\limits_{k=0}^{n-1} G(n-1;k) = (n-1)!$. Then \begin{align*} \sum\limits_{k = 1}^n G(n;k) &= \sum\limits_{k = 1}^n(n-1) G(n-1;k) + G(n-1;k-1) \\ &= (n-1)\sum\limits_{k=1}^{n-1} G(n-1;k) + \sum\limits_{k=2}^n G(n-1; k-1) \\ &= (n-1)*(n-1)! + (n-1)! \\ &= n! \end{align*}
• Super! One thing remains, can you give a combinatorial interpretation of the $sum = n!$ proof? – iJup Feb 14 '15 at 15:14
• We can do it via the recurrence relation (as above), or quite simply see that there are $n!$ permutations, and every permutation must appear in the sum exactly once, implying equality between the two numbers. – Marcus M Feb 14 '15 at 15:15


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## Spectrum of large random reversible Markov chains: heavy-tailed weights on the complete graph.(English)Zbl 1245.60008

The main object of study is the asymptotic behaviour of the spectral distribution of certain random matrices. The model under consideration depends on a single probability measure $$U$$ supported on the non-negative real numbers. The authors consider first a Wigner (i.e., with i.i.d. entries) $$N\times N$$ matrix with elements distributed according to $$U$$ and then normalize the rows so that the sum along each row is $$1$$. The resulting random matrix $$K$$ can be also viewed as a (random) Markov kernel of a random walk on the complete graph with $$N$$ vertices.
The authors consider the empirical spectral distribution $$\mu_K$$ of the matrix $$K$$ (i.e., the measure with atoms at eigenvalues of $$K$$) and its limit behavior (perhaps, after some rescaling) as $$N\to\infty$$. If the second moment of $$U$$ exists, then in the limit one uncovers the familiar Wigner semi-circle law, but the situation changes dramatically when $$U$$ has heavy tails. In the paper the case when $$U$$ has a tail of index $$\alpha$$, that is, when $$U([t,+\infty))$$ decays like $$t^{-\alpha}$$ up to some slowly changing function, is studied. The authors prove that when $$\alpha\in(1,2)$$, then $$\mu_K$$ converges to a non-random limit law which turns out to be the same as for Wigner (i.e., not normalized) random matrices with $$U$$-distributed entries. The intuitive reason for this fact is the law of large numbers for the sums of $$U$$-distributed random variables which means that the normalization constant in each row is approximately the same and does not change the shape of the limit distribution. This is no longer true when $$\alpha<1$$ and, indeed, the authors prove that there is a phase transition at $$\alpha=1$$. For $$\alpha\in(0,1)$$, the limit of $$\mu_K$$ still exists but no longer coincides with the one appearing in Wigner matrices.
The relation of the limiting law with Poisson weighted infinite trees and the Poisson-Dirichlet distribution is also explained in the article.

### MSC:

 60B20 Random matrices (probabilistic aspects) 15B52 Random matrices (algebraic aspects) 47A10 Spectrum, resolvent
Full Text:

### References:

 [1] Aldous, D. (1992). Asymptotics in the random assignment problem. Probab. Theory Related Fields 93 507-534. · Zbl 0767.60006 [2] Aldous, D. and Lyons, R. (2007). Processes on unimodular random networks. Electron. J. Probab. 12 1454-1508. · Zbl 1131.60003 [3] Aldous, D. and Steele, J. M. (2004). The objective method: Probabilistic combinatorial optimization and local weak convergence. In Probability on Discrete Structures. Encyclopaedia of Mathematical Sciences 110 1-72. Springer, Berlin. · Zbl 1037.60008 [4] Auffinger, A., Ben Arous, G. and Péché, S. (2009). Poisson convergence for the largest eigenvalues of heavy tailed random matrices. Ann. Inst. H. Poincaré Probab. Statist. 45 589-610. · Zbl 1177.15037 [5] Belinschi, S., Dembo, A. and Guionnet, A. (2009). Spectral measure of heavy tailed band and covariance random matrices. Comm. Math. Phys. 289 1023-1055. · Zbl 1221.15050 [6] Ben Arous, G. and Černý, J. (2008). The arcsine law as a universal aging scheme for trap models. Comm. Pure Appl. Math. 61 289-329. · Zbl 1141.60075 [7] Ben Arous, G. and Guionnet, A. (2008). The spectrum of heavy tailed random matrices. Comm. Math. Phys. 278 715-751. · Zbl 1157.60005 [8] Benjamini, I. and Schramm, O. (2001). Recurrence of distributional limits of finite planar graphs. Electron. J. Probab. 6 13 pp. (electronic). · Zbl 1010.82021 [9] Bertoin, J. (2006). Random Fragmentation and Coagulation Processes. Cambridge Studies in Advanced Mathematics 102 . Cambridge Univ. Press, Cambridge. · Zbl 1107.60002 [10] Bingham, N. H., Goldie, C. M. and Teugels, J. L. (1989). Regular Variation. Encyclopedia of Mathematics and Its Applications 27 . Cambridge Univ. Press, Cambridge. · Zbl 0667.26003 [11] Bordenave, C., Caputo, P. and Chafaï, D. (2010). Spectrum of large random reversible Markov chains: Two examples. ALEA Lat. Am. J. Probab. Math. Stat. 7 41-64. · Zbl 1276.15016 [12] Bordenave, C., Caputo, P. and Chafaï, D. (2010). Spectrum of non-Hermitian heavy tailed random matrices. Preprint. Available at . · Zbl 1235.60008 [13] Bordenave, C., Lelarge, M. and Salez, J. (2011). The rank of diluted random graphs. Ann. Probab. 39 1097-1121. · Zbl 1298.05283 [14] Bouchaud, J. P. (1992). Weak ergodicity breaking and aging in disordered systems. J. Phys. I France 2 1705-1713. [15] Bouchaud, J. P. and Cizeau, P. (1994). Theory of Lévy matrices. Phys. Rev. E 3 1810-1822. [16] Bovier, A. and Faggionato, A. (2005). Spectral characterization of aging: The REM-like trap model. Ann. Appl. Probab. 15 1997-2037. · Zbl 1086.60064 [17] Feller, W. (1971). An Introduction to Probability Theory and Its Applications. Vol. II , 2nd ed. Wiley, New York. · Zbl 0219.60003 [18] Fontes, L. R. G. and Mathieu, P. (2008). K -processes, scaling limit and aging for the trap model in the complete graph. Ann. Probab. 36 1322-1358. · Zbl 1154.60073 [19] Klein, A. (1998). Extended states in the Anderson model on the Bethe lattice. Adv. Math. 133 163-184. · Zbl 0899.60088 [20] LePage, R., Woodroofe, M. and Zinn, J. (1981). Convergence to a stable distribution via order statistics. Ann. Probab. 9 624-632. · Zbl 0465.60031 [21] Perman, M., Pitman, J. and Yor, M. (1992). Size-biased sampling of Poisson point processes and excursions. Probab. Theory Related Fields 92 21-39. · Zbl 0741.60037 [22] Pitman, J. and Yor, M. (1997). The two-parameter Poisson-Dirichlet distribution derived from a stable subordinator. Ann. Probab. 25 855-900. · Zbl 0880.60076 [23] Reed, M. and Simon, B. (1980). Methods of Modern Mathematical Physics. I : Functional Analysis , 2nd ed. Academic Press, New York. · Zbl 0459.46001 [24] Resnick, S. I. (2007). Heavy-tail Phenomena : Probabilistic and Statistical Modeling . Springer, New York. · Zbl 1152.62029 [25] Simon, B. (2005). Trace Ideals and Their Applications , 2nd ed. Mathematical Surveys and Monographs 120 . Amer. Math. Soc., Providence, RI. · Zbl 1074.47001 [26] Soshnikov, A. (2004). Poisson statistics for the largest eigenvalues of Wigner random matrices with heavy tails. Electron. Comm. Probab. 9 82-91 (electronic). · Zbl 1060.60013 [27] Talagrand, M. (2003). Spin Glasses : A Challenge for Mathematicians : Cavity and Mean Field Models. Ergebnisse der Mathematik und Ihrer Grenzgebiete . 3. Folge. A Series of Modern Surveys in Mathematics [ Results in Mathematics and Related Areas . 3 rd Series. A Series of Modern Surveys in Mathematics ] 46 . Springer, Berlin. · Zbl 1033.82002 [28] Zakharevich, I. (2006). A generalization of Wigner’s law. Comm. Math. Phys. 268 403-414. · Zbl 1147.82334 [29] Zhan, X. (2002). Matrix Inequalities. Lecture Notes in Math. 1790 . Springer, Berlin. · Zbl 1018.15016
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# Adaptive Concurrency¶

Adaptive Concurrency Control configuration overview.

## config.filter.http.adaptive_concurrency.v2alpha.GradientControllerConfig¶

[config.filter.http.adaptive_concurrency.v2alpha.GradientControllerConfig proto]

Configuration parameters for the gradient controller.

{
"sample_aggregate_percentile": "{...}",
"concurrency_limit_params": "{...}",
"min_rtt_calc_params": "{...}"
}

sample_aggregate_percentile

(type.Percent) The percentile to use when summarizing aggregated samples. Defaults to p50.

concurrency_limit_params
min_rtt_calc_params

## config.filter.http.adaptive_concurrency.v2alpha.GradientControllerConfig.ConcurrencyLimitCalculationParams¶

[config.filter.http.adaptive_concurrency.v2alpha.GradientControllerConfig.ConcurrencyLimitCalculationParams proto]

Parameters controlling the periodic recalculation of the concurrency limit from sampled request latencies.

{
"max_gradient": "{...}",
"max_concurrency_limit": "{...}",
"concurrency_update_interval": "{...}"
}

max_gradient

(DoubleValue) The maximum value the gradient is allowed to take. This influences how aggressively the concurrency limit can increase. Defaults to 2.0.

max_concurrency_limit

(UInt32Value) The allowed upper-bound on the calculated concurrency limit. Defaults to 1000.

concurrency_update_interval

(Duration) The period of time samples are taken to recalculate the concurrency limit.

## config.filter.http.adaptive_concurrency.v2alpha.GradientControllerConfig.MinimumRTTCalculationParams¶

[config.filter.http.adaptive_concurrency.v2alpha.GradientControllerConfig.MinimumRTTCalculationParams proto]

Parameters controlling the periodic minRTT recalculation.

{
"interval": "{...}",
"request_count": "{...}",
"jitter": "{...}"
}

interval

(Duration) The time interval between recalculating the minimum request round-trip time.

request_count

(UInt32Value) The number of requests to aggregate/sample during the minRTT recalculation window before updating. Defaults to 50.

jitter

(type.Percent) Randomized time delta that will be introduced to the start of the minRTT calculation window. This is represented as a percentage of the interval duration. Defaults to 15%.

Example: If the interval is 10s and the jitter is 15%, the next window will begin somewhere in the range (10s - 11.5s).

## config.filter.http.adaptive_concurrency.v2alpha.AdaptiveConcurrency¶

[config.filter.http.adaptive_concurrency.v2alpha.AdaptiveConcurrency proto]

{
"gradient_controller_config": "{...}",
"enabled": "{...}"
}

gradient_controller_config

(config.filter.http.adaptive_concurrency.v2alpha.GradientControllerConfig, REQUIRED) Gradient concurrency control will be used.

enabled

(core.RuntimeFeatureFlag) If set to false, the adaptive concurrency filter will operate as a pass-through filter. If the message is unspecified, the filter will be enabled.


text:
# An example of a divisible ideal

Let $R$ be a commutative ring (other than a field) with identity. I am looking for an example of a divisible ideal of $R$.

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What sense of divisibility? You mean it's an ideal that is a divisible group? Or it's a divisible as an $R$ module ala Lam's definition in Lectures on Modules and Rings? – rschwieb Feb 13 '13 at 18:01

Pick the ideal $I$ first. Then let $R = \mathbb{Z} \oplus I$, with multiplication $(m,i)(n,j) = (mn, mj+ni+ij)$.

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Not sure what notion of divisibility you're interested in.

If you are interested in it only being a divisible abelian group, then you can take any divisible abelian group $G$ and look at the matrix ring

$$R=\left\{\begin{bmatrix}n&g\\0&n\end{bmatrix}\mid n\in \Bbb Z, g\in G\right\}$$

The group $G$ is embedded as an ideal consisting of the set of strictly upper triangular matrices. (Actually this is a relative construction of Hurkyl's example. This is a subring of a "split-null extension", whereas his is a Dorroh extension.)

If you are interested in the definition Thomas Andrews mentions in the comments, (where you are checking for divisibility over non-zero divisors) then my matrix ring still works, since the zero divisors are exactly the elements with $n=0$. (This version of divisibility coincides with Lam's definition below when the ring is a domain.)

There is yet another good definition in Lam's Lectures on modules and rings:

An $R$ module $M$ is divisible if for any $m\in M$ and $r\in R$ such that $r.ann(r)\subseteq ann(m)$, there exists $n\in M$ such that $m=nr$.

You can also find it proven there that a ring is von Neumann regular iff all right $R$ modules are divisible (in this sense). In particular, all of its ideals would be divisible.

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Is that definition equivalent to the one here: mathworld.wolfram.com/DivisibleModule.html ? It seems related, but Lam's definition allows $r$ to be a zero-divisor as long as $m$ is annihilated by all annihilators of $r$. – Thomas Andrews Feb 13 '13 at 18:12
@ThomasAndrews It isn't the same, and Lam brings that up. He makes a case that this is a good definition in that book. I'm able to view the page in Lectures on Modules and rings on page 70, where he has a footnote addressing the definition you mentioned, so you might enjoy reading that :) – rschwieb Feb 13 '13 at 18:21
I am interested in the definition in Lam's Lectures on modules and rings when the ring is integral domain. Indeed, if $R$ is an integral domain (commutative ring) and $M$ is an $R$-module. Then it is said that $M$ is divisible provided that $RaM=M$ for every $a\in R$. Now, I have an example of a divisible ideal of $R$ as an $R$-module. – user62181 Feb 14 '13 at 5:57
@user62181 Oh, I just saw why it was obvious a nontrivial ideal of a domain can't be divisible. Let $a$ be nonzero in $I$. In particular for the choice of $a=r$, there must be a $b\in I$ such that $ab=a$, but this means that $a(b-1)=0$, and so $b=1$, a contradiction to $I$ being nontrivial. – rschwieb Feb 14 '13 at 16:02
@user62181 And if one hoped that a domain $R$ were divisible as a module over itself, then given any nonzero $a,b$ one could find $c$ such that $a=cb$, and in turn one could find $d$ such that $b=da$. But then $a=cda$ implies $cd=1$, and so every pair of nonzero elements are associates, but that means that $R$ is a field. There just aren't any divisible ideals in nonfield domains. – rschwieb Feb 14 '13 at 18:58


text:
stevendaryl
Staff Emeritus
To illustrate how confused it is to think that GR is needed to use coordinates in which an accelerated observer is at rest, take a look at Rindler coordinates. It's a coordinate transformation from

$(x,t) \Rightarrow (X,T)$

where $T = tanh^{-1}(\frac{ct}{x})$ and $X = \sqrt{x^2 - c^2 t^2}$

In terms of the coordinates $(X,T)$, you find that
1. Clocks at "rest" (that is, $X$ is constant) run faster the higher up they sit (larger values of $X$)
2. Light rays bend downwards (toward negative values of $X$).
3. An object dropped from "rest" will accelerate downward (decreasing $X$)
These facts don't require GR, they are derivable from SR alone.

There is no need for a "principle of equivalence" to allow us to use these coordinates, any more than there is a need for a principle of equivalence to use Newtonian physics in polar coordinates.

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Can you provide a reference for this "GR solution" that agrees with the SR comoving frames as an example?

I've seen it done in lots of places, over the years. And I've never seen its solution NOT agree with the co-moving inertial frames solution. I've never seen it agree with the Dolby&Gull solution. It always says that, anytime the traveler's rocket is off, the traveler concludes that the home twin is aging more slowly, as given by the standard time-dilation result. I.e., the GR solution always says that the traveler's perspective is the same as an inertial observer whenever the rocket is off. And that ALL of the fast aging of the home twin, according to the traveler, occurs only while the rocket is firing. That is different than the D&G solution. If there are multiple "equally good" SR solutions for the traveler's perspective, why aren't there multiple "equally good" GR solutions for the traveler's perspective?

I don't remember all the places I've seen that GR solution given, but I do remember that it is quite common. I think the Wikipedia page on the twin paradox does it (and it may give some specific references). And I think that the link given by JessM to John Baez's webpage on the twin paradox gives it (perhaps also with references). I think I might have also seen it in MTW's "Gravitation" book, and in Born's "Einstein's Theory of Relativity" book.

Dale
Mentor
2020 Award
I've seen it done in lots of places, over the years.
I need an example so that I can understand what you are referring to. I simply don't know what you mean by a "GR solution". If it has been done in lots of places then I am sure that I have read one, but I don't recognize it as what you are calling "a GR solution". Every solution that I am aware of is what I would call "an SR solution" since the spacetime is flat.

I'm not saying that you are wrong, I am just saying that I don't know what you are referring to. I suspect that we are talking about the same things and just using different words. Also, are you talking about a quantitative solution, or simply a hand-waving qualitative explanation?

A.T.
These facts don't require GR,
Why is it so important whether some explanation is called a "SR-explanation" or a "GR-explanation"? Isn't SR a special case of GR, and therefore any SR-explanation also a GR-explanation?

As I said, GR in the limit of vanishing spacetime curvature simply IS SR.
Doesn’t this make the equivalence principle part of SR, since it only applies when spacetime curvature is negligible?

stevendaryl
Staff Emeritus
Why is it so important whether some explanation is called a "SR-explanation" or a "GR-explanation"? Isn't SR a special case of GR, and therefore any SR-explanation also a GR-explanation?

Sure. Every SR explanation is also a GR explanation. But not the other way around. So some problems require GR, and others don't. My point is that no problem involving flat spacetime requires GR.

Doesn’t this make the equivalence principle part of SR, since it only applies when spacetime curvature is negligible?

If spacetime is perfectly flat, then you don't need the equivalence principle, because everything can be done using SR alone. What the equivalence principle allows is to solve problems in curved spacetime by breaking spacetime into small regions and then approximating those regions by flat sections of spacetime.

To me, the equivalence principle is really the same, as far as physical content, to the two claims below:
1. Spacetime is curved.
2. "Gravitational forces" are actually connection coefficients due to the choice of a noninertial coordinate system. (If spacetime is curved, then there are no inertial coordinate systems except in the limit of small regions of spacetime).
It's a guess about the nature of gravity. You don't need the equivalence principle to get "inertial forces" (connection coefficients) for a noninertial coordinate system; that's derivable from the equations of motion in inertial coordinates.

harrylin said: "As explained in the summary which was posted twice here: the challenge that was thrown at his feet, was how to produce the same prediction about the twins as in SR, while using coordinates in which the "traveler" is claimed to be in rest all the time. That should be possible according to Einstein's GR postulate, as he there also acknowledged. [..] Following your inclination, I'm unable to discern Maxwell's theory of electrodynamics from Einstein's."

My point is that both the challenge and the response are based on the misconception that GR is any more (or less) capable of using coordinates in which the traveling twin is at rest than SR.
[..]
Maxwell's theory was already invariant under Lorentz transformations. Einstein's contribution was to develop an analogous theory of mechanics. Before Einstein, we had Maxwell's equations, which were invariant in form under Lorentz transformations, and Newton's laws of motion, which were invariant in form under Galilean transformations. Einstein united the theories by modifying Newton's theory to get one that was invariant under Lorentz transformations, as well. He didn't need to modify Maxwell's equations.
[..]
If you know the equations of motion in one coordinate system, then you know the equations of motion in every coordinate system. You don't need an additional postulate that they are invariant under such and such a transformation, it's just a fact of the equations.[..]
Maxwell's theory assumed the use of Newton's transformations. I think that this is a good example to explain the difference between a theory and the equations of a theory, but apparently it is necessary to elaborate:

- the equations are the same
- the same coordinate transformations are mathematically possible with both theories

Nevertheless they were not the same theory because their physical assumptions differed on an essential point.
For MMX they even made contrary predictions!

Thus I clarified:
" the difference is not in Maxwell's equations; the difference is primarily in the postulates about the coordinate systems in which the equations are claimed to be valid."
"harrylin said: The equivalence principle of 1916 says that acceleration can be modeled as gravity"

That's a muddled way of thinking about it. [..]
[..] What you need the equivalence principle for is to predict that clocks and light beams work in a similar way on the surface of a massive planet. That's the physical content of the equivalence principle, that light curves near a massive planet, and that clocks run at different speeds at different altitudes. [..]
I certainly agree with that; however my dislike for GR "vintage 1916" (as the FAQ calls it) doesn't bring me to deny the facts about it. It appears that you even deny the meaning of the term "GR"! The new theory of gravitation was a fantastic spin-off of Einstein's theory of the general relativity of motion. He put it as follows:

"From these discussions we see, that the working out of the general relativity theory must, at the same time, lead to a theory of gravitation; for we can "create" a gravitational field by a simple variation of the co-ordinate system." (emphasis mine)

However, in the light of this discussion it strikes me that his phrasing here lacks precision, so that it is bound to be misunderstood if one does not already understand what he means. No doubt, if the issue had been brought up, he would have agreed that a mere coordinate transformation will not create a gravitational field. The creation of a gravitational field only occurs in his theory when one reinterprets the accelerating system as a non-accelerating system - one that is "in rest".

A similar situation occurs in SR when one Lorentz transforms from a "stationary" system S to a "moving" system S'. The transformation itself does not yet impose on an observer who is co-moving with S' to pretend not to be moving but to be in rest; however, the observer may do that if he/she so desires. This adaptation of interpretation to the used coordinate system is implied in most discussions.

There is no additional physics involved in interpreting "inertial forces" as gravity. It's just what you name terms in the equations of motion.
A sudden inertial force on a test particle doesn't create a gravitational wave, while - I think - a suddenly "induced gravitational field" certainly must do so. That is the central point of my criticism of Einstein's 1918 paper (which I did not yet present), and with that, of 1916 GR.

I consider that paragraph to be misleading, if it encourages people to think that you need GR to be able to reason as if the accelerated observer is at rest. That is purely SR + a coordinate transformation, and GR adds nothing. Well, what it adds is that you may need the inspiration of GR to get you thinking about the use of noninertial coordinate systems.
That does not merely encourage people to think that, it's what SR as well as 1916GR assert. The claim that the accelerated twin is not changing velocity so that the traveler can be considered to be all the time in rest, leads in SR to the twin paradox! We and others discussed this earlier in this same thread and you even brought it up in your reply. The following is according to SR not true in an SR "rest system":
"light beams appear to curve downward, and [..] clocks run at different speeds in different locations of the rocket."
To illustrate how confused it is to think that GR is needed to use coordinates in which an accelerated observer is at rest, take a look at Rindler coordinates. [..]
Once more: Einstein did not pretend to need GR in order to use coordinates in which an accelerated observer is at rest - quite the contrary, he used such SR coordinates as input for GR!
Perhaps the phrasing "at rest in an accelerated coordinate system" causes confusion. While one certainly can sit down and "rest" in an accelerating rocket, that does not change the physical interpretation of an accelerating system. It may be better to speak of "co-moving" with an accelerated coordinate system.

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stevendaryl
Staff Emeritus
Nevertheless they were not the same theory because their physical assumptions differed on an essential point.
For MMX they even made contrary predictions!

This is sort of a complicated subject, but the way I understand it is this:
• Maxwell's equations are invariant under Lorentz Transformations.
• Newton's equations are invariant under Galilean Transformations.
• Therefore, there is only one frame in which they both take on their usual forms.
What that means is that either Newton's equations or Maxwell's equations have to be modified if you change frames. I think that everyone assumed that it was Maxwell's equations that had to be modified; that they were only valid as is in the frame in which the "ether" was at rest.

But is it really true that the prediction of a nonnull result from the Michaelson-Moreley experiment was a prediction from the theory of electromagnetism? I don't think it was. I think it was a prediction from Newtonian physics. Newtonian physics says that if in one frame, you have an object that travels at speed $c$, then in another frame moving at speed $v$ relative to the first, that object travels at some speed between $c-v$ and $c+v$. The prediction of a nonnull result follows from Newtonian mechanics, not Maxwell's equations.

In any case, could you give a reaction to the following argument:

Assume that the usual equations of Special Relativity holds when written in terms of inertial coordinates $(x,y,t)$. Introduce new coordinates $(X,Y,T)$ related to the first two via:
• $X = \sqrt{x^2 - c^2 t^2}$
• $T = tanh^{-1}(\frac{ct}{x})$
• $x = X cosh(T)$
• $t = \frac{X}{c} sinh(T)$
• $Y = y$
Then when described in terms of these new coordinates,
• When an object is dropped from "rest", it accelerates downwards (towards smaller values of $X$)
• Clocks at "rest" that are "higher up" (at larger values of $X$) run faster.
• Light signals that are initially emitted in the $Y$ direction bend down (towards smaller values of $X$)
So the "general relativistic" effects of "gravitational time dilation" and "bending of light rays" and "objects falling under gravitational fields" seem to me to be purely effects of SR in noninertial coordinates. Not only do you not need the principle of equivalence to treat an accelerated observer as if he were at rest, the principle of equivalence plays no role, whatsoever. (Except maybe it allows you to use the word "gravity" when describing the above effects.)

So my questions are: (1) In what sense is the above discussion of Rindler coordinates not an "SR" argument? (2) How would the treatment be any different in a "GR" derivation?

[..] is it really true that the prediction of a nonnull result from the Michaelson-Moreley experiment was a prediction from the theory of electromagnetism? I don't think it was. I think it was a prediction from Newtonian physics. [..]The prediction of a nonnull result follows from Newtonian mechanics, not Maxwell's equations.
I referred to Maxwell's theory, in contrast with Maxwell's equations (in case you did not know it, Maxwell proposed MMX). And that was exactly my point: emphasizing the important difference between a theory of physics, and equations that are used as part of the theory.
Equations without the theory are like a piece of complicated equipment without the manual. :)

In any case, could you give a reaction to the following argument:
I will later, if it appears to still be useful (although I already provided a link to my reply, before you asked!). For could you please first react to :
- the remainder of my last post? (you apparently stopped reading at 1/3 of my post, while the next 2/3 gave a more elaborate explanation of the same)
- post #84? (it may well be that Phoebelasa explained it in a way that is clearer than the way I explained it; but apparently you overlooked it)

stevendaryl
Staff Emeritus
I referred to Maxwell's theory, in contrast with Maxwell's equations (in case you did not know it, Maxwell proposed MMX). And that was exactly my point: emphasizing the important difference between a theory of physics, and equations that are used as part of the theory.

Right, but as I said, the prediction of a nonnull result has almost nothing to do with Maxwell's theory. It follows from the fact that:
1. Light has speed c in at least one frame.
2. According to Newton's laws, velocity transforms as $c \Rightarrow c\pm v$
So a nonnull result for MMX was mostly a prediction about Newtonian physics, it seems to me.

Right, but as I said, the prediction of a nonnull result has almost nothing to do with Maxwell's theory. It follows from the fact that:
1. Light has speed c in at least one frame.
2. According to Newton's laws, velocity transforms as $c \Rightarrow c\pm v$
So a nonnull result for MMX was mostly a prediction about Newtonian physics, it seems to me.
Newton's theory would have predicted a null result, assuming his corpuscular light hypothesis.
Maxwell's theory predicted a positive result, because he assumed the validity of the Galilean transformations.

stevendaryl
Staff Emeritus
Here's another take on describing what was going on in the MMX:

If you know the equations of physics in one coordinate system $x^\alpha$ and you know how to transform from that coordinate system to a second $x'^\mu$, then you know the equations of physics in the second coordinate system. There is no additional physical assumption required to be able to use the new coordinate system. There is no empirical test as to whether the new coordinate system is "valid" or not.

But here's where physical assumptions come into play: Suppose you have two frames, $F$ and $F'$. You set up corresponding coordinate systems $x^\alpha$ and $x'^\mu$ using some physical convention for measuring distances and times. So you use standard clocks and metersticks at rest in $F$ to define the coordinate system $x^\alpha$, and you use standard clocks and metersticks at rest in $F'$ to define the coordinate system $x'^\mu$. Then you don't know what is the mathematical relationship between the primed and unprimed coordinate systems, without physical assumptions about the nature of clocks and metersticks. So you actually don't know what the equations look like in the new coordinate system until you perform empirical tests.

stevendaryl
Staff Emeritus
Newton's theory would have predicted a null result, assuming his corpuscular light hypothesis.
Maxwell's theory predicted a positive result, because he assumed the validity of the Galilean transformations.

Nobody is talking about Newton's theory of light--I'm talking about his laws of motion.

Anyway, I have to vigorously protest the phrase "the validity of the Galilean transformations". That is a meaningless phrase without some additional stipulations. There is no such thing as a valid or invalid coordinate transformation. You can use whatever coordinates are convenient; any are as "valid" as any other.

The real issue is the specific conventions for setting up coordinates in a frame. If you use physical objects, such as clocks, rods, light signals, or whatever, to measure distance and times, and you use those distances and times as the basis for a coordinate system, then it's an empirical question how such a coordinate system depends on the frame of rest of those clocks and rods. So it's not a matter of the Galilean transformations being "valid", but a matter of whether they correctly describe the relationship between two operationally defined coordinate systems.

PAllen
If you know the equations of physics in one coordinate system $x^\alpha$ and you know how to transform from that coordinate system to a second $x'^\mu$, then you know the equations of physics in the second coordinate system. There is no additional physical assumption required to be able to use the new coordinate system. There is no empirical test as to whether the new coordinate system is "valid" or not.
.
Don't you also have to assume transformation laws for the objects of the equations of physics? Or are you bundling that into what you mean by 'equations of physics'?

[..] I have to vigorously protest the phrase "the validity of the Galilean transformations". That is a meaningless phrase without some additional stipulations. [..].
The stipulations are the ones that the reader is supposed to know; else we can't even write "SR" in a discussion, because it is meaningless without a clarification of what it is an abbreviation.

[..] Einstein did not even consider the twin paradox as problematic at all [..] ?
I will now expand on my earlier comments.

Although the twin example was not an issue for him in connection with what he named the "special" theory of relativity, it became an issue for him in connection with his "general" theory. He discussed that in his 1918 paper https://en.wikisource.org/wiki/Dialog_about_Objections_against_the_Theory_of_Relativity , and PhoebeLasa gave a summary of that section as follows:
I think there was a specific reason Einstein used GR to resolve the twin paradox. He wanted to construct an analogous scenario (via the equivalence principle) in which the "rocket-twin" could say that he was absolutely stationary and unaccelerated during the whole time that the twins were separated. When he fired his rocket engine, he was doing it strictly to counteract the spatially-uniform gravitational field that is somehow momentarily switched on, so that the rocket-twin would remain stationary and unaccelerated. That momentarily switched-on gravitational field causes the "home twin" (the twin who has no rocket) to accelerate, reverse course, and move toward the "traveler". The resulting conclusion using this GR scenario is that the rocket-twin will say that the "home-twin" suddenly gets much older while that gravitational field is switched on.

The exact same result (regarding the rocket-twin's conclusion about the home-twin suddenly getting much older during the turnaround) is obtained without recourse to GR (and without any gravitational fields), purely from SR, using a non-inertial reference frame for the rocket-twin which is formed by piecing together multiple inertial frames that are each momentarily co-moving with the rocket-twin at different instants of his life. The rocket-twin is always at the spatial origin of his non-inertial reference frame, but he never contends that he doesn't accelerate. He knows that he accelerates, and reverses course, when he turns on his rocket. And he knows that it is the home-twin who is unaccelerated for the whole trip.

There is a difference between being "always absolutely at rest" (Einstein's GR scenario for the rocket-twin) versus "being always at the spatial origin of your own personal reference frame, but accelerating at will using your rocket engine" (the SR scenario). But what the rocket-twin says about the home-twin suddenly getting much older during the turnaround is exactly the same for both scenarios (even though it's a different twin doing the turnaround in the two cases).

The only glitch that I notice in the otherwise excellent summary above, is the following:
Einstein did not really "want to construct" a scenario in which the "rocket-twin" could say that he was absolutely stationary and unaccelerated during the whole time; instead it was asserted by critics that this must be possible according to Einstein's theory.
Also - and this is essential - he did not "use GR to resolve the twin paradox".
Instead he presented the twin scenario as one of the "Objections against the Theory of Relativity", that is, vintage 1916 GR. And this was his playful defence against accusations that 1916 GR is self-contradictory.

That criticism targeted the General postulate of relativity, according to which "The laws of physics must be so constituted that they should remain valid for any system of co-ordinates moving in any manner." - https://en.wikisource.org/wiki/The_...ain_the_extension_of_the_relativity-postulate.

Einstein accepted the challenge with the comment that the critic's 'assertion is of course indisputable'. And near the end of his defence, Einstein states that the theory 'means for a man who maintains consistency of thought a great satisfaction to see that the concept of absolute motion, to which kinematically no meaning can be attributed, does not have to enter physics'. No precise references are given at al, but almost certainly the objection was triggered by Langevin's 1911 paper which gives the first full "twin" example (using SR) to argue that a change of velocity is qualitatively "absolute". And this is just what GR was meant to make "relative".

In an earlier post I imprecisely (sorry!) stated that Moller and Builder criticized Einstein's 1918 paper. Moller's criticism is positive and in his 1952 textbook "The theory of relativity" (which I now again have at hand) he provides the calculation details that are missing in Einstein's paper. Professor Moller thus taught his students GR "vintage 1916". He surely understood* the issue, as he there explains the general principle of relativity and refers to the succession of key papers such as Einstein 1905, Langevin 1911, Einstein 1918.

Notwithstanding that great defence, I think that Builder's 1957 objection turns it into wood wreck - although he apparently never saw Einstein's paper and misunderstood the reason for the calculation; apparently he based his argument on his readings of Tolman and Moller. Historically, the "twin paradox" discussion is a continuous succession of misunderstandings.

Thus, in his paper "The resolution of the clock paradox" (G. Builder, Aust. J. Phys. 10, 246–262, 1957), Builder argues that GR can add nothing to the solution that SR already provides. And he argues that 'any application of the principle of equivalence [..] to such cases would be quite trivial", simply because the calculated fields predict effects from acceleration that were used to calculate those fields in the first place. But he next adds a different objection, in disagreement with his earlier triviality argument(!), and this one I consider pertinent:

The [accelerated] reference system Sm does not correspond to any physical system that is realizable even in principle. This conclusion is not affected by the introduction of the concept of the equivalent gravitational ﬁeld. On the contrary, nothing could demonstrate more clearly the artificiality of the reference system Sm, than the statement that its physical equivalent is a gravitational ﬁeld which is everywhere zero until the instant tm=T', has the potential gxm(1+gxm/2c2) from tm=T' to tm=T’+tau’2, and becomes zero everywhere again at tm=T’ +tau'2.
The concept of such a ﬁeld is completely incompatible with the limiting value c for all velocities measured in inertial reference systems; [..] so that the speciﬁed ﬁeld would have to be created simultaneously at all points in S' and be destroyed simultaneously at all points in So.
Thus the principle of equivalence [..] only accentuates the artiﬁciality of the description of our hypothetical experiment in terms of the coordinates of the accelerated reference system Sm.

Indeed, according to GR any "induced gravitational field" must propagate at the speed of light. On top of that, what Builder overlooked or didn't bother to mention: an infinite speed of induction is also not allowed in a gravitational field according to GR, and that still does not suffice to match SR's predicted Doppler effect, as also the speed of starlight is finite - all the stars Doppler-shift instantly at turnaround.

As far as I know, none of the involved authors (Einstein, Tolman, Moller, ...) addressed that self-contradiction.

*Note: Moller makes the statement that SR "only allows treatment of the physical phenomena in frames of reference in uniform motion". Perhaps he means that SR's laws of physics are only valid in those reference frames, but it is easy to see how it can bring unaware readers to the misunderstanding that SR cannot describe observations from accelerating rockets!

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stevendaryl
Staff Emeritus
The stipulations are the ones that the reader is supposed to know

I would say that this discussion is all about what it means for a coordinate system to be "valid", and that a lot of misconceptions about SR result from not being clear about what that means.

stevendaryl
Staff Emeritus
Don't you also have to assume transformation laws for the objects of the equations of physics? Or are you bundling that into what you mean by 'equations of physics'?

I realize that I'm on a little shaky grounds here. I'm not positive that what I said was wrong, but I'm not positive that what I said was correct, either.

The question, which I don't know the answer to, is: What goes wrong if you make the incorrect assumption about the transformation properties of scalars, vectors, tensor, etc.?

It might be worth while to work out an example.

Suppose you have an equation that is correct in one coordinate system, for example, the geodesic equation:

$m \frac{d^2 x^\mu}{ds^2} = F^\mu$

Let me introduce $U^\mu = \frac{dx^\mu}{ds}$. Then it's

$m \frac{d U^\mu}{ds} = F^\mu$

The difficulty is that although I've written $F^\mu$ as a vector, it may actually be a combination of a vector force together with connection coefficients.

So, change coordinates to $x'^\alpha$ and define $L^\mu_\alpha = \partial_\alpha x^\mu$. Then

$U^\mu = L^\mu_\alpha U^\alpha$ and we get an equation for $U^\alpha$:

$m (L^\mu_\alpha \frac{dU^\alpha}{ds} + (\partial_\beta L^\mu_\alpha) U^\alpha U^\beta) = F^\mu$

Multiplying by the inverse transformation matrix gives:
$m (\frac{dU^\alpha}{ds} + (L^{-1})^\alpha_\mu (\partial_\beta L^\mu_\gamma) U^\gamma U^\beta) = (L^{-1})^\alpha_\mu F^\mu$

So, we can define the "effective 4-force" in the new coordinates to be:

$F'^\alpha = (L^{-1})^\alpha_\mu F^\mu - m((L^{-1})^\alpha_\mu (\partial_\beta L^\mu_\gamma) U^\gamma U^\beta)$

So in the new coordinate system, we have the equation of motion:

$m \frac{dU^\alpha}{ds} = F'^\alpha$

So now I'm not sure--where is it necessary to know which things are vectors, tensors or scalars? I think the problem comes in with the physical meaning of $F'^\alpha$. Even if $F^\mu$ had some simple interpretation, in terms of scalar or vector fields, the new "force" $F'^\mu$ would have a very complicated definition. But the equations of motion would still work, wouldn't they, even though you've misidentified a connection coefficient term as a 4-vector?

PAllen
I realize that I'm on a little shaky grounds here. I'm not positive that what I said was wrong, but I'm not positive that what I said was correct, either.

The question, which I don't know the answer to, is: What goes wrong if you make the incorrect assumption about the transformation properties of scalars, vectors, tensor, etc.?

It might be worth while to work out an example.

Suppose you have an equation that is correct in one coordinate system, for example, the geodesic equation:

$m \frac{d^2 x^\mu}{ds^2} = F^\mu$

Let me introduce $U^\mu = \frac{dx^\mu}{ds}$. Then it's

$m \frac{d U^\mu}{ds} = F^\mu$

The difficulty is that although I've written $F^\mu$ as a vector, it may actually be a combination of a vector force together with connection coefficients.

So, change coordinates to $x'^\alpha$ and define $L^\mu_\alpha = \partial_\alpha x^\mu$. Then

$U^\mu = L^\mu_\alpha U^\alpha$ and we get an equation for $U^\alpha$:

$m (L^\mu_\alpha \frac{dU^\alpha}{ds} + (\partial_\beta L^\mu_\alpha) U^\alpha U^\beta) = F^\mu$

Multiplying by the inverse transformation matrix gives:
$m (\frac{dU^\alpha}{ds} + (L^{-1})^\alpha_\mu (\partial_\beta L^\mu_\gamma) U^\gamma U^\beta) = (L^{-1})^\alpha_\mu F^\mu$

So, we can define the "effective 4-force" in the new coordinates to be:

$F'^\alpha = (L^{-1})^\alpha_\mu F^\mu - m((L^{-1})^\alpha_\mu (\partial_\beta L^\mu_\gamma) U^\gamma U^\beta)$

So in the new coordinate system, we have the equation of motion:

$m \frac{dU^\alpha}{ds} = F'^\alpha$

So now I'm not sure--where is it necessary to know which things are vectors, tensors or scalars? I think the problem comes in with the physical meaning of $F'^\alpha$. Even if $F^\mu$ had some simple interpretation, in terms of scalar or vector fields, the new "force" $F'^\mu$ would have a very complicated definition. But the equations of motion would still work, wouldn't they, even though you've misidentified a connection coefficient term as a 4-vector?

What if you misinterpret your initial equation as a collection of 4 scalar equations and transform it? Then you get the same force at any point as you did in the original coordinates. I guess you could still try to argue that this 'works' with a force definition that says all forces must be measured by instruments at rest in in the starting coordinates. Do you really want to argue that?

stevendaryl
Staff Emeritus
What if you misinterpret your initial equation as a collection of 4 scalar equations and transform it? Then you get the same force at any point as you did in the original coordinates. I guess you could still try to argue that this 'works' with a force definition that says all forces must be measured by instruments at rest in in the starting coordinates. Do you really want to argue that?

That's sort of what I was getting at: You can do a coordinate change without understanding the nature of the terms involved to get equations in the new coordinate system, but it's the physical interpretation of terms in the new coordinate system that are obscure (or extremely convoluted).

stevendaryl
Staff Emeritus
What if you misinterpret your initial equation as a collection of 4 scalar equations and transform it? Then you get the same force at any point as you did in the original coordinates. I guess you could still try to argue that this 'works' with a force definition that says all forces must be measured by instruments at rest in in the starting coordinates. Do you really want to argue that?

That particular mixup doesn't make sense to me. I'm assuming that what IS known is the coordinates in the two systems. If you don't know that $x^\mu$ is a coordinate, then I don't know what it would mean to say that you know what the coordinate transformation is.

PAllen
That particular mixup doesn't make sense to me. I'm assuming that what IS known is the coordinates in the two systems. If you don't know that $x^\mu$ is a coordinate, then I don't know what it would mean to say that you know what the coordinate transformation is.
Sure it does. Consider Fx is a function of all 4 coordinates, so is Fy, so is Fz. Now, you just assume you substitute that new coordinate definitions, treating Fx, Fy, Fz as scalar functions. If you consider them as vector something else; as a covector, something else; as a density, something else.

stevendaryl
Staff Emeritus
Sure it does. Consider Fx is a function of all 4 coordinates, so is Fy, so is Fz. Now, you just assume you substitute that new coordinate definitions, treating Fx, Fy, Fz as scalar functions. If you consider them as vector something else; as a covector, something else; as a density, something else.

My derivation didn't make any assumptions about the nature of $F^\mu$.

Once again, if $U^\mu$ satisfies

$m \frac{dU^\mu}{ds} = F^\mu$

and

$U^\mu = L^\mu_\alpha U^\alpha$

then

$m (L^\mu_\alpha \frac{dU^\alpha}{ds} + (\partial_\beta L^\mu_\alpha) U^\alpha U^\beta) = F^\mu$

is an equation of motion for $U^\alpha$. You don't have to know anything about how $F^\mu$ transforms. Whether it's 4 scalars, or a 4-vector, or what not doesn't seem to matter.

PAllen
My derivation didn't make any assumptions about the nature of $F^\mu$.

Once again, if $U^\mu$ satisfies

$m \frac{dU^\mu}{ds} = F^\mu$

and

$U^\mu = L^\mu_\alpha U^\alpha$

then

$m (L^\mu_\alpha \frac{dU^\alpha}{ds} + (\partial_\beta L^\mu_\alpha) U^\alpha U^\beta) = F^\mu$

is an equation of motion for $U^\alpha$. You don't have to know anything about how $F^\mu$ transforms. Whether it's 4 scalars, or a 4-vector, or what not doesn't seem to matter.
But I'm considering F as a field, no expression in terms of U. For example, something like Maxwell's equations. You have these functions of coordinates. To transform the equation to other coordinates, at all, you have to make some assumptions.

I would say that this discussion is all about what it means for a coordinate system to be "valid", and that a lot of misconceptions about SR result from not being clear about what that means.
I thought that it was sufficiently clarified in this thread. And I don't recall ever having seen misconceptions about SR because people did not understand what it means for a coordinate system to be "valid" for the laws of physics.

In this context, it's basically the consequence of applying the principle of relativity (or of relative motion) as formulated by Einstein and earlier by Poincare.
- https://en.wikisource.org/wiki/Science_and_Hypothesis/Chapter_7
For good understanding: it was also expressed as the "impossibility to detect absolute motion", because the same laws of physics are observed in systems that are moving relative to each other.

According to the theory of Special Relativity, this principle is true for the inertial reference systems of classical mechanics, and SR's laws of physics are expressed with respect to a system of that class; similarly, its transformation equations are specified for systems of that class. Consequently the use of a reference system K' in arbitrary motion is "at your own risk". The twin "paradox" in SR illustrates nicely that non-inertial reference systems are not valid systems for application of the Lorentz transformations with SR.

However, according to 1916 GR, reference systems in any state of motion must be valid in that sense, with the physics of that theory - as Einstein defended in 1918:
"It is certainly correct that from the point of view of the general theory of relativity we can just as well use coordinate system K' as coordinate system K."
I discussed that issue in post #140 in this thread.

PS I came across a strange remark in Moller's textbook: he states that according to the special theory of relativity 'the special principle of relativity is valid for all physical laws', and adds the footnote: 'With the exception of the laws of gravitation [..]'.
I do think that gravitation works the same according to SR independent of the month of the year - the earth's gravitational field does not act differently in March as in September!

Last edited:
stevendaryl
Staff Emeritus


text:
# How to calculate Average Variance Extracted and Composite Reliability

2. Select the number of available indicators (see figure below). For example, we have four items or indicators measuring perceived quality of information in Wikipedia (Qu1, Qu2, Qu3 and Qu5), so we selected 4 indicators as shown below.


text:
First python project : HP41 emulator + hints of workflow
06-11-2021, 12:28 AM
Post: #41
 rprosperi Super Moderator Posts: 5,197 Joined: Dec 2013
RE: First python project : HP41 emulator + hints of workflow
(06-11-2021 12:12 AM)Steve Simpkin Wrote:  I would recommend the original HP-41C Owner's Handbook and Programming Guide. It assumes you know nothing about HP calculators and is a great example of the friendly, well written calculator manuals that HP used to write.

https://literature.hpcalc.org/community/...hpg-en.pdf

I endorse Steve's recommendation! The later CX manuals cover more of the functions, but the original 41C manual is in many ways the peak of the great old-school HP manuals, even including some of the semi-goofy little cartoons. Starting with the 41CX manual, they took on an ever-increasing tendency towards reference books, simply explaining how the functions work (parameters, stack use, results) but not explaining how to use them.

--Bob Prosperi
06-12-2021, 10:36 PM
Post: #42
 Liamtoh Resu Member Posts: 109 Joined: May 2021
RE: First python project : HP41 emulator + hints of workflow
New to the hp41c review and the pyhp41e emulator.

I was able to figure out the fast/slow switch, the roll down, xyzt rpn, rpn stack
and the trig/log/square/square root functions. ( I used the monkey typing on a
keyboard method without using any docs) I also figured out the virtual paper
tape printer.

Programming it will have to wait for now.

The emulator is a great rpn trainer for those who do not have any of the
classic hp calculators.

Thanks
06-13-2021, 03:29 AM (This post was last modified: 06-13-2021 01:22 PM by gfbci.)
Post: #43
 gfbci Junior Member Posts: 6 Joined: Mar 2016
RE: First python project : HP41 emulator + hints of workflow
First I would like to thank Olivier and this group for their generously shared knowledge.

Py41.e can work on G1 by editing lines 74-75 values of the py41.py file with something like:

...
aof=100 if k==-1 else 1
if aof>=500:
...

This apparently prevents the simulator from going into sleep mode when a key is released.
Don't trust my explanation, but it works!

Gianfranco

edit: Please disregard the values, what matters is to stop the increment of aof
06-13-2021, 02:00 PM
Post: #44
 Stevetuc Member Posts: 273 Joined: Jan 2014
RE: First python project : HP41 emulator + hints of workflow
(06-13-2021 03:29 AM)gfbci Wrote:  First I would like to thank Olivier and this group for their generously shared knowledge.

Py41.e can work on G1 by editing lines 74-75 values of the py41.py file with something like:

...
aof=100 if k==-1 else 1
if aof>=500:
...

This apparently prevents the simulator from going into sleep mode when a key is released.
Don't trust my explanation, but it works!

Gianfranco

edit: Please disregard the values, what matters is to stop the increment of aof

That's not working for me...
Can I check exactly what you have:
py41.py lines 74-75
And
nut.py line 893 (since Olivier advised changing delay here for G1)
Thanks
06-13-2021, 02:51 PM
Post: #45
 gfbci Junior Member Posts: 6 Joined: Mar 2016
RE: First python project : HP41 emulator + hints of workflow
(06-13-2021 02:00 PM)Stevetuc Wrote:
(06-13-2021 03:29 AM)gfbci Wrote:  First I would like to thank Olivier and this group for their generously shared knowledge.

Py41.e can work on G1 by editing lines 74-75 values of the py41.py file with something like:

...
aof=100 if k==-1 else 1
if aof>=500:
...

This apparently prevents the simulator from going into sleep mode when a key is released.
Don't trust my explanation, but it works!

Gianfranco

edit: Please disregard the values, what matters is to stop the increment of aof

That's not working for me...
Can I check exactly what you have:
py41.py lines 74-75
And
nut.py line 893 (since Olivier advised changing delay here for G1)
Thanks

Forgive me for the confusion, maybe I wasn't clear in editing the first post. No values need to be changed, just edit:

(74) aof+=300 if k==-1 else 1
to:
(74) aof=300 if k==-1 else 1

The file nut.py has not been touched.
Unfortunately I don't have a Windows machine at the moment to download the changed file, I would gladly send it.

Gianfranco
06-13-2021, 02:58 PM
Post: #46
 Stevetuc Member Posts: 273 Joined: Jan 2014
RE: First python project : HP41 emulator + hints of workflow
(06-13-2021 02:51 PM)gfbci Wrote:
(06-13-2021 02:00 PM)Stevetuc Wrote:  That's not working for me...
Can I check exactly what you have:
py41.py lines 74-75
And
nut.py line 893 (since Olivier advised changing delay here for G1)
Thanks

Forgive me for the confusion, maybe I wasn't clear in editing the first post. No values need to be changed, just edit:

(74) aof+=300 if k==-1 else 1
to:
(74) aof=300 if k==-1 else 1

The file nut.py has not been touched.
Unfortunately I don't have a Windows machine at the moment to download the changed file, I would gladly send it.

Gianfranco
Thanks Gianfranco you were clear, but I wanted to double check as unfortunately I tried this and it doesn't work on my G1.. not sure why
06-13-2021, 08:41 PM
Post: #47
 gfbci Junior Member Posts: 6 Joined: Mar 2016
RE: First python project : HP41 emulator + hints of workflow
(06-13-2021 02:58 PM)Stevetuc Wrote:
(06-13-2021 02:51 PM)gfbci Wrote:  Forgive me for the confusion, maybe I wasn't clear in editing the first post. No values need to be changed, just edit:

(74) aof+=300 if k==-1 else 1
to:
(74) aof=300 if k==-1 else 1

The file nut.py has not been touched.
Unfortunately I don't have a Windows machine at the moment to download the changed file, I would gladly send it.

Gianfranco
Thanks Gianfranco you were clear, but I wanted to double check as unfortunately I tried this and it doesn't work on my G1.. not sure why

The only action I had to take besides the mentioned change was the removal of all downloaded or copied apps, the reset of the native apps and the calculator itself. This was necessary after the installation of fimware 2021 05 05 to normalize its behavior.
Hardware version is "C".
I redid all the indicated procedure starting from scratch: it works!
I hope it helps.

Gianfranco
06-13-2021, 09:06 PM
Post: #48
 Stevetuc Member Posts: 273 Joined: Jan 2014
RE: First python project : HP41 emulator + hints of workflow
(06-13-2021 08:41 PM)gfbci Wrote:
(06-13-2021 02:58 PM)Stevetuc Wrote:  Thanks Gianfranco you were clear, but I wanted to double check as unfortunately I tried this and it doesn't work on my G1.. not sure why

The only action I had to take besides the mentioned change was the removal of all downloaded or copied apps, the reset of the native apps and the calculator itself. This was necessary after the installation of fimware 2021 05 05 to normalize its behavior.
Hardware version is "C".
I redid all the indicated procedure starting from scratch: it works!
I hope it helps.

Gianfranco

Thanks. It's now working but a slightly different way. I stopped the program using the "On" key and it invited me to edit py41.py
I found that even though I thought I had sent the edited file previously, line 74 still contained
aof+=300 if ... etc
I removed the + and restarted. Now it works!
06-24-2021, 10:25 PM
Post: #49
 Liamtoh Resu Member Posts: 109 Joined: May 2021
RE: First python project : HP41 emulator + hints of workflow
Is it possible to switch the py41e simulator into radians mode and back to degrees
mode? And of course how would I do it?

Thanks
08-05-2021, 07:29 PM (This post was last modified: 08-06-2021 06:13 PM by Guenter Schink.)
Post: #50
 Guenter Schink Senior Member Posts: 462 Joined: Dec 2013
RE: First python project : HP41 emulator + hints of workflow
Edit:
I did of course forget something IMPORTANT.
in py41.py line 24 or so, after "def loop():" and before "k=0"add proper indented a line "global addonState" I've added that to the description below.
/EDIT

The HP41 emulator certainly is pretty cool. It is however not so easy to remember the assignments of the keys. One solution is to have some notes handy, but that's not really cool.

I tinkered around a little bit to have something available right away, by using the unused estate left and right of the printout. The display of this is toggled on/off by <SHIFT><VIEW>. Have a look at the picture.

In order to make it work you have to manually do some editing yourself. Please use an editor (e.g. notepad++) that is is suitable to edit text files.

2. open nuts.py to edit as text file, goto line 918 "def infos(self,c)" add as next line (proper indented) "return" it should look like
PHP Code:
def infos(self,c):    return    col=(0xFF0000,0x00FF00,0x0000FF)[c]    if self.theme==1:      hpprime.fillrect(0,0,20,50,13,0xffffff,0xffffff)    else:      hpprime.fillrect(0,0,20,50,13,0x181818,0x181818)    hpprime.textout(0,0,20,hex(self.stack[0]),col)
3. save nuts.py
4. open Py41.py to edit as text file
5. on top, right after "import hpprime" add a line "import AddOn"
5a.At line 24 or so, after "def loop():" and before "k=0"add proper indented a line "global addonState"
7. look upwards and find line 68 "elif k==4:" followed by "pass"
6 after that line insert a new line "elif k==9 and display.lE&0x0080:" folowed by "addonState=AddOn.addon(addonState)" followed by "cpu.togshift()" observe the indentation as in the lines already there. That now should look like:
PHP Code:
###                 elif k==4:    # Esc key        pass      elif k==9 and display.lE&0x0080:        addonState=AddOn.addon(addonState)        cpu.togShift()      elif k==10:   # Cas key

8. save Py41e.hpappdir and send it to your real and virtual Prime.

I hope I haven't forgotten anything. And remember this help is invoked (toggled) by <SHIFT><VIEW>

I must admit that's not as elegant as the fine art of Olivier, but may be of great help using the 41cx emulator on the Prime.

Olivier, if you read this perhaps it motivates you to include something like this in your distribution.

Günter

Attached File(s) Thumbnail(s)

08-06-2021, 12:39 AM
Post: #51
 Liamtoh Resu Member Posts: 109 Joined: May 2021
RE: First python project : HP41 emulator + hints of workflow
I still need some kind of documentation to use the hp41 emulator.

Does a user really need to do a line by line review of the source code to
use the HP41 emulator to its fullest extent? Should this emulator be
reguarded as a basic rpn toy?

I looked at the key.py file and see a list of binary MOD files with no docs on
how to use them.

A short internal help file may be of help to a small number of potential users.

I would really like to reguard the HP41 emulator as a usefull tool.

Thanks.
08-06-2021, 02:32 AM
Post: #52
 rprosperi Super Moderator Posts: 5,197 Joined: Dec 2013
RE: First python project : HP41 emulator + hints of workflow
(08-06-2021 12:39 AM)Liamtoh Resu Wrote:  I looked at the key.py file and see a list of binary MOD files with no docs on
how to use them.

The ROM modules for an HP-41 are generally well-known in the community, and manuals for many of the modules HP sold are included in the MoHPC Document set.

Other manuals can also be found on the 41-specific site:

http://www.ho41.org (substitute a "p" for the "o" - the actual url can't be used here for historical reasons)

And still other modern modules, from Angel Martin, Håkan Thörngren, and a few others can be found on the HP-41CL site here:

http://www.systemyde.com/hp41/documents.html

--Bob Prosperi
08-06-2021, 09:06 PM
Post: #53
 Liamtoh Resu Member Posts: 109 Joined: May 2021
RE: First python project : HP41 emulator + hints of workflow
Thanks for your response and information, rprosperi.

I seem to have problems mapping some of the HP41 emulator keys to the real calculator key functions and vice versa.

For instance I still can't work with radians on the emulator. If you had an example
of accessing one of the modules, I could figure outhow to access the rest.

I will check hout the "ho41,org" site.

I have looked at the docs at "literature.hpcalc.org"

Thanks.
08-06-2021, 10:35 PM
Post: #54
 Guenter Schink Senior Member Posts: 462 Joined: Dec 2013
RE: First python project : HP41 emulator + hints of workflow
(08-06-2021 09:06 PM)Liamtoh Resu Wrote:  I seem to have problems mapping some of the HP41 emulator keys to the real calculator key functions and vice versa.
That's what my extension above is for. Alternatively go back to the download site of P41, at the bottom of that page is a layout that shows the key assignments.

Quote:re: radians - as long as you don't familiarize yourself with the HP41 itself the emulator is almost useless " XEQ RAD"

All the necessary references were given above.
Günter
08-07-2021, 03:10 AM
Post: #55
 Liamtoh Resu Member Posts: 109 Joined: May 2021
RE: First python project : HP41 emulator + hints of workflow
The closest thing to HP41 calculator is the V41 emulator by Warren Furlow etc.

I admit I will have spend more time with documents that are available.

I was hoping someone from the Prime/HP41 communities could give me
some mentoring help besides "read the manual".

I looked at the "help" screen and I could not map it so I could use
the PY41e emulator with the shifted keys.

For a beginning, how do I access " XEQ RAD" with the PY41e emulator.

It would be usefull to post even a little bit of tips to help other users
who want to make dual use of the HP Prime calculator. The number
of current experienced HP41 users are not getting any younger.

Thanks.
08-07-2021, 03:47 AM (This post was last modified: 08-07-2021 03:49 AM by Liamtoh Resu.)
Post: #56
 Liamtoh Resu Member Posts: 109 Joined: May 2021
RE: First python project : HP41 emulator + hints of workflow
Ok Thanks Guenter for the "XEQ RAD" tip.

I used the V41 as a trainer.

So on the Py41e emulator:

[toolbox/Mem] brings up XEQ

[Alpha] puts the Py41e in ALPHA mode.

I typed [R] [A] [D]

I pressed the [Alpha] to take the Py41e emulator out of Alpha mode.

I pressed [ENTER] to put the Py41e into radian mode :)

I tested the Py41e with
3.1415926535 [Enter] 6 /

The Py41e gave the expected 0.5 result.

I was then able to switch the Py41e back to degrees mode.

Thanks again.
10-07-2021, 11:15 PM (This post was last modified: 10-10-2021 09:06 AM by Peet.)
Post: #57
 Peet Member Posts: 130 Joined: Oct 2020
RE: First python project : HP41 emulator + hints of workflow
Amazing program, makes the Prime an almost perfect HP-41clone. I only miss the BEEP a bit, HP should give the Prime sound output

If I see that correctly, to load a .mod or .raw currently only works via the config.py? Does anyone already have an idea for an easy way to load a .raw?

With the current routine, it reloads the .raw(s) every time I start it and fills up the memory.

"if '<PRG>' not in focal.listPrgms(cpu):" seems not to work. Maybe because of an type-error since focal.listPrgms(cpu) isn't a string?

My calculators - former: CBM PR100, HP41CV, HP11C, HP28S - current: HP48G, HP35S, Prime, DM41X, DM42
10-16-2021, 11:31 PM
Post: #58
 rogeriol Junior Member Posts: 7 Joined: Mar 2015
RE: First python project : HP41 emulator + hints of workflow
(06-11-2021 12:28 AM)rprosperi Wrote:
(06-11-2021 12:12 AM)Steve Simpkin Wrote:  I would recommend the original HP-41C Owner's Handbook and Programming Guide. It assumes you know nothing about HP calculators and is a great example of the friendly, well written calculator manuals that HP used to write.

https://literature.hpcalc.org/community/...hpg-en.pdf

I endorse Steve's recommendation! The later CX manuals cover more of the functions, but the original 41C manual is in many ways the peak of the great old-school HP

As a fan of HP manuals myself, as I learned a lot of programming on the HP28s thick manuals, I find it really funny on such excerpts:

"Should you see MEMORY LOSTin the display the first time you turn the HP-41C on, do not
worry —it means that power to the continuous memory of the calculator has been inter-
rupted at some time. Merely press (5 (the correction key) to clearthe error, then continue.
When power to continuous memory is interrupted, all information you placed into the
HP-41C is lost."

Yeah, do not worry. Everything you programmed and your data is gone forever if you did not export it someway

No need to say I'm reading hp41c manual for the first time, I never had the chance to own this beast.
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text:
Login Join Maker Pro

# Design of MP3

Z

#### ziro_axis

Jan 1, 1970
0
Hello All;
i have one class in the University is miniProject, my idea is to build
MP3 player, i have started searching the net until now i found ATMEL
is the suitable MCU i can use, but really i need help, i need some
guide about the basic structure of MP3, I appreciate any help, web
reference, book suggestion, any thing can help.
Thanks a lot

B

#### Bob

Jan 1, 1970
0
Hello All;
i have one class in the University is miniProject, my idea is to build
MP3 player, i have started searching the net until now i found ATMEL
is the suitable MCU i can use, but really i need help, i need some
guide about the basic structure of MP3, I appreciate any help, web
reference, book suggestion, any thing can help.
Thanks a lot

Learn to use google. The wikipedia page for MP3 would be a good
start.

ISO/IEC 11172-3 costs about 140 euro's to buy. It's not hard to
find a copy, see google.

http://www.mp3licensing.com/ lists a load of patents that
apply to mp3 and the royalty rates the patent holder demand.
$0.75 per hardware mp3 player, minimum$15000 per year.

Bob

J

#### jasen

Jan 1, 1970
0
Hello All;
i have one class in the University is miniProject, my idea is to build
MP3 player, i have started searching the net until now i found ATMEL
is the suitable MCU i can use, but really i need help,
i need some guide about the basic structure of MP3, I appreciate any help,
web reference, book suggestion, any thing can help.

The AT89C51SND2C chip will do most of the work for you - you don't need to
know how the data is encoded, if you're after the format of the ID3 / ID3v2
tags the get the source of a free taging tool.

Bye.
Jasen

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text:
Introductory Statistics

# 9.3Distribution Needed for Hypothesis Testing

Introductory Statistics9.3 Distribution Needed for Hypothesis Testing

Earlier in the course, we discussed sampling distributions. Particular distributions are associated with hypothesis testing. Perform tests of a population mean using a normal distribution or a Student's t-distribution. (Remember, use a Student's t-distribution when the population standard deviation is unknown and the distribution of the sample mean is approximately normal.) We perform tests of a population proportion using a normal distribution (usually n is large).

If you are testing a single population mean, the distribution for the test is for means:

$X ¯ ~N( μ X , σ X n ) X ¯ ~N( μ X , σ X n )$ or $t df t df$

The population parameter is μ. The estimated value (point estimate) for μ is $x ¯ x ¯$, the sample mean.

If you are testing a single population proportion, the distribution for the test is for proportions or percentages:

$P ′ ~N( p, p⋅q n ) P ′ ~N( p, p⋅q n )$

The population parameter is p. The estimated value (point estimate) for p is p′. p′ = $x n x n$ where x is the number of successes and n is the sample size.

### Assumptions

When you perform a hypothesis test of a single population mean μ using a Student's t-distribution (often called a t-test), there are fundamental assumptions that need to be met in order for the test to work properly. Your data should be a simple random sample that comes from a population that is approximately normally distributed. You use the sample standard deviation to approximate the population standard deviation. (Note that if the sample size is sufficiently large, a t-test will work even if the population is not approximately normally distributed).

When you perform a hypothesis test of a single population mean μ using a normal distribution (often called a z-test), you take a simple random sample from the population. The population you are testing is normally distributed or your sample size is sufficiently large. You know the value of the population standard deviation which, in reality, is rarely known.

When you perform a hypothesis test of a single population proportion p, you take a simple random sample from the population. You must meet the conditions for a binomial distribution which are: there are a certain number n of independent trials, the outcomes of any trial are success or failure, and each trial has the same probability of a success p. The shape of the binomial distribution needs to be similar to the shape of the normal distribution. To ensure this, the quantities np and nq must both be greater than five (np > 5 and nq > 5). Then the binomial distribution of a sample (estimated) proportion can be approximated by the normal distribution with μ = p and $σ= pq n σ= pq n$. Remember that q = 1 – p.

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# How many structural isomers can be proposed for a formula of C_4H_8?

Feb 4, 2017

${C}_{4} {H}_{8}$ has one degree of unsaturation, and thus whatever the compound we have the one degree of unsaturation, i.e. one olefinic bond, or 1 ring junction.
So I count 2 ring structures: $\text{cyclobutane}$; and $\text{methylcyclopropane.}$
And I count 3 olefinic structural isomers: $\text{1-butylene}$; $\text{2-butylene}$; $\text{2-methylpropene}$. $\text{2-butylene}$ can exist as 2 geometric isomers, which are?


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# Eulers Formula variant

Eulers formula says that e^ix = cosx + isinx

but in my text book theres another formula its e^ikx = coskx + isinkx
i still can't figure out how they got that. Is this still eulers formula? and how do u get it in that form