Wikipedia:Reference desk/Archives/Mathematics/2016 May 14

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May 14

Neighborly polytopes

Let P be the Cartesian product of two simplices (not necessarily the same dimension) and let D be the dual of P. When is D neighborly? For example the dual of the product of two triangles is neighborly, but the dual of the product of a line segment and a tetrahedron is not. --RDBury (talk) 05:46, 14 May 2016 (UTC)

When does a "multiple comparisons problems" occur (statistical inference / t-tests)

Hello!

I am wondering whether I am facing a Multiple comparisons problem in my analysis. I set up a "global" (compound/intersection) hypothesis for 2 independent groups, each tested with a t-test. I want to accept the global hypothesis in case both (indepedent) groups are improving on one variable (outcome).

$H_{AB}=H_{A}\cap H_{B}:\mu _{A}>0\quad {\text{and}}\quad \mu _{B}>0$

So the way I see this: both single t-test are highly significant. To accept the global hypothesis I don't have to add anything up. Can I infer that the intersection hypothesis is accepted since each of the two independent tests are significant?

Thanks for advice, --WissensDürster (talk) 08:40, 14 May 2016 (UTC)

I don't think you can infer that. See Simpson paradox. Loraof (talk) 14:21, 14 May 2016 (UTC)

Can you explain this in more detail please? I have two interventions given to two randomized groups. Simpson-Paradox: "in which a trend appears in different groups of data but disappears or reverses when these groups are combined". Combing the data (groups) is not what I want to do (nor test). Combing data of different interventions on different groups isn't my goal. I will rephrase my problem: I just need the correct form of hypothesis to express the following: "There is an intervention on a group and it will increase the performance in the outcome variable and there is another intervention and another group, and their performance on the outcome will also increase" (outcome variable is the same for both groups). kind regards --WissensDürster (talk) 16:02, 17 May 2016 (UTC)

@User:WissensDürster I see. My reading of the multiple comparisons article leads me to believe that you have to be careful about this. You say you want to have a hypothesis that both treatments are effective. I don't think there's a way to do that. You can have a null hypothesis that neither is effective, and the probability of observing what you've observed given that null is (assuming you've used alpha=.05 and they are independent groups) .05×.05=.0025. Loraof (talk) 20:59, 18 May 2016 (UTC)

@User:Loraof. I just need a way of writing this down in mathematical terms. In case it's rather difficult to combine those two:

${\begin{aligned}H_{1}:\mu _{G1|\delta }&>0\quad (H_{0}:\mu _{G1|\delta }\leq 0)\\H_{1}:\mu _{G2|\delta }&>0\quad (H_{0}:\mu _{G2|\delta }\leq 0).\end{aligned}}$

maybe I can separate them? Is it legitimate to call them "SHa" and "SHb". I am relatively sure this won't change the research pattern or results, I just need a precise notation. Subsequent correction for p-value have to be multiplication? I just thought division by 2 is what those Bonferoni methods are trying to do.. I will think more about this. --WissensDürster (talk) 10:14, 19 May 2016 (UTC)

Each null hypothesis has to have an equal sign in it, not a less than or equal to sign. The only possible joint null hypothesis has two equal signs: one effect equals zero AND the other effect equals zero. The alternative hypothesis then would be that they are not both zero. Loraof (talk) 14:33, 19 May 2016 (UTC)

Stupid complex number question

Hi,

I apologize in advance for this probably being a stupid question:

I want to convert (1-i) into polar form. Now, I believe the magnitude of this number is sqrt(1^2 - i^2) = 0??? How is this possible? Also, I get from trig that the angle would be arctan(-i)? What's going on?? — Preceding unsigned comment added by 140.233.174.42 (talk) 14:25, 14 May 2016 (UTC)

Not a stupid question! The point (1-i) in the complex plane is one unit to the right (in the real direction) and one unit down (in the imaginary direction).So its angle is minus 45°, or equivalently 360° - 45° = 315°. Its magnitude is found as the square root of the sum of squares of the coefficients (of 1 and -1, not of 1 and i). So the magnitude is ${\sqrt {1^{2}+(-1)^{2}}}={\sqrt {1+1}}={\sqrt {2}}.$ Loraof (talk) 14:36, 14 May 2016 (UTC)

And the angle can be written as arctan(-1/1) = arctan(-1). Loraof (talk) 14:41, 14 May 2016 (UTC)

oh! thank you! — Preceding unsigned comment added by 140.233.174.42 (talk) 14:45, 14 May 2016 (UTC)

Are complex numbers in polar coords really meaningful ? While it's useful for regular old X and Y coords, to tell us the distance and direction from the starting point, using it for complex numbers seems like saying 1 apple and 1 orange equals 1.41 fruit at a 45 degree angle. StuRat (talk) 14:50, 14 May 2016 (UTC)

They most certainly are, which is most obvious when you multiply them: multiplying a complex number by (1+i) multiplies its magnitude by the square root of 2, and adds 45 degrees to the angle it makes with the x-axis. Double sharp (talk) 14:57, 14 May 2016 (UTC)

There is also no denying that it is useful, as can be seen if you try to work out (1+i)¹⁷ both ways. Double sharp (talk) 15:00, 14 May 2016 (UTC)

So is it just useful as a calculation method, or do complex numbers in polar coords relate to anything in the real world ? If so, what does the angle and magnitude represent ? StuRat (talk) 15:02, 14 May 2016 (UTC)

you're getting philosophical, which will be dismissed by mathematicians..the idea being that all math is just a calculating method, so to speak..and none of it directly relates to the real world (though it can be used for applications in the real world)...68.48.241.158 (talk) 16:19, 14 May 2016 (UTC)

Very much yes! Putting a complex number in its modulus-argument form is frequently very useful, and is no or less meaningful than representing it in its real and complex components. For an example, look at complex number § electromagnetism and electrical engineering, where the argument is often used to represent phase. —  crh ₂₃  (Talk) 16:38, 14 May 2016 (UTC)

In linear difference equations, the characteristic roots may be complex. If so, the magnitude of the largest pair determines whether the real-world quantity being modeled converges to a steady state (magnitude less than 1) or diverges (magnitude greater than 1). The departure of the magnitude from 1 determines how fast the convergence or divergence is. The additive contribution of this pair of complex conjugate roots to the solution for the dynamic variable is $2m^{t}\cdot \cos(\theta t+\delta )$ where m is the magnitude of the roots, t is time, $\theta$ is the angle of one of the complex numbers, and $\delta$ depends on the real and imaginary parts of the complex numbers scaled by their magnitude. Something similar happens with linear differential equations. So the magnitude and angle appear in a solution equation that contains only real elements. Loraof (talk) 16:54, 14 May 2016 (UTC)

Thanks all, for providing examples. StuRat (talk) 22:42, 14 May 2016 (UTC)

A formalism rigorous

When is a formalism rigorous? Is a rigorous formalism just another word for exhaustive or explicit? --Llaanngg (talk) 20:51, 14 May 2016 (UTC)

Presumably this question comes with context. You will get better answers if you provide the context than if you obscure it. --JBL (talk) 22:27, 14 May 2016 (UTC)

It depends with time. I'm sure the Egyptians and Babylonians and Greeks thought what they did was rigorous. However new axioms have had to be added to even Euclids Geometry to fix things missed out in it. Nowadays a formal system would be one that can be checked by a proof checker on a computer. In the future, and I for one welcome our new AI overlords, ;-) artificial intelligences will presumably think our idea of rigor is naive and primitive. Dmcq (talk) 22:52, 14 May 2016 (UTC)

For example: "The objective of this work is to present a rigorous formalism for the solution of engineering problems on vibrations in which the vibrating structure has a discrete distribution of loads." But there are plenty of mathematical or technical texts setting their objective to outline/present/develop a "rigorous formalism" for some problem/field/issue. What would be the difference if their formalism were not rigorous?--Llaanngg (talk) 23:07, 14 May 2016 (UTC)

From formalism (philosophy of mathematics):

Formalism is associated with rigorous method. In common use, a formalism means the out-turn of the effort towards formalisation of a given limited area.

Assuming this is the concept of formalism that is relevant to your sources, it seems to me that this quote implies that "rigorous formalism" is a redundancy meaning "formalism". Loraof (talk) 23:55, 14 May 2016 (UTC)

I think your question is more about the language being used as opposed to what "formalism" is/means...I think the "rigorous" word is just tagged on here to emphasize the formal nature of a formalism...68.48.241.158 (talk) 14:18, 15 May 2016 (UTC)

Couldn't it be that "rigorous" means from basic principles, explicitly describing each step? Llaanngg (talk) 17:52, 15 May 2016 (UTC)

I think that's more along the lines of what "formal" means here...the "rigorous" just meaning here that it's carefully formalized..68.48.241.158 (talk) 17:56, 15 May 2016 (UTC)