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# American Institute of Mathematical Sciences

December  2016, 9(6): 1899-1912. doi: 10.3934/dcdss.2016077

## A global existence and blow-up threshold for Davey-Stewartson equations in $\mathbb{R}^3$

 1 School of Mathematics, South China University of Technology, Guangzhou, Guangdong 510640, China 2 Department of Mathematics, South China University of Technology, Guangzhou, Guangdong 510640 3 College of Mathematics and Information Science, Henan Normal University, Xinxiang, Henan 453007

Received  July 2015 Revised  September 2016 Published  November 2016

In this paper we study the threshold of global existence and blow-up for the solutions to the generalized 3D Davey-Stewartson equations \begin{equation*} \left\{ \begin{aligned} & iu_t + \Delta u + |u|^{p-1} u + E_1(|u|^2)u = 0, \quad t > 0, \ \ x\in \mathbb{R}^3, \\ & u(0,x) = u_0(x) \in H^1(\mathbb{R}^3), \end{aligned} \right. \end{equation*} where $1 < p < \frac{7}{3}$ and the operator $E_1$ is given by $E_1(f) = \mathcal {F}^{-1} \left( \frac{\xi_1^2}{|\xi|^2} \mathcal{F}(f) \right)$. We construct two kinds of invariant sets under the evolution flow by analyzing the property of the upper bound function of the energy. Then we show that the solution exists globally for the initial function $u_0$ in first kind of the invariant sets, while the solution blows up in finite time for $u_0$ in another kind. We remark that the exponent $p$ is subcritical for the nonlinear Schrödinger equations for which blow-up solutions would not occur. The result shows that the occurrence of blow-up phenomenon is caused by the coupling mechanics of the Davey-Stewartson equations.
Citation: Shiming Li, Yongsheng Li, Wei Yan. A global existence and blow-up threshold for Davey-Stewartson equations in $\mathbb{R}^3$. Discrete & Continuous Dynamical Systems - S, 2016, 9 (6) : 1899-1912. doi: 10.3934/dcdss.2016077
##### References:
 [1] D. Anker and N. C. Freeman, On the soliton solutions of the Davey-Stewartson equation for long waves,, Proc. Roy. Soc. London, 360 (1978), 529. doi: 10.1098/rspa.1978.0083. Google Scholar [2] M. J. Ablowitz and A. S. Fokas, On the inverse scattering transform of multidimensional nonlinear equations related to first-order systems in the plane,, J. Math. Phys., 25 (1984), 2494. doi: 10.1063/1.526471. Google Scholar [3] R. Cipolatti, On the existence of standing waves for a Davey-Stewartson system,, Commun. Partial Diff. Eqns., 17 (1992), 967. doi: 10.1080/03605309208820872. Google Scholar [4] R. Cipolatti, On the instability of ground states for a Davey-Stewartson system,, Ann. Inst. H. Poincaré, 58 (1993), 85. Google Scholar [5] V. D. Djordjevic and L. G. Redekopp, On two-dimensional packets of capillary-gravity waves,, J. Fluid Mech., 79 (1977), 703. doi: 10.1017/S0022112077000408. Google Scholar [6] A. Davey and S. K. Stewartson, On three-dimensional packets of surface waves,, Proc. R. Soc. London, 338 (1974), 101. doi: 10.1098/rspa.1974.0076. Google Scholar [7] Z. H. Gan and J. Zhang, Sharp threshold of global existence and instability of standing wave for a Davey-Stewartson system,, Commun. Math. Phys., 283 (2008), 93. doi: 10.1007/s00220-008-0456-y. Google Scholar [8] J. M. Ghidaglia and J. C. Saut, On the initial value problem for the Davey-Stewartson systems,, Nonlinearity, 3 (1990), 475. doi: 10.1088/0951-7715/3/2/010. Google Scholar [9] N. Godet, A lower bound on the blow-up rate for the Davey-Stewartson system on the torus,, Ann. Inst. H. Poincaré - AN, 30 (2013), 691. doi: 10.1016/j.anihpc.2012.12.001. Google Scholar [10] B. L. Guo and B. X. Wang, The Cauchy problem for Davey-Stewartson systems,, Commun. Pure Appl. Math., 52 (1999), 1477. doi: 10.1002/(SICI)1097-0312(199912)52:12<1477::AID-CPA1>3.0.CO;2-N. Google Scholar [11] N. Hayashi, Local existence in time of small solutions to the Davey-Stewartson systems,, Ann. Inst. H. Poincaré, 65 (1996), 313. Google Scholar [12] N. Hayashi and H. Hirata, Global existence and asymptotic behavior in time of small solutions to the elliptic-hyperbolic Davey-Stewartson system,, Nonlinearity, 9 (1996), 1387. doi: 10.1088/0951-7715/9/6/001. Google Scholar [13] N. Hayashi and J. C. Saut, Global existence of small solutions to the Davey-Stewartson and the Ishimori systems,, Diff. and Integ. Eqns., 8 (1995), 1657. Google Scholar [14] T. Hmidi and S. Keraani, Blowup theory for the critical nonlinear Schrödinger equations revisited,, Intern. Math. Res. Notices, 46 (2005), 2815. doi: 10.1155/IMRN.2005.2815. Google Scholar [15] X. G. Li, J. Zhang, S. Y. Lai and Y. H. Wu, The sharp threshold and limiting profile of blow-up solutions for a Davey-Stewartson system,, J. Diff. Eqns., 250 (2011), 2197. doi: 10.1016/j.jde.2010.10.022. Google Scholar [16] F. Linares and G. Ponce, On the Davey-Stewartson systems,, Ann. Inst. H. Poincaré, 10 (1993), 523. Google Scholar [17] J. Lu and Y. F. Wu, Sharp threshold for scattering of a generalized Davey-Stewartson system in three dimension,, Commun. Pure Appl. Anal., 14 (2015), 1641. doi: 10.3934/cpaa.2015.14.1641. Google Scholar [18] T. Ozawa, Exact blow-up solutions to Cauchy problem for the Davey-Stewartson system,, Proc. R. Soc. Lond. Ser. A, 436 (1992), 345. doi: 10.1098/rspa.1992.0022. Google Scholar [19] M. Ohta, Stability of standing waves for the generalized Davey-Stewartson system,, J. Dyn. Diff. Eqns., 6 (1994), 325. doi: 10.1007/BF02218533. Google Scholar [20] G. Richards, Mass concentration for the Davey-Stewartson system,, Diff. and Integ. Eqns., 24 (2011), 261. Google Scholar [21] J. Shu and J. Zhang, Sharp conditions of global existence for the generalized Davey-Stewartson system,, IMA J. Appl. Math., 72 (2007), 36. doi: 10.1093/imamat/hxl029. Google Scholar [22] M. Tsutsumi, Decay of weak solutions to the Davey-Stewartson systems,, J. Math. Anal. Appl., 182 (1994), 680. doi: 10.1006/jmaa.1994.1113. Google Scholar [23] B. X. Wang and B. L. Guo, On the initial value problem and scattering of solutions for the generalized Davey-Stewartson systems,, Sci. China Ser. A, 44 (2001), 994. doi: 10.1007/BF02878975. Google Scholar [24] M. I. Weinstein, Nonlinear Schrödinger equations and sharp interpolation estimates,, Commun. Math. Phys., 87 (1983), 567. Google Scholar [25] H. Yang, X. M. Fan and S. H. Zhu, Global analysis for rough solutions to the Davey- Stewartson system,, Abstract and Appl. Anal., 2012 (2012). Google Scholar [26] J. Zhang and S. Zhu, Sharp blow-up criteria for the Davey-Stewartson system in $\mathbbR^3$,, Dyn. Partial Diff. Eqns., 8 (2011), 239. doi: 10.4310/DPDE.2011.v8.n3.a4. Google Scholar [27] S. H. Zhu, Blow-up dynamics of $L^2$ solutions for the Davey-Stewartson system,, Acta Math. Sinica, 31 (2015), 411. doi: 10.1007/s10114-015-4349-7. Google Scholar

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##### References:
 [1] D. Anker and N. C. Freeman, On the soliton solutions of the Davey-Stewartson equation for long waves,, Proc. Roy. Soc. London, 360 (1978), 529. doi: 10.1098/rspa.1978.0083. Google Scholar [2] M. J. Ablowitz and A. S. Fokas, On the inverse scattering transform of multidimensional nonlinear equations related to first-order systems in the plane,, J. Math. Phys., 25 (1984), 2494. doi: 10.1063/1.526471. Google Scholar [3] R. Cipolatti, On the existence of standing waves for a Davey-Stewartson system,, Commun. Partial Diff. Eqns., 17 (1992), 967. doi: 10.1080/03605309208820872. Google Scholar [4] R. Cipolatti, On the instability of ground states for a Davey-Stewartson system,, Ann. Inst. H. Poincaré, 58 (1993), 85. Google Scholar [5] V. D. Djordjevic and L. G. Redekopp, On two-dimensional packets of capillary-gravity waves,, J. Fluid Mech., 79 (1977), 703. doi: 10.1017/S0022112077000408. Google Scholar [6] A. Davey and S. K. Stewartson, On three-dimensional packets of surface waves,, Proc. R. Soc. London, 338 (1974), 101. doi: 10.1098/rspa.1974.0076. Google Scholar [7] Z. H. Gan and J. Zhang, Sharp threshold of global existence and instability of standing wave for a Davey-Stewartson system,, Commun. Math. Phys., 283 (2008), 93. doi: 10.1007/s00220-008-0456-y. Google Scholar [8] J. M. Ghidaglia and J. C. Saut, On the initial value problem for the Davey-Stewartson systems,, Nonlinearity, 3 (1990), 475. doi: 10.1088/0951-7715/3/2/010. Google Scholar [9] N. Godet, A lower bound on the blow-up rate for the Davey-Stewartson system on the torus,, Ann. Inst. H. Poincaré - AN, 30 (2013), 691. doi: 10.1016/j.anihpc.2012.12.001. Google Scholar [10] B. L. Guo and B. X. Wang, The Cauchy problem for Davey-Stewartson systems,, Commun. Pure Appl. Math., 52 (1999), 1477. doi: 10.1002/(SICI)1097-0312(199912)52:12<1477::AID-CPA1>3.0.CO;2-N. Google Scholar [11] N. Hayashi, Local existence in time of small solutions to the Davey-Stewartson systems,, Ann. Inst. H. Poincaré, 65 (1996), 313. Google Scholar [12] N. Hayashi and H. Hirata, Global existence and asymptotic behavior in time of small solutions to the elliptic-hyperbolic Davey-Stewartson system,, Nonlinearity, 9 (1996), 1387. doi: 10.1088/0951-7715/9/6/001. Google Scholar [13] N. Hayashi and J. C. Saut, Global existence of small solutions to the Davey-Stewartson and the Ishimori systems,, Diff. and Integ. Eqns., 8 (1995), 1657. Google Scholar [14] T. Hmidi and S. Keraani, Blowup theory for the critical nonlinear Schrödinger equations revisited,, Intern. Math. Res. Notices, 46 (2005), 2815. doi: 10.1155/IMRN.2005.2815. Google Scholar [15] X. G. Li, J. Zhang, S. Y. Lai and Y. H. Wu, The sharp threshold and limiting profile of blow-up solutions for a Davey-Stewartson system,, J. Diff. Eqns., 250 (2011), 2197. doi: 10.1016/j.jde.2010.10.022. Google Scholar [16] F. Linares and G. Ponce, On the Davey-Stewartson systems,, Ann. Inst. H. Poincaré, 10 (1993), 523. Google Scholar [17] J. Lu and Y. F. Wu, Sharp threshold for scattering of a generalized Davey-Stewartson system in three dimension,, Commun. Pure Appl. Anal., 14 (2015), 1641. doi: 10.3934/cpaa.2015.14.1641. Google Scholar [18] T. Ozawa, Exact blow-up solutions to Cauchy problem for the Davey-Stewartson system,, Proc. R. Soc. Lond. Ser. A, 436 (1992), 345. doi: 10.1098/rspa.1992.0022. Google Scholar [19] M. Ohta, Stability of standing waves for the generalized Davey-Stewartson system,, J. Dyn. Diff. Eqns., 6 (1994), 325. doi: 10.1007/BF02218533. Google Scholar [20] G. Richards, Mass concentration for the Davey-Stewartson system,, Diff. and Integ. Eqns., 24 (2011), 261. Google Scholar [21] J. Shu and J. Zhang, Sharp conditions of global existence for the generalized Davey-Stewartson system,, IMA J. Appl. Math., 72 (2007), 36. doi: 10.1093/imamat/hxl029. Google Scholar [22] M. Tsutsumi, Decay of weak solutions to the Davey-Stewartson systems,, J. Math. Anal. Appl., 182 (1994), 680. doi: 10.1006/jmaa.1994.1113. Google Scholar [23] B. X. Wang and B. L. Guo, On the initial value problem and scattering of solutions for the generalized Davey-Stewartson systems,, Sci. China Ser. A, 44 (2001), 994. doi: 10.1007/BF02878975. Google Scholar [24] M. I. Weinstein, Nonlinear Schrödinger equations and sharp interpolation estimates,, Commun. Math. Phys., 87 (1983), 567. Google Scholar [25] H. Yang, X. M. Fan and S. H. Zhu, Global analysis for rough solutions to the Davey- Stewartson system,, Abstract and Appl. Anal., 2012 (2012). Google Scholar [26] J. Zhang and S. Zhu, Sharp blow-up criteria for the Davey-Stewartson system in $\mathbbR^3$,, Dyn. Partial Diff. Eqns., 8 (2011), 239. doi: 10.4310/DPDE.2011.v8.n3.a4. Google Scholar [27] S. H. Zhu, Blow-up dynamics of $L^2$ solutions for the Davey-Stewartson system,, Acta Math. Sinica, 31 (2015), 411. doi: 10.1007/s10114-015-4349-7. Google Scholar
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# 2.7 Box-Sizing

By default, a box’s width, padding, border, and margin are all added together to determine the full dimensions of the box. In this lesson, you’ll learn about the box-sizing property, which allows you to alter this default behavior.

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## Posts

A problem with "playing chicken with the universe" as an approach to UDT 2013-03-08T02:34:27.571Z · score: 16 (19 votes)
Rational vs. real utilities in the cousin_it-Nesov model of UDT 2012-01-29T04:43:17.705Z · score: 6 (7 votes)
Is it possible to build a safe oracle AI? 2011-04-20T12:54:55.934Z · score: 0 (7 votes)

Comment by karl on Introducing Corrigibility (an FAI research subfield) · 2014-10-24T21:12:20.965Z · score: 3 (3 votes) · LW · GW

Firstly, the important part of my modification to the indifference formalism is not about conditioning on the actual o but it's the fact that in evaluating the expectation of UN it take the action in A2 (for a given pair (a1,o)) which maximize UN instead of the action which maximize U (note that U is equal to US in the case that o is not in Press.).

Secondly an agent which chose a1 by simply maximizing E[UN | NotPress; a1] + E[US | Press; a1] do exhibit pathological behaviors. In partcular, there will still be incentives to manage the news, but from both sides now (there is an incentive to cause the button to be pressed in the event of an information which is bad news from the point of view of UN and incentives to cause the button to not be pressed in the events of information which is bad news from the point of view of US.

Comment by karl on Introducing Corrigibility (an FAI research subfield) · 2014-10-24T04:00:01.715Z · score: 2 (2 votes) · LW · GW

In the indifference formalism the agent in selecting A1 act like a UN agent that believe that the shutdown button will not be pressed, therefore it create perverse incentives to "manage the news". Which means that if the agent can cause his shutdown button to be pressed in the event of bad news, it will.

My formulation avoid this pathological behavior by instead making the agent select A1 as if it was a UN-agent which believed that it would continue to optimize according to UN even in the vent of the button being pressed which avoid the perverse incentives to "manage the news", while still not having any incentives to avoid the button being pressed because the agent will act like it believe that pressing the button will not cause it to initiate a shutdown.

Comment by karl on Introducing Corrigibility (an FAI research subfield) · 2014-10-24T02:38:51.075Z · score: 4 (4 votes) · LW · GW

Why not make it so that the agent in selecting A1 act as a UN-agent that believe that it will continue to optimize according to UN even in the event of the button being pressed rather than a UN agent that believe that the button will never be pressed: that is pick U such that

U(a1,o,a2) = UN(a1,o,a2) if o is in Press or US(a1,o,a2) + f(a1,o) - g(a1,o) if o is not in Press

where f(a1,o) is the maximum value of UN(a1,o,b) for b in A2 and g(a1,o) is the maximum value of US(a1,o,b) for b in A2.

This would avoid the perverse manipulation incentives problem detailed on section 4.2 of the paper.

Comment by karl on Reduced impact AI: no back channels · 2013-11-11T19:23:34.699Z · score: 2 (2 votes) · LW · GW

Apart from the obvious problems with this approach, (The AI can do a lot with the output channel other than what you wanted it to do, choosing an appropriate value for λ, etc.) I don't see why this approach would be any easier to implement than CEV.

Once you know what a bounded approximation of an ideal algorithm is supposed to look like, how the bounded version is supposed to reason about it's idealised version and how to refer to arbitrary physical data, as the algorithm defined in your post assume, then implementing CEV really doesn't seem to be that hard of a problem.

So could you explain why you believe that implementing CEV would be so much harder than what you propose in your post?

Comment by karl on An argument against indirect normativity · 2013-07-24T23:39:03.066Z · score: 3 (3 votes) · LW · GW

By that term I simply mean Eliezer's idea that the correct decision theory ought to use a maximization vantage points with a no-blackmail equilibrium.

Comment by karl on An argument against indirect normativity · 2013-07-24T23:14:21.748Z · score: 0 (0 votes) · LW · GW

If the NBC (No Blackmail Conjecture) is correct, then that shouldn't be a problem.

Comment by karl on The Empty White Room: Surreal Utilities · 2013-07-24T14:37:42.241Z · score: 1 (1 votes) · LW · GW

So you wouldn't trade whatever amount of time Frank as left, which is at most measured in decades, against a literal eternity of Fun?

If I was Frank in this scenario, I would tell the other guy to accept the deal.

Comment by karl on An Attempt at Preference Uncertainty Using VNM · 2013-07-16T16:48:04.596Z · score: 3 (3 votes) · LW · GW

Hmm. I'll have to take a closer look at that. You mean that the uncertainties are correlated, right?

No. To quote your own post:

A similar process allows us to arbitrarily set exactly one of the km.

I meant that the utility function resulting from averaging over your uncertainty over the km's will depend on which km you chose to arbitrarily set in this way. I gave an example of this phenomenon in my original comment.

Comment by karl on An Attempt at Preference Uncertainty Using VNM · 2013-07-16T15:02:49.099Z · score: 4 (4 votes) · LW · GW

You can't simply average the km's. Suppose you estimate .5 probability that k2 should be twice k1 and .5 probability that k1 should be twice k2. Then if you normalize k1 to 1, k2 will average to 1.25, while similarly if you normalize k2 to 1, k1 will average to 1.25.

In general, to each choice of km's will correspond a utility function and the utility function we should use will be a linear combination of those utility functions and we will have renormalization parameters k'm and, if we accept the argument given in your post, those k'm ought to be just as dependant on your preferences, so you're probably also uncertain about the values that those parameters should take and so you obtain k''m's and so on ad infinitum. So you obtain an infinite tower of uncertain parameters and it isn't obvious how to obtain a utility function out of this mess.

Comment by karl on Robust Cooperation in the Prisoner's Dilemma · 2013-07-15T03:24:37.907Z · score: 2 (2 votes) · LW · GW

What do you even mean by "is a possible outcome" here? Do you mean that there is no proof in PA of the negation of the proposition?

The formula of a modal agent must be fully modalized, which means that all propositions containing references to actions of agents within the formula must be within the scope of a provability operator.

Comment by karl on Robust Cooperation in the Prisoner's Dilemma · 2013-07-15T03:13:22.972Z · score: 3 (3 votes) · LW · GW

Proof without using Kripke semantic: Let X be a modal agent and Phi(...) it's associated fully modalized formula. Then if PA was inconsistent Phi(...) would reduce to a truth value independent of X opponent and so X would play the same move against both FairBot and UnfairBot (and this is provable in PA). But PA cannot prove it's own consistency so PA cannot both prove X(FairBot) = C and X(UnfairBot) = D and so we can't both have FairBot(X) = C and UnfairBot(X) = C. QED

Comment by karl on Robust Cooperation in the Prisoner's Dilemma · 2013-07-14T13:18:56.198Z · score: 1 (1 votes) · LW · GW

Proof: Let X be a modal agent, Phi(...) it's associated fully modalized formula, (K, R) a GL Kripke model and w minimal in K. Then, for all statement of the form ◻(...) we have w |- ◻(...) so Phi(...) reduce in w to a truth value which is independent of X opponent. As a result, we can't have both w |- X(FairBot) = C and w |- X(UnfairBot) = D and so we can't have both ◻(X(FairBot) = C) and ◻(X(UnfairBot) = D) and so we can't both have FairBot(X) = C and UnfairBot(X) = C. QED

Comment by karl on Robust Cooperation in the Prisoner's Dilemma · 2013-07-14T02:53:57.898Z · score: 1 (1 votes) · LW · GW

no modal agent can get both FairBot and UnfairBot to cooperate with it.

TrollDetector is not a modal agent.

Comment by karl on Robust Cooperation in the Prisoner's Dilemma · 2013-07-14T00:18:02.896Z · score: 3 (3 votes) · LW · GW

UnfairBot defect against PrudentBot.

Proof: For UnfairBot to cooperate with PrudentBot, PA would have to prove that PrudentBot defect against UnfairBot which would require PA to prove that "PA does not prove that UnfairBot cooperate with PrudentBot or PA+1 does not prove that UnfairBot defect against DefectBot" but that would require PA to prove it's own consistency which it cannot do. QED

Comment by karl on Robust Cooperation in the Prisoner's Dilemma · 2013-07-10T22:08:21.702Z · score: 2 (2 votes) · LW · GW

Here is another obstacle to an optimality result: define UnfairBot as the agent which cooperate with X if and only if PA prove that X defect against UnfairBot, then no modal agent can get both FairBot and UnfairBot to cooperate with it.

Comment by karl on Robust Cooperation in the Prisoner's Dilemma · 2013-06-11T02:09:58.113Z · score: 1 (1 votes) · LW · GW

Both of those agents are modal agents of rank 0 and so the fact that they defect against CooperateBot imply that FairBot defects against them by theorem 4.1.

Comment by karl on Prisoner's Dilemma (with visible source code) Tournament · 2013-06-11T00:24:31.219Z · score: 0 (0 votes) · LW · GW

You're right, and wubbles's agent can easily be exploited by a modal agent A defined by A(X)=C <-> [] (X(PB)=C) (where PB is PrudentBot).

Comment by karl on Prisoner's Dilemma (with visible source code) Tournament · 2013-06-10T21:07:00.772Z · score: 0 (0 votes) · LW · GW

The agent defined by wubbles is actually the agent called JustBot in the robust cooperation paper and which is proven to be non-exploitable by modal agents.

Comment by karl on A problem with "playing chicken with the universe" as an approach to UDT · 2013-03-09T16:11:33.123Z · score: 0 (0 votes) · LW · GW

Well, depend how different the order is...

If there are theorem in PA of the form "If there are theorems of the form A()≠a and of the form A'()≠a' then the a and the a' such that the corresponding theorem come first in the appropriate ordering must be identical." then you should be okay in the prisoner dilemma setting but otherwise there will be a model of PA in which the two players end up playing different actions and we end up in the same situation as in the post.

More generally, no matter how you try to cut it, there will always be a model of PA in which all theorems are provable and your agent behavior will always depend on what happen in that model because PA will never be able to prove anything which is false in that model.

Comment by karl on A problem with "playing chicken with the universe" as an approach to UDT · 2013-03-08T23:31:44.648Z · score: 1 (1 votes) · LW · GW

If I understand what you're proposing correctly then that doesn't work either.

Suppose it is a theorem of PA that this algorithm return an error, then all theorems of the form A()=a => U()>=u are demonstrable in PA and so there is no action with the highest u and the algorithm return an error, because that is demonstrable in PA the algorithm must return an error by Löb's theorem.

Comment by karl on A problem with "playing chicken with the universe" as an approach to UDT · 2013-03-08T15:06:01.421Z · score: 1 (1 votes) · LW · GW

I didn't like the way the first paragraph ended. It seemed excessively confrontational and made me not want to read the rest of the post.

My intention was not to appear confrontational. It actually seemed obvious when I began thinking about this problem that the order in which we check actions in step 1 shouldn't matter but that ended up being completely wrong. That was what I was trying to convey though I admit I might have done so in a clumsy manner.

Comment by karl on A problem with "playing chicken with the universe" as an approach to UDT · 2013-03-08T14:56:15.471Z · score: 4 (4 votes) · LW · GW

To quote step 2 of the original algorithm:

For every possible action a, find some utility value u such that S proves that A()=a ⇒ U()=u. If such a proof cannot be found for some a, break down and cry because the universe is unfair.

Comment by karl on Harry Potter and the Methods of Rationality discussion thread, part 18, chapter 87 · 2012-12-23T21:44:09.798Z · score: 2 (2 votes) · LW · GW

Timeless physics predate Barbour.

Comment by karl on Ontological Crisis in Humans · 2012-12-20T05:14:40.156Z · score: 1 (3 votes) · LW · GW

If something is not ontologically fundamental and doesn't reduce to anything which is, then that thing isn't real.

Comment by karl on Ontological Crisis in Humans · 2012-12-20T02:29:42.480Z · score: 0 (0 votes) · LW · GW

Given that we are able to come to agreement about certain moral matters and the existence of moral progress, I do think that the evidence favor the existence of a well-behaved idealized reasoning system that we are approximating when we do moral reasoning.

Can you give a detailed example of this?

This for a start.

Comment by karl on Ontological Crisis in Humans · 2012-12-20T02:07:05.767Z · score: 0 (0 votes) · LW · GW

Are you saying you think qualia is ontologically fundamental or that it isn't real or what?

Comment by karl on Ontological Crisis in Humans · 2012-12-19T21:40:29.535Z · score: 0 (0 votes) · LW · GW

there still isn't any Scientifically Accepted Unique Solution for the moral value of animals

There isn't any SAUS for the problem of free will either. Nonetheless, it is a solved problem. Scientists are not in the business of solving that kind of problems, those problems generally being considered philosophical in nature.

the question is whether the solution uniquely follows from your other preferences, or is somewhat arbitrary?

It certainly appear to uniquely follow.

see the post "The "Scary problem of Qualia".

That seems easy to answer. Modulo a reduction of computation of course but computation seems like a concept which ought to be canonically reducible.

Comment by karl on The "Scary problem of Qualia" · 2012-12-19T21:13:10.135Z · score: 0 (0 votes) · LW · GW

Did the person come into existence:

Ve came into existence whenever a computation isomorphic to verself was performed.

Comment by karl on Harry Potter and the Methods of Rationality discussion thread, part 17, chapter 86 · 2012-12-19T17:10:03.463Z · score: 6 (6 votes) · LW · GW

For Harry had only loaned his Cloak, not given it

That seems like it answer your question: his invisible copies aren't borrowing the cloak from him because they are him.

Comment by karl on Ontological Crisis in Humans · 2012-12-19T17:05:32.433Z · score: 1 (1 votes) · LW · GW

You seem to be taking a position that's different from Eliezer's, since AFAIK he has consistently defended this approach that you call "wrong" (for example in the thread following my first linked comment).

Well, Eliezer_2009 do seem to underestimate the difficulty of the extrapolation problem.

If you have some idea of how to "derive the computation that we approximate when we talk about morality by examining the dynamic of how people reacts to moral arguments" that doesn't involve just "looking at where people moralities concentrate as you present moral arguments to them" then I'd be interested to know what it is.

Have I solved that problem? No. But humans do seem to be able to infer from the way pebblesorters reason that they are referring to primality and also it seem that by looking at mathematicians reasoning about Goldbach's Conjecture we should be able to infer what they refer to when they speak about Goldbach's Conjecture and we don't do that by looking at what position they end up concentrating at when we present them with all possible arguments. So the problem ought to be solvable.

Assuming "when we do moral reasoning we are approximating some computation", what reasons do we have for thinking that the "some computation" will allow us to fully reduce "pain" to a set of truth conditions? What are some properties of this computation that you can cite as being known, that leads you to think this?

Are you trying to imply that it could be the case that the computation that we approximate is only defined over a fixed ontology which doesn't correspond to the correct ontology and simply return a domain error when one try to apply it to the real world? Well that doesn't seem to be the case because we are able to do moral reasoning at a more fine-grained level than the naive ontology and a lot of the concept that seems relevant to moral reasoning, like qualia for example, seems like they ought to have canonical reductions. I detail the way in which I see us doing that kind of elucidation in this comment.

Comment by karl on Harry Potter and the Methods of Rationality discussion thread, part 17, chapter 86 · 2012-12-19T05:52:00.799Z · score: 0 (0 votes) · LW · GW

Quiditch, the lack of adequate protection on time turners before Harry gave them the idea put protective shell on them... Seriously, just reread the fic.

Comment by karl on Ontological Crisis in Humans · 2012-12-19T05:37:19.763Z · score: 0 (0 votes) · LW · GW

Eliezer seems to think that moral arguments are meaningful, but their meanings are derived only from how humans happen to respond to them (or more specifically to whatever coherence humans may show in their responses to moral arguments).

No. What he actually says is that when we do moral reasoning we are approximating some computation in the same ways that the pebblesorters are approximating primality. What make moral arguments valid or invalid is whether the arguments actually establish what they where trying to establish in the context of the actual concept of rightness which is being approximated in the same way that an argument by a pebblesorter that 6 is an incorrect heap because 6=2*3 is judged valid because that is an argument establishing 6 is not a prime number. Of course, to determine what computation we actually are trying to approximate or to establish that we actually are approximating something, looking at the coherence human show in their response to moral arguments is an excellent idea, but it is not how you define morality.

Looking at the first link you provided, I think that looking at where people moralities concentrate as you present moral arguments to them is totally the wrong way to look at this problem. Consider for example Goldbach's Conjecture. If you where to take peoples and present random arguments to them about the conjecture it doesn't seem to necessarily be the case, depending on what distribution you use on possible arguments, that people opinions will concentrate to the correct conclusion concerning the validity of the conjecture. That doesn't mean that people can't talk meaningfully about the validity of Goldbach's Conjecture. Should we be able to derive the computation that we approximate when we talk about morality by examining the dynamic of how people reacts to moral arguments? The answer is yes, but it isn't a trivial problem.

As for the second link you provided, you argue that the way we react to moral arguments could be totally random or depend on trivial details, which doesn't appear to be the case and which seems in contradiction with the fact that we do manage to agree about a lot of thing concerning morality and that moral progress do seems to be a thing.

Comment by karl on Ontological Crisis in Humans · 2012-12-19T03:04:33.370Z · score: -1 (1 votes) · LW · GW

huge, unsolved debates over morality

One shouldn't confuse there being a huge debate over something with the problem being unsolved, far less unsolvable (look at the debate over free will or worse p-zombies). I have actually solved the problem of the moral value of animals to my satisfaction (my solution could be wrong, of course). As for the problem of dealing with peoples having multiple copies this really seems like the problem of reducing "magical reality fluid" which while hard seems like it should be possible.

also in math, it might be possible to generalize things, but not necessarily, and not always uniquely

Well, yes. But in general if you're trying to elucidate some concept in your moral reasoning you should ask yourself the specific reason why you care about that specific concept until you reach concepts that looks like they should have canonical reductions, then you reduce them. If in doing so you end up with multiple possible reductions that probably mean you didn't go deep enough and should be asking why you care about that specific concept some more so that you can pinpoint the reduction you are actually interested in. If after all that you're still left with multiple possible reductions for a certain concept, that you appear to value terminally, and not for any other reasons, then you should still be able to judge between possible reductions using the other things you care about: elegance, tractability, etc. (though if you end up in this situation it probably means you made an error somewhere...)

generalize n-th differentials over real numbers

I'm not sure what you're referring to here...

Also, looking at the possibilities you enumerate again, 3 appear incoherent. Contradictions are for logical systems, if you have a component of your utility function which is monotone increasing in the quantity of blue in the universe and another component which is monotone decreasing in the quantity of blue in the universe, they partially or totally cancel one another but that doesn't result in a contradiction.

Comment by karl on Ontological Crisis in Humans · 2012-12-18T23:48:21.141Z · score: 1 (1 votes) · LW · GW

Why exactly do you call 1 unlikely? The whole metaethics sequence argue in favor in 1 (If I understand what you mean by 1 correctly), so what part of that argument do you think is wrong specifically?

Comment by karl on Ontological Crisis in Humans · 2012-12-18T23:15:47.226Z · score: 0 (0 votes) · LW · GW

There is an entire sequence dedicated to how to define concepts and the specific problem of categories as they matter for your utility function is studied in this post where it is argued those problems should be solved by moral arguments and the whole metaethics sequence argue for the fact that moral arguments are meaningful.

Now if you're asking me if we have a complete reduction of some concept relevant to our utility function all the way down to fundamental physics then the answer is no. That doesn't mean that partial reductions of some concepts potentially relevant to our utility function has never been accomplished or that complete reduction is not possible.

So are you arguing that the metaethics sequence is wrong and that moral arguments are meaningless or are you arguing that the GRT is wrong and that reduction of the concept which appear in your utility function is impossible despite them being real or what? I still have no idea what it is that you're arguing for exactly.

Comment by karl on Ontological Crisis in Humans · 2012-12-18T19:26:58.286Z · score: 2 (2 votes) · LW · GW

Actually, dealing with a component of your ontology not being real seems like a far harder problem than the problem of such a component not being fundamental.

According to the Great Reductionist Thesis everything real can be reduced to a mix of physical reference and logical reference. In which case, if every component of your ontology is real, you can obtain a formulation of your utility function in terms of fundamental things.

The case where some components of your ontology can't be reduced because they're not real and where your utility function refer explicitly to such entity seem considerably harder, but that is exactly the problem that someone who realize God doesn't actually exist is confronted with, and we do manage that kind of ontology crisis.

So are you saying that the GRT is wrong or that none of the things that we value are actually real or that we can't program a computer to perform reduction (which seems absurd given that we have managed to perform some reductions already) or what? Because I don't see what you're trying to get at here.

Comment by karl on By Which It May Be Judged · 2012-12-11T03:52:51.875Z · score: 3 (3 votes) · LW · GW

Also I totally think there was a respectable hard problem

So you do have a solution to the problem?

Comment by karl on The Useful Idea of Truth · 2012-10-03T01:41:17.029Z · score: 1 (1 votes) · LW · GW

Proposition p is meaningful relative to the collection of possible worlds W if and only if there exist w, w' in W such that p is true in the possible world w and false in the possible world w'.

Then the question become: to be able to reason in all generality what collection of possible worlds should one use?

That's a very hard question.

Comment by karl on Harry Potter and the Methods of Rationality discussion thread, part 13, chapter 81 · 2012-04-02T20:19:34.213Z · score: 1 (1 votes) · LW · GW

It's explained in detail in chapter 25 that the genes that make a person a wizard do not do so by building some complex machinery which allow you to become a wizard; the genes that make you a wizard constitute a marker which indicate to the source of magic that you should be allowed to cast spells.

Comment by karl on Harry Potter and the Methods of Rationality discussion thread, part 13, chapter 81 · 2012-04-02T19:31:24.389Z · score: 0 (0 votes) · LW · GW

I don't think taking polyjuice modify your genetic code. If that was the case, using polyjuice to take the form of a muggle or a squib would leave you without your magical powers.

Comment by karl on Formulas of arithmetic that behave like decision agents · 2012-02-03T21:34:09.482Z · score: 1 (1 votes) · LW · GW

I'm confused. What's the distinction between x and $\\underline\{x\}$ here?

Comment by karl on Rational vs. real utilities in the cousin_it-Nesov model of UDT · 2012-01-29T19:32:44.982Z · score: 1 (1 votes) · LW · GW

On the other hand, as I have shown, if you chose t sufficiently large the algorithm I recommend in my post will necessarily end up one boxing if the formal system used is sound.

This is incorrect, as Zeno had shown more than 2000 years ago. It could be that your inference system generates an infinite sequence of statements of the form A()=1 ⇒ U()≥Si with sequence {Si} tending to, say, 100, but with all Si<100, so that A()=1 loses to A()=2 no matter how large the timeout is.

That's why you enumerate all proofs of statement of the form A()=a ⇒ U()≥u (where u is rational number in canonical form). It's a well known fact that it is possible to enumerate all the provable statements in a given formal system without skipping any.

This is a possible behavior, even if the formal system used is sound, if one use rational intervals as you recommend.

Not if we use a Goedel statement/chicken rule failsafe like the one discussed in Slepnev's article you linked to.

There are some subtleties about doing this in the interval setting which made me doubt that it could be done, but after thinking about it some more I must admit that it is possible.

But I think that my algorithm for the non-oracle setting is still valuable.

Comment by karl on Rational vs. real utilities in the cousin_it-Nesov model of UDT · 2012-01-29T17:13:13.219Z · score: 0 (0 votes) · LW · GW

That doesn't actually work. Take the Newcomb's problem. Suppose that your formal system quickly prove that A()≠1. Then it conclude that A()=1 ⇒ U()∈[0,0], on the other it end up concluding correctly that A()=2 ⇒ U()∈[1000,1000] so it end up two boxing. This is a possible behavior, even if the formal system used is sound, if one use rational intervals as you recommend. On the other hand, as I have shown, if you chose t sufficiently large the algorithm I recommend in my post will necessarily end up one boxing if the formal system used is sound. (using intervals was actually the first idea I had when coming to term with the problem detailed in the post, but it simply doesn't work.)

Comment by karl on Rational vs. real utilities in the cousin_it-Nesov model of UDT · 2012-01-29T16:45:32.925Z · score: 1 (1 votes) · LW · GW

Rationals are dense in the reals so there are always a rational value between any two real numbers. So for example, if it can be proven in your formal system that A()=1 ⇒ U()=π and this happen to be the maximal utility attainable, there will be a rational number x greater than the utility that can be achieved by any other action and such that x≤π, so you will be able to prove that A()=1 ⇒ U()≥x and so the first action will end up being taken because by definition of x it is greater than the utility that can be obtained by taking any other action.

Comment by karl on Harry Potter and the Methods of Rationality discussion thread, part 8 · 2011-08-29T17:22:30.052Z · score: 2 (2 votes) · LW · GW

I don't think the part about summoning Death is a reference to anything. After all, we already know what the incarnations of Death are in MOR. And it looks like the conterspell to dismiss Death is lost no more thanks to Harry...

Comment by karl on Dungeons and Discourse implementation · 2011-07-30T14:21:38.828Z · score: 0 (0 votes) · LW · GW

What if that square is now occupied?

Comment by karl on Dungeons and Discourse implementation · 2011-07-28T17:43:41.568Z · score: 1 (1 votes) · LW · GW

So you're atheists are actually nay theists... If that's the case I have difficulty imagining how a group containing both atheists and theists could work at all...

Comment by karl on Dungeons and Discourse implementation · 2011-07-25T14:15:38.152Z · score: 1 (1 votes) · LW · GW

Color me interested.

I think the character creation rules really should be collected together. Has is, I can't figure out how your supposed to determine many of your initial statistics (wealth level, hit points, speed...). Also I don't like the fact that the number of points you have to distribute among the big five and among the skills are random. And of course a system where you simply divide X points among your stats however you wish is completely broken. You should really think about introducing some limit on the number of point you can put into one your stat and some sort of diminishing return.

But even if many part of the design are open to criticism what you've created is still very awesome.

ETA: There are a lot of references in the rules to a Faith stat, except that there is no such stat! Also, the righteousness stat is often called morality in the rules.

Comment by karl on Self-improving AGI: Is a confrontational or a secretive approach favorable? · 2011-07-06T15:36:06.385Z · score: 1 (1 votes) · LW · GW

Here in Spain, France, the UK, the majority of people are Atheists.

I would be interested in knowing where you got your numbers because the statistics I found definitively disagreed with this.

Comment by karl on People who want to save the world · 2011-05-18T01:56:22.549Z · score: 1 (1 votes) · LW · GW

Congratulation for raising the expected utility of the future!


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# find the Local minimum or Maximum Of the Function

• Dec 11th 2010, 08:59 AM
ZOOZ
find the Local minimum or Maximum Of the Function
how i find the Local minimum or Maximum?

(there is an option that there is no Local minimum or Maximum point in this Function)

BUT i need to show the way.

thanks.
• Dec 11th 2010, 09:41 AM
Soroban
Hello, ZOOZ!

Quote:

$\text{Find the local minimum or maximum (if any): }\;f(x) \:=\:\cot x - \tan x$

I did some simplifying first . . .

$\displaystyle f(x) \;=\; \cot x - \tan x \;=\;\frac{\cos x}{\sin x} - \frac{\sin x}{\cos x} \;=\;\frac{\cos^2\!x - \sin^2\!x}{\sin x\cos x}$

. . . . $\displaystyle =\;\frac{\cos2x}{\frac{1}{2}\sin2x} \;=\;2\cot2x$

Set the derivative equal to zero: . $f'(x) \;=\;-4\csc^2\!2x \;=\;0$

. . But $|\csc2x| \:\ge\:1$ . . . $f'(x)$ can never equal zero.

Therefore, $\,f(x)$ has no maximum or minimum.

• Dec 12th 2010, 01:11 AM
ZOOZ
i think you mistake with Absolute minimum and Absolute maximum .

BUT to find Local minimum or Maximum we not must that the derivative will equal to zero.


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# Step-by-step Solution

Go!
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asin
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sinh
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## Step-by-step explanation

Problem to solve:

$\frac{d}{dx}\left(\frac{\pi}{2}-\arctan\left(x\right)\right)$

Learn how to solve sum rule of differentiation problems step by step online.

$\frac{d}{dx}\left(\frac{\pi}{2}\right)+\frac{d}{dx}\left(-\arctan\left(x\right)\right)$

Learn how to solve sum rule of differentiation problems step by step online. Find the derivative (d/dx)(pi/2-arctan(x)) using the sum rule. The derivative of a sum of two functions is the sum of the derivatives of each function. The derivative of the constant function (\frac{\pi}{2}) is equal to zero. The derivative of a function multiplied by a constant is equal to the constant times the derivative of the function. Taking the derivative of arctangent.

$\frac{-1}{1+x^2}$

### Problem Analysis

$\frac{d}{dx}\left(\frac{\pi}{2}-\arctan\left(x\right)\right)$

### Main topic:

Sum rule of differentiation

~ 0.12 seconds


text:
SEARCH HOME
 Math Central Quandaries & Queries
 Question from Simran: what is difference between f(x) compared to f(-x) on graph

Hi Simran,

I want to illustrate using the graph of a function.

Suppose $x = 2.5$ then from the graph it looks like $f(x) = f(2.5)$ is approximately $\large \frac12$ and $f(-x) = f(-2.5)$ is approximately $3 \large \frac{1}{2}.$ Also if $x=-5$ then $f(x) = f(-5)$ is a little less than $-5$ and $f(-x) = f(5)$ is a little less than $2 \large \frac{1}{2}.$

The point is that the relationship between $f(x)$ and $f(-x)$ depends very much on the value of $x$ and the shape of the graph.

I hope this helps,
Penny


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# Random Rectangles

Write a class named RectangleDrawer that extends a TwoButtonBreadboard. When executed, this class should prompt the user to enter some number of rectangles to draw, and then -- when the user clicks the "Go" button -- this number of rectangles should be drawn, with one rectangle drawn each 100 milliseconds. The rectangles themselves should be randomly colored red, blue, green, or yellow. The sizes of the rectangles should be randomly chosen to be somewhere between 20 and 100 pixels. The location of each rectangle should be randomly chosen to be somewhere in the central drawing area. However, each rectangle must be entirely visible in the window (i.e., there should be no rectangles "hanging off the edge of the window").

Check out the video below to see what this program should look like when you are done...

### Hint:

Having trouble knowing when your rectangle is vertically "in the canvas area"? Check out the code below with puts a rectangle in the canvas with a 10 pixel border:

import java.awt.Color;
import acm.graphics.GRect;

public class AcmFun extends OneButtonBreadboard {
public void run() {
int canvasHeight = this.getRegionPanel(CENTER).getHeight();
int canvasWidth = this.getWidth();
GRect rect = new GRect(10, 10,canvasWidth - 20,canvasHeight - 20);
rect.setFilled(true);
rect.setFillColor(Color.RED);


text:
# Publications of B. Sudakov

BACK TO COATI PUBLICATION INDEX

## Publications of B. Sudakov

 Articles in journal or book's chapters
1. B. Reed and B. Sudakov. List colouring when the chromatic number is close to the order of the graph. Combinatorica, 25(1):117--123, 2005. [bibtex-entry]

2. B. Reed and B. Sudakov. Asymptotically the list colouring constants are 1. J. Combin. Theory Ser. B, 86(1):27--37, 2002. [bibtex-entry]

 Conference's articles
1. B. Reed and B. Sudakov. List colouring of graphs with at most $(2-o(1))\chi$ vertices. In Proceedings of the International Congress of Mathematicians, Vol. III (Beijing, 2002), Beijing, pages 587--603, 2002. Higher Ed. Press. [bibtex-entry]

BACK TO COATI PUBLICATION INDEX


text:
# The use-once join (or matching in bipartite graphs with Spark)

A bipartite graph is a graph whose vertices can be divided into two sets such that no two vertices in the same set are connected by an edge in the graph. A matching is subset of the edges in the graph such that no two edges in the subset share an endpoint.

Bipartite graphs naturally occur implicitly when working with dataframes. Suppose we are given two dataframes and we want to compute a join between them for some join condition. Then the rows of the two dataframes are the disjoint sets of vertices and there is an edge between two vertices if the join condition between the two corresponding rows is satisfied.

Two dataframes and a join condition induce a bipartite graph and the join type decides how to use the graph to produce the result of the join from the graph. For example, the result of an inner join has exactly as many rows as there are edges in the implicit graph. Similarly, the result of a left join has one row for each edge in the graph and each vertex in the left set with degree 0.

So what type of join is equivalent to a bipartite matching? It corresponds to a join where each row can be used only once in the join. In this post we show how to compute a join corresponding to a matching in a bipartite graph in Spark and describe a use case for this.

## Matching in bipartite graphs

We need to ensure that each row of each dataframe occurs exactly once in the join. To achieve this we group the rows by the join conditions, i.e., each row in a group has the same values for the columns affected by the join condition. We can do this without actually reducing the dataframes using a window function.

The following example demonstrates how this might look in Spark. Suppose that we are given two dataframes dfA and dfB, both with a column val, and we want to join them on dfA("val") === dfB("val"). Then we may compute a matching as follows.

val window = Window.partitionBy($"val") val matching = dfA.withColumn("_rn", row_number.over(window)) .join( dfB.withColumn("_rn", row_number.over(window)),$"val" === $"val" &&$"_rn" === $"_rn", "inner" ) .drop($"_rn")

Notice that there may be several matchings and the above method finds some matching. The above is in fact a greedy matching algorithm: If there are more than one choice of edges for a vertex, then we pick the first edge (the one connecting the vertex with a vertex with the same _rn). Interestingly, the greedy matching algorithm also finds a maximal matching. Consequently, for simple join conditions like dfA("val") === dfB("val"), this method produces a result using as many rows from each dataframe as possible. For more elaborate join conditions, this does not necessarily hold true.

## Matching entries in log files

We will now describe a use case for the matching. Suppose we are given an event log with the columns: user_idtimestamp, and event_type. There are two event types: A and B. We would like to find all the events of type A immediately followed an event of type B generated by the same user.

To do this, we apply the matching method to a self join of the event log.

val window = Window.partitionBy($"user_id").orderBy(asc("timestamp")) val groupedLogs = logs.withColumn("_rn", row_number.over(window)) val matching = groupedLogs.alias("left") .join( groupedLogs.alias("right"),$"user_id" === $"user_id" &&$"left.event_type" === "A" &&
$"right.event_type" === "B" &&$"_rn" === $"_rn" - 1, "inner" ) .drop($"_rn")

First we order the events generated by each user by the time of occurrence and assign each group row numbers. In the join we specify that we want to match all events of type A (line 7) with an event of type B (line 8) generated by the same user (line 6) if the events occur with no other events between them (line 9).

This post first appeared May 19, 2019 on Medium.


text:
# Calculating initial velocity (horizontal launch)

I need help for the following question:

In a long jump event, an athlete jumps a horizontal distance of 7.56m. The athlete was airborne for 3.04 seconds. The acceleration due to gravity is taken as 9.81m/s^2. Assume that air resistance is negligible, calculate his take-off speed (initial velocity) in m/s.

Any comments and feedback on how to tackle the question will be much appreciated.

Let initial velocity v have horizontal component v1 m/s and vertical component v2 m/s. v2 = 9.81 * 3.04/2 = 14.9112. v1 = 7.56/3.04 = 2.48684211. Speed I guess will be norm of v = $\sqrt{v1^2 + v2^2} = \sqrt{2.48684211^2 + 14.9112^2} = 15.1171515 m/s$.


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# Semantic Image Inpainting with Deep Generative Models

Raymond A. Yeh*   Chen Chen*   Teck Yian Lim   Alexander G. Schwing   Mark Hasegawa-Johnson   Minh N. Do
* indicating equal contribution.

## Abstract

Semantic image inpainting is a challenging task where large missing regions have to be filled based on the available visual data. Existing methods which extract information from only a single image generally produce unsatisfactory results due to the lack of high level context. In this paper, we propose a novel method for semantic image inpainting, which generates the missing content by conditioning on the available data. Given a trained generative model, we search for the closest encoding of the corrupted image in the latent image manifold using our context and prior losses. This encoding is then passed through the generative model to infer the missing content. In our method, inference is possible irrespective of how the missing content is structured, while the state-of-the-art learning based method requires specific information about the holes in the training phase. Experiments on three datasets show that our method successfully predicts information in large missing regions and achieves pixel-level photorealism, significantly outperforming the state-of-the-art methods.

## Citation

@inproceedings{
yeh2017semantic,
title={Semantic Image Inpainting with Deep Generative Models},
author={Yeh$^\ast$, Raymond A. and Chen$^\ast$, Chen and Lim, Teck Yian and Schwing Alexander G. and Hasegawa-Johnson, Mark and Do, Minh N.},
booktitle={Proceedings of the IEEE Conference on Computer Vision and Pattern Recognition},
year={2017},
note = {$^\ast$ equal contribution},
}


text:
# Lesson 5

Construction Techniques 3: Perpendicular Lines and Angle Bisectors

## 5.1: Two Circles (5 minutes)

### Warm-up

The purpose of this activity is to prepare students for constructing a perpendicular line through a point on the line. This figure of two circles of the same radius intersecting also plays a role in the angle bisector construction later in the lesson.

### Launch

Arrange students in groups of 2. After quiet work time, ask students to compare their conjectures to their partner’s and decide whether they are both correct, even if they are different. Follow with a whole-class discussion.

### Student Facing

Points $$A$$ and $$B$$ are each at the centers of circles of radius $$AB$$.

1. Compare the distance $$EA$$ to the distance $$EB$$. Be prepared to explain your reasoning.
2. Compare the distance $$FA$$ to the distance $$FB$$. Be prepared to explain your reasoning.
3. Draw line $$EF$$ and write a conjecture about its relationship with segment $$AB$$.

### Activity Synthesis

The goal of this discussion is to connect this diagram to perpendicular lines. Ask for students to share their conjectures. Connect these ideas back to the previous activity, Human Perpendicular Bisector, in which students conjectured that the set of points the same distance from points $$A$$ and $$B$$ is the perpendicular bisector of segment $$AB$$. In this case, that would mean that line $$EF$$ would be the perpendicular bisector of segment $$AB$$.

## 5.2: Make It Right (10 minutes)

### Activity

This activity invites students to play with straightedge and compass moves to construct a line perpendicular to a given line through a point on the given line. This leads to the next lesson in which students will extend this skill to construct a square inscribed in a circle.

Making dynamic geometry software available gives students an opportunity to choose appropriate tools strategically (MP5).

### Launch

Representation: Internalize Comprehension. Begin with a physical demonstration to remind students of valid straightedge and compass moves. For example, demonstrate how to create a circle centered at a marked point going through another marked point. Ask students how this straightedge and compass move could be used to construct a line perpendicular to $$l$$ that goes through $$C$$.
Supports accessibility for: Conceptual processing; Visual-spatial processing

### Student Facing

Here is a line $$\ell$$ with a point labeled $$C$$:

Use straightedge and compass tools to construct a line perpendicular to $$\ell$$ that goes through $$C$$.

### Launch

Representation: Internalize Comprehension. Begin with a physical demonstration to remind students of valid straightedge and compass moves. For example, demonstrate how to create a circle centered at a marked point going through another marked point. Ask students how this straightedge and compass move could be used to construct a line perpendicular to $$l$$ that goes through $$C$$.
Supports accessibility for: Conceptual processing; Visual-spatial processing

### Student Facing

Here is a line $$\ell$$ with a point labeled $$C$$. Use straightedge and compass moves to construct a line perpendicular to $$\ell$$ that goes through $$C$$.

### Anticipated Misconceptions

Some students may get stuck for longer than is productive. Ask these students to think about how they could use their tools to construct points $$A$$ and $$B$$ like in the warm-up.

### Activity Synthesis

The goal of this discussion is to establish a method for constructing a perpendicular line. Ask 2 or 3 students to explain how they were able to construct a perpendicular line. Connect these methods with the figure students explored in the warm-up (two circles where each center is on the other circle).

## 5.3: Bisect This (20 minutes)

### Activity

In this activity, students take turns using a straightedge and compass to construct an angle bisector. Students trade roles, explaining their thinking and listening, providing opportunities to explain their reasoning and critique the reasoning of others (MP3).

Making dynamic geometry software available gives students an opportunity to choose appropriate tools strategically (MP5).

### Launch

Arrange students in groups of 2. Explain that they will practice their construction technique by drawing and bisecting more angles once they have bisected angle $$ABC$$. Point out that the instructions to “create a ray that divides angle $$CBA$$ into 2 congruent angles” are instructions to bisect that angle. Encourage students to draw angles that look different from the given angle when they are practicing.

Reading, Writing: MLR 3 Clarify, Critique, Correct. Present a first draft of an explanation for how to bisect another angle: “I made three circles, and then I drew a line through the middle.” Prompt discussion by asking, “What were the steps that the author took?” Ask students to clarify and correct the statement. Improved statements should include some of the following: the center and radius of any circle constructed, two specific points that were used to construct a line, the order or time transition words (first, next, then, etc.), and the reasons for decisions made during steps. This will help students develop and understand explanations of how to carry out constructions. Design Principle(s): Maximize meta-awareness; Optimize output (for explanation)
Engagement: Internalize Self Regulation. Demonstrate giving and receiving constructive feedback. Use a structured process and display sentence frames to support productive feedback. For example, “This method works/doesn’t work because…,” “We can agree that…,” and “_____’s idea reminds me of….”
Supports accessibility for: Social-emotional skills; Organization; Language

### Student Facing

Here is an angle:

1. Estimate the location of a point $$D$$ so that angle $$ABD$$ is approximately congruent to angle $$CBD$$.
2. Use compass and straightedge tools to create a ray that divides angle $$CBA$$ into 2 congruent angles. How close is the ray to going through your point $$D$$?
3. Take turns with your partner, drawing and bisecting other angles.
1. For each angle that you draw, explain to your partner how each straightedge and compass move helps you to bisect it.

2. For each angle that your partner draws, listen carefully to their explanation. If you disagree, discuss your thinking and work to reach an agreement.

### Student Facing

#### Are you ready for more?

For thousands of years since the ancient Greeks started playing with straightedge and compass constructions, people strived to find a construction to trisect an arbitrary angle into three equal angles. Many claimed to have found such a construction, but there was always some flaw in their reasoning. Finally, in 1837, Pierre Wantzel used a new field of mathematics to prove it was impossible—which still did not stop some from claiming to have found a construction. If we allow other tools besides just a straightedge and compass, though, it is possible. For example, try this method of using origami (paper folding) to trisect an angle.

### Launch

Arrange students in groups of 2. Explain that they will practice their construction technique by drawing and bisecting more angles once they have bisected angle $$ABC$$. Point out that the instructions to “create a ray that divides angle $$CBA$$ into 2 congruent angles” are instructions to bisect that angle. Encourage students to draw angles that look different from the given angle when they are practicing.

Reading, Writing: MLR 3 Clarify, Critique, Correct. Present a first draft of an explanation for how to bisect another angle: “I made three circles, and then I drew a line through the middle.” Prompt discussion by asking, “What were the steps that the author took?” Ask students to clarify and correct the statement. Improved statements should include some of the following: the center and radius of any circle constructed, two specific points that were used to construct a line, the order or time transition words (first, next, then, etc.), and the reasons for decisions made during steps. This will help students develop and understand explanations of how to carry out constructions. Design Principle(s): Maximize meta-awareness; Optimize output (for explanation)
Engagement: Internalize Self Regulation. Demonstrate giving and receiving constructive feedback. Use a structured process and display sentence frames to support productive feedback. For example, “This method works/doesn’t work because…,” “We can agree that…,” and “_____’s idea reminds me of….”
Supports accessibility for: Social-emotional skills; Organization; Language

### Student Facing

Here is an angle:

1. Estimate the location of a point $$D$$ so that angle $$ABD$$ is approximately congruent to angle $$CBD$$.
2. Use compass and straightedge moves to create a ray that divides angle $$CBA$$ into 2 congruent angles. How close is the ray to going through your point $$D$$?
3. Take turns with your partner, drawing and bisecting other angles.
1. For each angle that you draw, explain to your partner how each straightedge and compass move helps you to bisect it.

2. For each angle that your partner draws, listen carefully to their explanation. If you disagree, discuss your thinking and work to reach an agreement.

### Student Facing

#### Are you ready for more?

For thousands of years since the ancient Greeks started playing with straightedge and compass constructions, people strived to find a construction to trisect an arbitrary angle into three equal angles. Many claimed to have found such a construction, but there was always some flaw in their reasoning. Finally, in 1837, Pierre Wantzel used a new field of mathematics to prove it was impossible—which still did not stop some from claiming to have found a construction. If we allow other tools besides just a straightedge and compass, though, it is possible. For example, try this method of using origami (paper folding) to trisect an angle.

### Anticipated Misconceptions

Some students may get stuck for longer than is productive. Ask these students how they could use their tools to construct a segment they know how to cut in half like they did in the warm-up and in the last activity. If they need further help, ask them to construct a circle centered at $$B$$ and highlight the segment created when connecting the two intersection points.

### Activity Synthesis

The purpose of this discussion is to connect the previous construction of a perpendicular line with the construction of an angle bisector. Select 2 or 3 students to display their responses for all to see. Ask them to explain how they were able to construct a ray that divides the angle into two congruent parts.

Introduce the term angle bisector. Recall that “bi” means “two” and “sect” means “cut,” and so bisecting an angle literally means to cut the angle into two congruent angles. Consider making a connection to the previous activity by demonstrating the construction using geometry software and increasing the measure of the angle until it reaches 180 degrees.

If time allows, ask students to trace the angle along with their estimation point $$D$$ and overlay several tracing papers to show a cluster of points. Compare the class’s estimation cluster with the angle bisector.

## Lesson Synthesis

### Lesson Synthesis

The purpose of this discussion is to connect the basic construction of two circles going through each other’s center with the constructions for a line perpendicular to a given line and an angle bisector. Display images of the construction from each activity for all to see. Ask students:

• “What was the first step in each of these constructions?” (In each construction, the first step was to create a circle that intersects the given line or angle in two places.)
• “The first circle creates two points that are the same distance away from the center. Now, where do you see the two circles that go through each other’s center in each of the constructions?” (In each construction, the 2 points created from the first step become centers of circles that go through each other’s center.)

Highlight the 2 points created from the first step of each construction, and then highlight the 2 circles that go through each other’s centers in the second step.

Explain that each construction is really a perpendicular bisector. For the angle bisector, connecting the two equidistant points forms an isosceles triangle. This surfaces a surprising and important connection. The angle bisector of an isosceles triangle seems to also be the perpendicular bisector of the base. Tell students that this is only a conjecture at this point, but this connection between the angle bisector of an isosceles triangle and the perpendicular bisector of its base will play an important role in the next unit.


text:
## Stream: maths

### Topic: horner polynomials

#### Chris Hughes (Nov 29 2018 at 19:10):

I remember @Johannes Hölzl saying something in Orsay about having an interface for polynomials in horner form? What exactly did he mean by this? Is something like this worth having, and is it worth proving the equation lemmas for this, and proving it in a semiring? It does make some proofs easier.

@[elab_as_eliminator] def rec_on_horner {α : Type*}
[nonzero_comm_ring α] [decidable_eq α]
{M : polynomial α → Sort*} : Π (p : polynomial α),
M 0 →
(Π p a, coeff p 0 = 0 → a ≠ 0 → M p → M (p + C a)) →
(Π p, p ≠ 0 → M p → M (p * X)) →
M p
| p := λ M0 MC MX,
if hp : p = 0 then eq.rec_on hp.symm M0
else
have wf : degree (p /ₘ X) < degree p,
from degree_div_by_monic_lt _ monic_X hp (by rw degree_X; exact dec_trivial),
by rw [← mod_by_monic_add_div p monic_X, mod_by_monic_X, ← coeff_zero_eq_eval_zero,
exact
if hcp0 : coeff p 0 = 0
then by rw [hcp0, C_0, add_zero];
exact MX _ (λ h : p /ₘ X = 0, by simpa [h, hcp0] using hp)
(rec_on_horner _ M0 MC MX)
else MC _ _ (coeff_mul_X_zero _) hcp0 (if hpX0 : p /ₘ X = 0
then show M (p /ₘ X * X), by rw [hpX0, zero_mul]; exact M0
else MX (p /ₘ X) hpX0 (rec_on_horner _ M0 MC MX))
using_well_founded {dec_tac := tactic.assumption}

#### Johannes Hölzl (Nov 30 2018 at 09:01):

My idea was a little bit different. Instead of using p + C a and assuming coeff p 0 = 0 we only use

def horner p a := p * X + C a

My hope is that some proofs are easier using this construction, especially proofs about coeff degree etc.
And it would not be limited to a recursor, but would also include a lot of rewrite rules for horner.

I also tried to define polynomials using the horner scheme, i.e. subtype on list with an additional non-leading zero assumption.

Last updated: May 06 2021 at 19:30 UTC


text:
Compute the CDF of a Random Variable - Maple Programming Help

Compute the CDF of a Random Variable

 Description Compute the CDF (cumulative distribution function) of a random variable.

Define a random variable using the built-in probability distributions or by creating a custom distribution. The parameters of the distributions can be symbolic, numeric, or a mix.

 > $X:=\mathrm{Statistics}\left[\mathrm{RandomVariable}\right]\left(\mathrm{Weibull}\left({a},{b}\right)\right):$

Evaluate the CDF of the random variable at a point. The point can be numeric or symbolic.

 > $s:=\mathrm{Statistics}\left[\mathrm{CDF}\right]\left(X,{t}\right)$
 $\colorbox[rgb]{0,0,0}{{s}}\colorbox[rgb]{0,0,0}{{:=}}\colorbox[rgb]{0,0,0}{{{}}\begin{array}{cc}\colorbox[rgb]{0,0,0}{{0}}& \colorbox[rgb]{0,0,0}{{t}}\colorbox[rgb]{0,0,0}{{<}}\colorbox[rgb]{0,0,0}{{0}}\\ \colorbox[rgb]{0,0,0}{{1}}\colorbox[rgb]{0,0,0}{{-}}{\colorbox[rgb]{0,0,0}{{ⅇ}}}^{\colorbox[rgb]{0,0,0}{{-}}{\left(\frac{\colorbox[rgb]{0,0,0}{{t}}}{\colorbox[rgb]{0,0,0}{{a}}}\right)}^{\colorbox[rgb]{0,0,0}{{b}}}}& \colorbox[rgb]{0,0,0}{{\mathrm{otherwise}}}\end{array}$ (1)
 Commands Used


text:
# Flow Through Pipes

 Question 1
A high velocity water jet of cross section area $= 0.01 m^2$ and velocity $= 35 m/s$ enters a pipe filled with stagnant water. The diameter of the pipe is 0.32 m. This high velocity water jet entrains additional water from the pipe and the total water leaves the pipe with a velocity 6m/s as shown in the figure.

The flow rate of entrained water is _______litres/s (round off to two decimal places).
 A 152.45 B 125.36 C 132.55 D 148.45
GATE ME 2021 SET-2   Fluid Mechanics
Question 1 Explanation:
Applying continuity equation:
\begin{aligned} 35 \times 0.01+\dot{q} &=6 \times \frac{\pi}{4} \times 0.32^{2} \\ \dot{q} &=0.13255 \mathrm{~m}^{3} / \mathrm{s} \\ \dot{q} &=132.55 \; \mathrm{lit} / \mathrm{s} \end{aligned}
 Question 2
Water flows out from a large tank of cross-sectional area $A_t=1\; m^2$ through a small rounded orifice of cross-sectional area $A_0=1\; cm^2$, located at $y=0$. Initially the water level, measured from $y=0$, is $H=1m$. The acceleration due to gravity is $9.8 \; m/s^2$.

Neglecting any losses, the time taken by water in the tank to reach a level of $y=H/4$ is ________seconds (round off to one decimal place).
 A 2146.2 B 2258.8 C 1245.6 D 3251.2
GATE ME 2021 SET-2   Fluid Mechanics
Question 2 Explanation:

\begin{aligned} \dot{m}_{\text {in }}-\dot{m}_{\text {out }} &=\frac{d m_{w}}{d t} \;\;\;\; \left(\dot{m}_{\text {in }}=0\right)\\ -\rho A_{0} \sqrt{2 g H} &=\frac{d}{d t}[A \times H \times \rho] \\ d t &=-\frac{A}{A_{0}} \times \frac{1}{\sqrt{2 g}} \times \frac{d H}{\sqrt{H}} \\ t &=-\frac{A}{A_{0}} \times \frac{1}{\sqrt{2 g}} \int_{1}^{0.25} \frac{d H}{\sqrt{H}} \\ t &=\frac{A}{A_{0}} \times \frac{1}{\sqrt{2 g}}(2 \sqrt{H})_{0.25}^{1} \\ t &=\frac{1}{10^{-4}}(2-1) \times \frac{1}{\sqrt{2 \times 9.81}} \\ t &=2258.8 \mathrm{~s} \end{aligned}
 Question 3
Water flows through two different pipes A and B of the same circular cross-section but at different flow rates. The length of pipe A is 1.0 m and that of pipe B is 2.0 m. The flow in both the pipes is laminar and fully developed. If the frictional head loss across the length of the pipes is same, the ratio of volume flow rates $Q_B/Q_A$ is ______ (round off to two decimal places).
 A 0.24 B 0.5 C 0.96 D 0.68
GATE ME 2019 SET-2   Fluid Mechanics
Question 3 Explanation:
Given data,
Water is flowing in two different pipes A and B.

Given that, $A_{A} = A_{B} \Rightarrow d_{A} = d_{B}$ Flow is laminar in both pipes.
In laminar flow,
$\Delta \mathrm{P}=\frac{32 \mu \mathrm{VL}}{\mathrm{d}^{2}}=\frac{128 \mu \mathrm{LQ}}{\pi \mathrm{d}^{4}}$
and $\mathrm{h}_{\mathrm{f}}=\frac{\Delta \mathrm{P}}{\rho \mathrm{g}}=\frac{128 \mu \mathrm{LQ}}{\rho \pi \mathrm{gd}^{4}}$
since, $\mu, \pi, \mathrm{g} \text{and } \rho$ are all same for both pipes, we can write
$\mathrm{h}_{\mathrm{f}} \propto \mathrm{L} \mathrm{Q}(\text { as d is same })$
or $\quad \frac{\mathrm{h}_{\mathrm{fA}}}{\mathrm{h}_{\mathrm{fB}}}=\frac{\mathrm{L}_{\mathrm{A}}}{\mathrm{L}_{\mathrm{B}}} \times \frac{\mathrm{Q}_{\mathrm{A}}}{\mathrm{Q}_{\mathrm{B}}}$
$\text{since,}\quad h_{\mathrm{fA}}=\mathrm{h}_{\mathrm{fB}}$
$\frac{Q_{B}}{Q_{A}}=\frac{L_{A}}{L_{B}}=\frac{1}{2}=0.50$
 Question 4
The wall of a constant diameter pipe of length 1 m is heated uniformly with flux q'' by wrapping a heater coil around it. The flow at the inlet to the pipe is hydrodynamically fully developed. The fluid is incompressible and the flow is assumed to be laminar and steady all through the pipe. The bulk temperature of the fluid is equal to $0^{\circ}C$ at the inlet and $50^{\circ}C$ at the exit. The wall temperatures are measured at three locations, P, Q and R, as shown in the figure. The flow thermally develops after some distance from the inlet. The following measurements are made:

Among the locations P, Q and R, the flow is thermally developed at
 A P, Q and R B P and Q only C Q and R only D R only
GATE ME 2019 SET-1   Fluid Mechanics
Question 4 Explanation:

Energy balance: between points 1 and 5:
$\dot{\mathbf{q}} \times \pi \mathrm{d} \times \mathrm{L}=\dot{\mathrm{m}} \mathrm{c}_{\mathrm{p}}\left(\mathrm{T}_{\mathrm{s}}-\mathrm{T}_{\mathrm{i}}\right)$
$\dot{\mathbf{q}} \times \pi \times \mathrm{d} \times 1=50 \mathrm{m} \mathrm{c}_{\mathrm{p}}$
$\dot{\mathrm{q}}=\frac{50 \mathrm{mc}_{\mathrm{p}}}{\pi \mathrm{d}}$
Similarly,
Energy balance between points 1 and 2:
$\mathrm{\dot{m}c}_{\mathrm{p}}\left(\mathrm{T}_{2}-\mathrm{T}_{1}\right)=\dot{\mathrm{q}} \times \pi \times \mathrm{d} \times 0.4$
$\dot{\mathrm{m}} \times \mathrm{c}_{\mathrm{p}} \times\left(\mathrm{T}_{2}-0\right)=\frac{50 \mathrm{m} \mathrm{c}_{\mathrm{p}}}{\pi \mathrm{d}} \times \pi \mathrm{d} \times 0.4$
$\mathrm{T}_{2}=20^{\circ} \mathrm{C}$
Energy balance between points 1 and 3:
$\dot{\mathrm{m}} \mathrm{c}_{\mathrm{p}}\left(\mathrm{T}_{3}-\mathrm{T}_{1}\right)=\dot{\mathrm{q}} \times \pi \times \mathrm{d} \times 0.6$
$\dot{\mathrm{m}} \times \mathrm{c}_{p} \times\left(\mathrm{T}_{3}-0\right)=\frac{50 \mathrm{mc}_{\mathrm{p}}}{\pi \mathrm{d}} \times \pi \mathrm{d} \times 0.6 \\ \mathrm{T}_{3}=30^{\circ} \mathrm{C}$
Energy balance between points 1 and 4
$\mathrm{mc}_{p}\left(\mathrm{T}_{4}-\mathrm{T}_{1}\right)=\dot{\mathrm{q}} \times \pi \times \mathrm{d} \times 0.8$
$\dot{\mathrm{m}} \times \mathrm{c}_{\mathrm{p}} \times\left(\mathrm{T}_{4}-0\right)=\frac{50 \mathrm{mc}_{\mathrm{p}}}{\pi \mathrm{d}} \times \pi \mathrm{d} \times 0.8$
$\mathrm{T}_{4}=40^{\circ} \mathrm{C}$
In constant heat flux condition:
$\mathrm{Nu}=4.36(\text { constant })$
$\frac{\mathrm{hd}}{\mathrm{k}}=4.36$
$\mathrm{h}=4.36 \frac{\mathrm{k}}{\mathrm{d}}=\text { constant (because k and d are constant )}$
Heat flux $\mathrm{q}=\mathrm{h} \Delta \mathrm{T}$
q = constant
h = constant
that means, $\Delta \mathrm{T}$= constant (for steady and fully developed flow).
$(\Delta \mathrm{T})_{\mathrm{P}}=\mathrm{T}_{\mathrm{p}}-\mathrm{T}_{2}=50-20=30^{\circ} \mathrm{C}$
$(\Delta \mathrm{T})_{\mathrm{Q}}=\mathrm{T}_{\mathrm{Q}}-\mathrm{T}_{3}=80-30=50^{\circ} \mathrm{C}$
$(\Delta \mathrm{T})_{\mathrm{R}}=\mathrm{T}_{\mathrm{R}}-\mathrm{T}_{4}=90-40=50^{\circ} \mathrm{C}$
$(\Delta \mathrm{T})_{\mathrm{Q}}=(\Delta \mathrm{T})_{\mathrm{R}}=50^{\circ} \mathrm{C}$
The flow is fully developed at locations Q and R

 Question 5
Water flows through a pipe with a velocity given by $\vec{V}=\left ( \frac{4}{t}+x+y \right )\hat{j}$ m/s, where $\hat{j}$ is the unit vector in the y direction, $t(\gt 0)$ is in seconds, and x and y are in meters. The magnitude of total acceleration at the point (x,y)=(1,1) at t=2s is ________ $m/s^2$.
 A 1 B 2 C 3 D 4
GATE ME 2019 SET-1   Fluid Mechanics
Question 5 Explanation:
$\begin{array}{l} \overrightarrow{\mathrm{V}}=\left(\frac{4}{\mathrm{t}}+\mathrm{x}+\mathrm{y}\right) \hat{\mathrm{j}} \\ \therefore \mathrm{u}=0 \\ \mathrm{v}=\left(\frac{4}{\mathrm{t}}+\mathrm{x}+\mathrm{y}\right) \\ \mathrm{w}=0 \\ \end{array}$
\begin{aligned} \mathrm{a}_{\mathrm{x}}&=\mathrm{u} \frac{\partial \mathrm{u}}{\partial \mathrm{x}}+\mathrm{v} \frac{\partial \mathrm{u}}{\partial \mathrm{y}}+\mathrm{w} \frac{\partial \mathrm{u}}{\partial \mathrm{z}}+\frac{\partial \mathrm{u}}{\partial \mathrm{t}}=0+0+0+0=0 \\ \mathrm{a}_{\mathrm{z}}&=\mathrm{u} \frac{\partial \mathrm{w}}{\partial \mathrm{x}}+\mathrm{v} \frac{\partial \mathrm{w}}{\partial \mathrm{y}}+\mathrm{w} \frac{\partial \mathrm{w}}{\partial \mathrm{z}}+\frac{\partial \mathrm{w}}{\partial \mathrm{t}}=0+0+0+0=0 \\ \mathrm{a}_{\mathrm{y}}&=\mathrm{a}_{\mathrm{y}}= \mathrm{u} \frac{\partial \mathrm{v}}{\partial \mathrm{x}}+\mathrm{v} \frac{\partial \mathrm{v}}{\partial \mathrm{y}}+\mathrm{w} \frac{\partial \mathrm{v}}{\partial \mathrm{z}}+\frac{\partial \mathrm{v}}{\partial \mathrm{t}} \\ & =\mathrm{v} \frac{\partial \mathrm{v}}{\partial \mathrm{y}}+\frac{\partial \mathrm{v}}{\partial \mathrm{t}} \\ & =\left(\frac{4}{\mathrm{t}}+\mathrm{x}+\mathrm{y}\right) \cdot(1)+\left(\frac{-4}{\mathrm{t}^{2}}\right) \end{aligned}
$\begin{array}{l} =x+y+\frac{4}{t}-\frac{4}{t^{2}} \\ =1+1+\frac{4}{2}-\frac{4}{2^{2}} \\ =3 \\ |\vec{a}|=\sqrt{a_{x}^{2}+a_{y}^{2}+a_{z}^{2}}=\sqrt{0^{2}+3^{2}+0^{2}}=3 \mathrm{m} / \mathrm{s}^{2} \end{array}$
 Question 6
Air flows at the rate of 1.5 $m^{3}/s$ through a horizontal pipe with a gradually reducing crosssection as shown in the figure. The two cross-sections of the pipe have diameters of 400 mm and 200 mm. Take the air density as 1.2 kg/$m^{3}$ and assume inviscid incompressible flow. The change in pressure$p_{2}-p_{1}$ (in kPa) between sections 1 and 2 is
 A -1.28 B 2.56 C -2.13 D 1.28
GATE ME 2018 SET-2   Fluid Mechanics
Question 6 Explanation:
\begin{aligned} \frac{P_{1}}{\rho g}+\frac{V_{1}^{2}}{2 g}+Z_{1} &=\frac{P_{2}}{\rho g}+\frac{V_{2}^{2}}{2 g}+Z_{2} \qquad \left\{Z= comstant\right\}\\ A_{1} &=\frac{\pi}{4} \times 0.4^{2} \\ A_{1} &=0.1256 \mathrm{m}^{2} \\ A_{2} &=\frac{\pi}{4} \times 0.2^{2} \\ \mathrm{A} 2 &=0.0314 \mathrm{m}^{2} \\ \frac{P_{2}-P_{1}}{\rho g} &=\frac{V_{1}^{2}-V_{2}^{2}}{2 g}\\ \frac{P_{2}-P_{1}}{1.2} &=\frac{1}{2}\left[\frac{Q^{2}}{A_{1}^{2}}-\frac{Q^{2}}{A_{2}^{2}}\right] \\ P_{2}-P_{1} &=0.6 \times(1.5)^{2}\left[\frac{1}{A_{1}^{2}}-\frac{1}{A_{2}^{2}}\right] \\ P_{2}-P_{1} &=1.35\left[\frac{1}{0.01577}-\frac{1}{0.00985}\right] \\ &=1.35[63.41-1015.22] \\ &=-1284.94=-1.28 \mathrm{kPa} \end{aligned}
 Question 7
Consider steady flow of an incompressible fluid through two long and straight pipes of diameters $d_{1}$ and $d_{2}$ arranged in series. Both pipes are of equal length and the flow is turbulent in both pipes. The friction factor for turbulent flow though pipes is of the form. $f=k(Re)^{-n}$, where K and n are known positive constants and Re is the Reynolds number. Neglecting minor losses, the ratio of the frictional pressure drop in pipe 1 to that in pipe 2,$(\frac{\Delta P_{1}}{\Delta P_{2}})$, is given by
 A $(\frac{d_{2}}{d_{1}})^{(5-n)}$ B $(\frac{d_{2}}{d_{1}})^{5}$ C $(\frac{d_{2}}{d_{1}})^{(3-n)}$ D $(\frac{d_{2}}{d_{1}})^{(5+n)}$
GATE ME 2017 SET-1   Fluid Mechanics
Question 7 Explanation:

For series connection,
$Q_{1}=Q_{2}=Q$
As given $\quad l_{1}=l_{2}=l$
For horizontal pipe,
\begin{aligned} h_{f}&=\Delta p \\ h_{f_{1}}&=\frac{f_{1} L Q^{2}}{12.1 d_{1}^{5}} ; \quad h_{f_{2}}=\frac{f_{2} L Q^{2}}{12.1 d_{2}^{5}} \\ \frac{\Delta P_{1}}{\Delta P_{2}}&=\frac{f_{1}}{d_{1}^{5}} \times \frac{d_{2}^{5}}{f_{2}} \\ f&=\frac{k}{\operatorname{Re}^{n}}=\frac{k}{\left(\frac{V D}{v}\right)^{n}} \\ Q&=\frac{\pi}{4} D^{2} \times V \\ V &\propto \frac{1}{D^{2}} \\ f &\propto \frac{1}{\left(\frac{1}{D^{2}} D\right)^{n}} \end{aligned}
By equation (i) $f \propto D^{n}$
$\frac{\Delta P_{1}}{\Delta P_{2}}=\frac{d_{1}^{n}}{d_{1}^{5}} \times \frac{d_{2}^{5}}{d_{2}^{n}}$
$\frac{\Delta P_{1}}{\Delta P_{2}}=\frac{d_{2}^{5-n}}{d_{1}^{5-n}}$
 Question 8
Water (density=1000 kg/$k^{3}$ ) at ambient temperature flows through a horizontral pipe of uniform cross section at the rate of kg/s. If the pressure drop across the pipe is 100 kPa, the minimum power required to pump the water across the pipe, in watts. is_________
 A 10 B 100 C 1000 D 10000
GATE ME 2017 SET-1   Fluid Mechanics
Question 8 Explanation:
\begin{aligned} \rho &=1000 \mathrm{kg} / \mathrm{m}^{3} \\ m &=1 \mathrm{kg} / \mathrm{s} \\ \Delta p &=100 \mathrm{kPa}=100 \times 10^{3} \mathrm{Pa} \\ P &=\rho Q g h_{f} \\ &=Q \Delta p \\ &=\frac{m}{\rho} \Delta p=\frac{1}{1000} \times 100 \times 10^{3} \\ &=100 \mathrm{W} \end{aligned}
 Question 9
A channel of width 450 mm branches into two sub-channels having width 300 mm and 200 mm as shown in figure. If the volumetric flow rate (taking unit depth) of an incompressible flow through the main channel is 0.9 $m^{3}$/s and the velocity in the sub-channel of width 200 mm is 3 m/s, the velocity in the sub-channel of width 300 mm is _____________ m/s.
Assume both inlet and outlet to be at the same elevation.

 A 2m/s B 1m/s C 4m/s D 5m/s
GATE ME 2016 SET-3   Fluid Mechanics
Question 9 Explanation:

Given data:
\begin{aligned} b_{1}&=450 \mathrm{mm}=0.45 \mathrm{m} \\ b_{2}&=300 \mathrm{mm}=0.30 \mathrm{m} \\ b_{3}&=200 \mathrm{mm}=0.20 \mathrm{m} Depth: \quad h&=1 \mathrm{m} \\ Q_{1}&=0.9 \mathrm{m}^{3} / \mathrm{s} \\ V_{3}&=3 \mathrm{m} / \mathrm{s} \\ V_{2}&=? \end{aligned}
Apply equation of continuity, we get
\begin{aligned} Q_{1}&=Q_{2}+Q_{3}\\ Q_{1}&=A_{2} V_{2}+A_{3} V_{3} \\ Q_{1}&=b_{2} h V_{2}+b_{3} h V_{3} \\ 0.9&=0.30 \times 1 \times V_{2}+0.20 \times 1 \times 3 \\ 0.9&=0.30 \mathrm{V}_{2}+0.60\\ or \quad 0.30 \mathrm{V}_{2}&=0.9-0.6=0.30 \\ or\qquad v_{2}=\frac{0.30}{0.30}&=1 \mathrm{m} / \mathrm{s} \end{aligned}
 Question 10
Three parallel pipes connected at the two ends have flow-rates $Q_{1}$ , $Q_{2}$ and $Q_{3}$ respectively, and the corresponding frictional head losses are $h_{L1},h_{L2} \; and \; h_{L3}$ respectively. The correct expressions for total flow rate (Q) and frictional head loss across the two ends ($h_{L}$ ) are
 A $Q=Q_{1}+Q_{2}+Q_{3};$ $h_{L}=h_{L1}+h_{L2}+h_{L3}$ B $Q=Q_{1}+Q_{2}+Q_{3};$ $h_{L}=h_{L1}=h_{L2}=h_{L3}$ C $Q=Q_{1}=Q_{2}=Q_{3};$ $h_{L}=h_{L1}+h_{L2}+h_{L3}$ D $Q=Q_{1}=Q_{2}=Q_{3};$ $h_{L}=h_{L1}=h_{L2}=h_{L3}$
GATE ME 2015 SET-3   Fluid Mechanics
Question 10 Explanation:
Total discharge,
$Q=Q_{1}+Q_{2}+Q_{2}$
Head loss: $h_{L}=h_{L_{1}}=h_{L_{2}}=h_{L_{3}}$

There are 10 questions to complete.


text:
# What method and tools should I use for AI that suggests/assigns a person for a task?

I'm trying to learn AI and thinking to apply it to our system. We have an application for Translation industry. What we are doing now is the Coordinator assigns a file to a Translator, the Coordinator usually considers this criteria (but not limited to):

• the deadline of the file and availability of translator
• the language pair that the translator can translate
• is the translator already reached his target? (maybe we can give the file to other translator to reach their target)
• the difficulty level of the file for translator(basic translation, medical field, IT field)
• accuracy of translator
• speed of translator

Given the following, is it possible to make a recommendation to the Coordinator to whom she can assign a particular file?

What are the methods/topics that I need to research?

I'm considering javascript as the primary tool and maybe python if javascript will be more of a hindrance in implementation.

EDIT:

In addition to suggesting a Translator we are also looking into suggesting the deadline of the translator. Basically, we have deadline of customer and deadline of translator

The reason for this is that, if the Translators are occupied throughout the day, it makes sense to suggest it to a busy translator but allow him to finish it until next day.

• This sounds like Task Allocation, an old area of AI. I will write up an answer when I'm not on my phone. – John Doucette Sep 10 '18 at 13:11
• I have adjusted the tags as you asked. I don't think your choice of language will be very relevant to answers at the high level you are asking. When you get into implementation, and get stuck, then depending on the nature of the problem you could post here or even on Stack Overflow (SO would be for specific error messages in your code, or post here if your code works but you think that you misunderstood some concept in AI). – Neil Slater Sep 10 '18 at 14:07

What you have could be well described as a Task Allocation problem, which is studied as part of the planning subfield of AI. Chapters 10 & 11 of Russell & Norvig provide a good overview of this area, although I think they don't talk too much about Task Allocation in particular.

There are two basic approaches to this problem: centralized approaches, and decentralized approaches.

In centralized approaches, the properties of each task (or sub-task) and the skills of each processing entity are recorded in a central database. The task is phrased as an optimization problem. For example, given the skills of the processors and the tasks' types, find the schedule that minimizes average processing time (or cost, or usage of rare-resource types, or whatever you're interested in). Common approaches include phrasing the optimization task as a linear-programming problem; phrasing the problem as a graph and using something like the graphplan algorithm; or phrasing the problem as a constraint satisfaction problem and using some kind of heuristic-guided local search.

There are all kinds of other more modern techniques too. I'm not aware of a survey paper for translation tasks in particular, but there are lots of examples in robotics and distributed computing.

Manuel Rodriguez's answer is partially correct as well: although good AI techniques exist for scheduling the task, they are predicated on being able to quantify the tasks' properties and the abilities of the agents, and on the translators accepting the decisions of the system. If you want an interactive system, you may need to look at techniques from Natural Language Processing. The work on Mixed Initiative Scheduling Systems might also be relevant if you have to go that route.

• thank you for taking the time answering. I've made an edit to my post to add additional feature (suggesting deadline of the translator to finish the file). Will the same method applies in this context? – Jaime Sangcap Sep 10 '18 at 14:55
• Yep. If you want a more "mixed initiative" system, you can ask the user how long it might take before making the assignment, or you can generate plans that have estimated minimum and maximum processing times using some more modern techniques. I'm pretty sure there's an annual challenge for Task Allocation where the good techniques are showcased each year, but I can't seem to find it right now. Maybe it's a AAAI workshop? – John Doucette Sep 10 '18 at 15:00
• I think the number of pages of the file is a good start. Do you consider this problems extremely difficult? Atleast in an entry level? – Jaime Sangcap Sep 10 '18 at 15:04
• I expect that you can get reasonably good schedules pretty easily. The bigger problem is actually on the software engineering side: can you get your users to accept the system? What factors do the users consider essential when doing this kind of scheduling? Users probably won't accept a system that makes decisions they consider poor, so you may find that a lot of requirements gathering is needed. The AI part of this is probably not so bad though. There are lots of off-the-shelf scheduling tools that will do at least an okay job. I'd probably start with local search-based approaches. – John Doucette Sep 10 '18 at 15:19
• Good point on the usability part, we are still under discussion actually. Good to know that at least its feasible. – Jaime Sangcap Sep 10 '18 at 15:41

Signaling games

Bayesian Stackelberg games are signaling games. The agents can send and receive messages. Sometimes it is called a dynamic Bayesian game. In it's easiest form, the message coding is binary, that means agent1 can send a code to agent2. It is not defined, that the coding must contain natural language, this special aspect is discussed in the next chapter.

Natural language agent communication

The topic is called workflow management system, it is mostly a manual issue which has to do with communication in a workgroup. That means, there is a list of 10 people who are working on the same goal and each of them is doing a subtask. Because human people are involved, it has to do with natural language communication. A possible speech act might be “Hi, do you want to do the translation, or should i do it?”.

To describe this as an agent based workflow system the first step is to create a simulated environment. That means, the task “translation of a text” has to be converted into a formal model. A possible description language would be RDF and linked data. In the next step, this model has to be converted into a game, in which actions are possible. For example, one agent can take a request or he can decline a request. If the game works quite well, it is possible to write an Artificial Intelligence for that game who controls the agents.

The sad news is, that in the literature this topic is not very well discussed. Converting a workflow from real life into a simulation with the aim to create an AI for it, can be called Advanced AI-research.

• -1 because scheduling of this kind has been studied for a long time, and so is not "Advanced AI". Even the NLP-ish side is pretty well studied. See the DARPA funded work of Scerri, Tambe, et al. in the early 2000's, for example. – John Doucette Sep 10 '18 at 14:49
• @ManuelRodriguez thank you for answering, I will also research regarding the topics you've mentioned – Jaime Sangcap Sep 10 '18 at 14:57
• Teams of software agents who are interacting with humans in natural language isn't Advanced AI? Perhaps you're right, I'm reading right now the paper of Tambe et. al. and he has figured out everything: CAP, C4.5, SOAR and so on. It's my fault, i've ignored the research of SRI and DARPA, the -1 downvote is correct. – Manuel Rodriguez Sep 10 '18 at 15:39
• @ManuelRodriguez I think the key term is "Natural Language". Nowhere in OP's post is NLP actually mentioned as a requirement. In fact, NLP for narrowly constrained subdomains has been a solved problem since the 1970's (see, e.g. "frames"), so even if OP wanted that, it is solvable. You don't need generalized NLP to solve scheduling. – John Doucette Sep 11 '18 at 11:29
• @JohnDoucette I don't think so. Task allocation = multi agent system = shared vocabulary = natural language processing (especially the OP was about task allocation to real persons) – Manuel Rodriguez Sep 11 '18 at 11:51