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October  2019, 12(6): 1685-1708. doi: 10.3934/dcdss.2019113

## Efficient high-order implicit solvers for the dynamic of thin-walled beams with open cross section under external arbitrary loadings

 1 Laboratoire d'Ingénierie et Matériaux (LIMAT), Faculté des Sciences Ben M'Sik, Hassan Ⅱ University of Casablanca, Avenue Cdt Driss El Harti B.P. 7955, Sidi Othman, Casablanca, Morocco 2 Laboratoire de Génie Mécanique (LGM), Faculté des Sciences et Techniques, Fès, Université Sidi Mohamed Ben Abdellah, Route d'Imouzzer B.P. 2202, Fès, Maroc 3 Laboratoire d'Étude des Microstructures et de Mécanique des Matériaux (LEM3), Université de Lorraine, Metz, CNRS UMR 7239, Ile du Saulcy, 57057, France

* Corresponding author: Bouazza Braikat

Received  November 2017 Revised  April 2018 Published  November 2018

This paper aims to investigate, in large displacement and torsion context, the nonlinear dynamic behavior of thin-walled beams with open cross section subjected to various loadings by high-order implicit solvers. These homotopy transformations consist to modify the nonlinear discretized dynamic problem by introducing an arbitrary invertible pre-conditioner $[K^\star]$ and an arbitrary path following parameter. The nonlinear strongly coupled equations of these structures are derived by using a $3D$ nonlinear dynamic model which accounts for large displacements and large torsion without any assumption on torsion angle amplitude. Coupling complex structural phenomena such that warping, bending-bending, and flexural-torsion are taken into account.

Two examples of great practical interest of nonlinear dynamic problems of various thin-walled beams with open section are presented to validate the efficiency and accuracy of high-order implicit solvers. The obtained results show that the proposed homotopy transformations reveal a few number of matrix triangulations. A comparison with Abaqus code is presented.

Citation: Ahmed El Kaimbillah, Oussama Bourihane, Bouazza Braikat, Mohammad Jamal, Foudil Mohri, Noureddine Damil. Efficient high-order implicit solvers for the dynamic of thin-walled beams with open cross section under external arbitrary loadings. Discrete & Continuous Dynamical Systems - S, 2019, 12 (6) : 1685-1708. doi: 10.3934/dcdss.2019113
##### References:
 [1] Abaqus, Version 6.11 Documentation, Dassautt Systemes Simulia Corp, Providence, RI, USA, 2011. Google Scholar [2] E. L. Allgower and K. Georg, Numerical Continuation Methods: An Introduction, Springer series in Computational Mathematics, 1990. doi: 10.1007/978-3-642-61257-2.  Google Scholar [3] R. D. Ambrosini, J. D. Riera and R. F. Danesi, Dynamic analysis of thin-walled and variable open section beams with shear flexibility, International Journal for Numerical Methods in Engineering, 38 (1995), 2867-2885.   Google Scholar [4] K. J. Bathe, Finite Elements Procedures, Prentice-Hall, New Jersey, 1996. Google Scholar [5] J. L. Batoz and G. Dhatt, Modélisation des structures par éléments finis, Hermès, Paris, 1990. Google Scholar [6] K. Behdinan, M. C. Stylianou and B. Tabarrok, Co-rotational dynamic analysis of flexible beams, Computer Methods in Applied Mechanics and Engineering, 154 (1998), 151-161.   Google Scholar [7] P. Betsch and P. Steinmann, Constrained dynamics of geometrically exact beams, Computational Mechanics, 31 (2003), 49-59.   Google Scholar [8] O. Bourihane, B. Braikat, M. Jamal, F. Mohri and N. Damil, Dynamic analysis of a thin-walled beam with open cross section subjected to dynamic loads using a high-order implicit algorithm, Engineering Structures, 120 (2016), 133-146.   Google Scholar [9] S. Boutmir, B. Braikat, M. Jamal, N. Damil, B. Cochelin and M. Potier-Ferry, Des solveurs implicites d'ordre supérieurs pour les problèmes de dynamique non linéaire des structures, Revue Européenne des Eléments Finis, 13 (2004), 449-460.   Google Scholar [10] M. A. Crisfield, Nonlinear Finite Elements Analysis of Solids and Structures, John Willey and Sons, 1991. Google Scholar [11] E. Dale Martin, A technique for accelerating iterative convergence in numerical integration with application in transonic aerodynamics, Lectures notes in Physics, 47 (1976), 123-139.   Google Scholar [12] A. Ed-dinari, H. Mottaqui, B. Braikat, M. Jamal, F. Mohri and N. Damil, Large torsion analysis of thin-walled open sections beams by the asymptotic numerical method, Engineering Structures, 81 (2014), 240-255.   Google Scholar [13] Y. Guevel, G. Girault and J. M. Cadou, Numerical comparisons of high-order nonlinear solvers for the transient $\textbf{N}$avier-$\textbf{S}$tokes equations based on homotopy and perturbation techniques, Journal of Computational and Applied Mathematics, 289 (2015), 356-370.  doi: 10.1016/j.cam.2014.12.008.  Google Scholar [14] D. Haijuan, Nonlinear free vibration analysis of asymmetric thin-walled circularly curved beams with open section, Thin-Walled Structures, 46 (2008), 107-112.   Google Scholar [15] M. Jamal, B. Braikat, S. Boutmir, N. Damil and M. Potier-Ferry, A high order implicit algorithm for solving instationary nonlinear problems, Computational Mechanics, 28 (2002), 375-380.  doi: 10.1007/s00466-002-0301-7.  Google Scholar [16] T. N. Le, J. M. Battini and M. Hjiaj, Efficient formulation for dynamics of corotational 2D beams, Computational Mechanics, 48 (2011), 153-161.  doi: 10.1007/s00466-011-0585-6.  Google Scholar [17] T. N. Le, J. M. Battini and M. Hjiaj, Corotational formulation for nonlinear dynamics of beams with arbitrary thin-walled open cross-sections, Computer and Structures, 134 (2014), 112-127.   Google Scholar [18] S. Mesmoudi, A. Timesli, B. Braikat, H. Lahmam and H. Zahrouni, A 2D mechanical--thermal coupled model to simulate material mixing observed in friction stir welding process, Engineering with Computers, (2017), 1-11.   Google Scholar [19] F. Mohri, N. Damil and M. Potier Ferry, Large torsion finite element model for thin-walled beams, Computers and Structures, 86 (2008), 671-683.   Google Scholar [20] F. Mohri, L. Azrar and M. Potier-Ferry, Vibration analysis of buckled thin-walled beams with open sections, Journal of Sound and Vibration, 275 (2004), 434-446.   Google Scholar [21] F. Mohri, N. Damil and M. Potier-Ferry, Linear and nonlinear stability analyses of thin-walled beams with monsymmetric sections, Thin-Walled Structures, 48 (2010), 299-315.   Google Scholar [22] F. Mohri, N. Damil and M. Potier-Ferry, Large torsion finite element model for thin-walled beams, Computers and Structures, 86 (2008), 671-683.   Google Scholar [23] F. Mohri, A. Ed-dinari and N. Damil, A beam finite element for nonlinear analysis of thin-walled elements, Thin Walled Structures, 46 (2008), 981-990.   Google Scholar [24] H. Mottaqui, B. Braikat and N. Damil, Discussion about parameterization in the asymptotic numerical method: Application to nonlinear elastic shells, Computer Methods in Applied Mechanics and Engineering, 199 (2010), 1701-1709.  doi: 10.1016/j.cma.2010.01.020.  Google Scholar [25] H. Mottaqui, B. Braikat and N. Damil, Local parameterization and the asymptotic numerical method, Mathematical Modelling of Natural Phenomena, 5 (2010), 16-22.   Google Scholar [26] N. Newmark, A method of computation for structural dynamics, Journal of the Engineering Mechanics Division, Proceeding of ASCE, (1959), 67-94.   Google Scholar [27] E. J. Sapountzakis and I. C. Dikaros, Nonlinear flexural-torsional dynamic analysis of beams of variable doubly symmetric cross section-application to wind turbine towers, Nonlinear Dynamics, 73 (2013), 199-227.  doi: 10.1007/s11071-013-0779-x.  Google Scholar [28] A. Timesli, B. Braikat, H. Lahmam and H. Zahrouni, A new algorithm based on moving least square method to simulate material mixing in friction stir welding, Engineering Analysis with Boundary Elements, 50 (2015), 372-380.   Google Scholar [29] V. Z. Vlasov, Thin walled elastic beams, Eyrolles, French translation: Pièces longues en voiles minces, Paris, 1965. Google Scholar [30] O. C. Zienkiewicz and R. Taylor, The Finite Element Method, Solid and Fluid Mechanics and Non-linearity, Book Company, 1987. Google Scholar

show all references

##### References:
 [1] Abaqus, Version 6.11 Documentation, Dassautt Systemes Simulia Corp, Providence, RI, USA, 2011. Google Scholar [2] E. L. Allgower and K. Georg, Numerical Continuation Methods: An Introduction, Springer series in Computational Mathematics, 1990. doi: 10.1007/978-3-642-61257-2.  Google Scholar [3] R. D. Ambrosini, J. D. Riera and R. F. Danesi, Dynamic analysis of thin-walled and variable open section beams with shear flexibility, International Journal for Numerical Methods in Engineering, 38 (1995), 2867-2885.   Google Scholar [4] K. J. Bathe, Finite Elements Procedures, Prentice-Hall, New Jersey, 1996. Google Scholar [5] J. L. Batoz and G. Dhatt, Modélisation des structures par éléments finis, Hermès, Paris, 1990. Google Scholar [6] K. Behdinan, M. C. Stylianou and B. Tabarrok, Co-rotational dynamic analysis of flexible beams, Computer Methods in Applied Mechanics and Engineering, 154 (1998), 151-161.   Google Scholar [7] P. Betsch and P. Steinmann, Constrained dynamics of geometrically exact beams, Computational Mechanics, 31 (2003), 49-59.   Google Scholar [8] O. Bourihane, B. Braikat, M. Jamal, F. Mohri and N. Damil, Dynamic analysis of a thin-walled beam with open cross section subjected to dynamic loads using a high-order implicit algorithm, Engineering Structures, 120 (2016), 133-146.   Google Scholar [9] S. Boutmir, B. Braikat, M. Jamal, N. Damil, B. Cochelin and M. Potier-Ferry, Des solveurs implicites d'ordre supérieurs pour les problèmes de dynamique non linéaire des structures, Revue Européenne des Eléments Finis, 13 (2004), 449-460.   Google Scholar [10] M. A. Crisfield, Nonlinear Finite Elements Analysis of Solids and Structures, John Willey and Sons, 1991. Google Scholar [11] E. Dale Martin, A technique for accelerating iterative convergence in numerical integration with application in transonic aerodynamics, Lectures notes in Physics, 47 (1976), 123-139.   Google Scholar [12] A. Ed-dinari, H. Mottaqui, B. Braikat, M. Jamal, F. Mohri and N. Damil, Large torsion analysis of thin-walled open sections beams by the asymptotic numerical method, Engineering Structures, 81 (2014), 240-255.   Google Scholar [13] Y. Guevel, G. Girault and J. M. Cadou, Numerical comparisons of high-order nonlinear solvers for the transient $\textbf{N}$avier-$\textbf{S}$tokes equations based on homotopy and perturbation techniques, Journal of Computational and Applied Mathematics, 289 (2015), 356-370.  doi: 10.1016/j.cam.2014.12.008.  Google Scholar [14] D. Haijuan, Nonlinear free vibration analysis of asymmetric thin-walled circularly curved beams with open section, Thin-Walled Structures, 46 (2008), 107-112.   Google Scholar [15] M. Jamal, B. Braikat, S. Boutmir, N. Damil and M. Potier-Ferry, A high order implicit algorithm for solving instationary nonlinear problems, Computational Mechanics, 28 (2002), 375-380.  doi: 10.1007/s00466-002-0301-7.  Google Scholar [16] T. N. Le, J. M. Battini and M. Hjiaj, Efficient formulation for dynamics of corotational 2D beams, Computational Mechanics, 48 (2011), 153-161.  doi: 10.1007/s00466-011-0585-6.  Google Scholar [17] T. N. Le, J. M. Battini and M. Hjiaj, Corotational formulation for nonlinear dynamics of beams with arbitrary thin-walled open cross-sections, Computer and Structures, 134 (2014), 112-127.   Google Scholar [18] S. Mesmoudi, A. Timesli, B. Braikat, H. Lahmam and H. Zahrouni, A 2D mechanical--thermal coupled model to simulate material mixing observed in friction stir welding process, Engineering with Computers, (2017), 1-11.   Google Scholar [19] F. Mohri, N. Damil and M. Potier Ferry, Large torsion finite element model for thin-walled beams, Computers and Structures, 86 (2008), 671-683.   Google Scholar [20] F. Mohri, L. Azrar and M. Potier-Ferry, Vibration analysis of buckled thin-walled beams with open sections, Journal of Sound and Vibration, 275 (2004), 434-446.   Google Scholar [21] F. Mohri, N. Damil and M. Potier-Ferry, Linear and nonlinear stability analyses of thin-walled beams with monsymmetric sections, Thin-Walled Structures, 48 (2010), 299-315.   Google Scholar [22] F. Mohri, N. Damil and M. Potier-Ferry, Large torsion finite element model for thin-walled beams, Computers and Structures, 86 (2008), 671-683.   Google Scholar [23] F. Mohri, A. Ed-dinari and N. Damil, A beam finite element for nonlinear analysis of thin-walled elements, Thin Walled Structures, 46 (2008), 981-990.   Google Scholar [24] H. Mottaqui, B. Braikat and N. Damil, Discussion about parameterization in the asymptotic numerical method: Application to nonlinear elastic shells, Computer Methods in Applied Mechanics and Engineering, 199 (2010), 1701-1709.  doi: 10.1016/j.cma.2010.01.020.  Google Scholar [25] H. Mottaqui, B. Braikat and N. Damil, Local parameterization and the asymptotic numerical method, Mathematical Modelling of Natural Phenomena, 5 (2010), 16-22.   Google Scholar [26] N. Newmark, A method of computation for structural dynamics, Journal of the Engineering Mechanics Division, Proceeding of ASCE, (1959), 67-94.   Google Scholar [27] E. J. Sapountzakis and I. C. Dikaros, Nonlinear flexural-torsional dynamic analysis of beams of variable doubly symmetric cross section-application to wind turbine towers, Nonlinear Dynamics, 73 (2013), 199-227.  doi: 10.1007/s11071-013-0779-x.  Google Scholar [28] A. Timesli, B. Braikat, H. Lahmam and H. Zahrouni, A new algorithm based on moving least square method to simulate material mixing in friction stir welding, Engineering Analysis with Boundary Elements, 50 (2015), 372-380.   Google Scholar [29] V. Z. Vlasov, Thin walled elastic beams, Eyrolles, French translation: Pièces longues en voiles minces, Paris, 1965. Google Scholar [30] O. C. Zienkiewicz and R. Taylor, The Finite Element Method, Solid and Fluid Mechanics and Non-linearity, Book Company, 1987. Google Scholar
Thin-walled beam with open cross section, co-ordinates of the point $M$ on the cross section contour
Axial force $N$, bending moments $M_{y}$ and $M_{z}$, bimoment $B_{\omega}$ and St-Venant torsion moment $M_{sv}$
Section beam under concentrated and distributed forces
External dynamical loading and its time evolution applied on the U-mono-symmetrical thin-walled beam with open cross section
Geometrical characteristics of sections $A$ and $B$
Response curves obtained by the high-order implicit solver $Alg_3$ and by Abaqus code, Time evolution of displacement components $(u(L, t), v(L, t), w(L, t), \theta_x(L, t))$
Cantilever bi-symmetrical beam with steel I cross section under eccentric loading and its time evolution
Thin-walled beam with steel I cross section under transverse eccentric force $F_{z}(t)$ and its point of application
Response curves obtained by the high-order implicit solver $Alg_3$, by Abaqus code and by Sapountzakis: Time evolution of components $(u(L, t), v(L, t), w(L, t), \theta_x(L, t))$
Comparison between three solvers $Alg_1$, $Alg_2$ and $Alg_3$: Influence of time step
 Solvers $Alg_1$ $Alg_2$ $Alg_3$ $\Delta t$ Optimal order $Log|Res|$ Optimal order $Log|Res|$ Optimal order $Log|Res|$ $10^{-3}$ $10$ $-3.73$ $9$ $-3.71$ $8$ $-3.71$ $2\, 10^{-3}$ $12$ $-3.72$ $10$ $-3.71$ $9$ $-3.70$ $3\, 10^{-3}$ $13$ $-3.69$ $11$ $-3.67$ $10$ $-3.65$
 Solvers $Alg_1$ $Alg_2$ $Alg_3$ $\Delta t$ Optimal order $Log|Res|$ Optimal order $Log|Res|$ Optimal order $Log|Res|$ $10^{-3}$ $10$ $-3.73$ $9$ $-3.71$ $8$ $-3.71$ $2\, 10^{-3}$ $12$ $-3.72$ $10$ $-3.71$ $9$ $-3.70$ $3\, 10^{-3}$ $13$ $-3.69$ $11$ $-3.67$ $10$ $-3.65$
Comparison between three solvers $Alg_1$, $Alg_2$ and $Alg_3$: Effect of truncation order
 Solver $Alg_1$ $Alg_2$ $Alg_3$ $p$ $IM$ $RHS$ $CPU(s)$ $IM$ $RHS$ $CPU(s)$ $IM$ $RHS$ $CPU(s)$ $7$ $a_{max}<1$ $a_{max}<1$ $a_{max}<1$ $8$ $a_{max}<1$ $a_{max}<1$ $2995$ $32000$ $3890$ $9$ $a_{max}<1$ $2810$ $36000$ $4252$ $2711$ $36000$ $4102$ $10$ 2850 40000 4900 $2600$ $40000$ $4470$ $2480$ $40000$ $4262$ $15$ 630 60000 12376 $612$ $60000$ $12023$ $520$ $60000$ $10210$ $20$ $320$ $80000$ $25896$ $309$ $80000$ $25000$ $280$ $80000$ $22640$
 Solver $Alg_1$ $Alg_2$ $Alg_3$ $p$ $IM$ $RHS$ $CPU(s)$ $IM$ $RHS$ $CPU(s)$ $IM$ $RHS$ $CPU(s)$ $7$ $a_{max}<1$ $a_{max}<1$ $a_{max}<1$ $8$ $a_{max}<1$ $a_{max}<1$ $2995$ $32000$ $3890$ $9$ $a_{max}<1$ $2810$ $36000$ $4252$ $2711$ $36000$ $4102$ $10$ 2850 40000 4900 $2600$ $40000$ $4470$ $2480$ $40000$ $4262$ $15$ 630 60000 12376 $612$ $60000$ $12023$ $520$ $60000$ $10210$ $20$ $320$ $80000$ $25896$ $309$ $80000$ $25000$ $280$ $80000$ $22640$
Comparison between three solvers $Alg_1$, $Alg_2$ and $Alg_3$: Influence of time step
 Solver $Alg_1$ $Alg_2$ $Alg_3$ $\Delta t$ Optimal order $Log|Res|$ Optimal order $Log|Res|$ Optimal order $Log|Res|$ $10^{-3}$ $6$ $-5.23$ $4$ $-5.2$ $3$ $-5.13$ $2\, 10^{-3}$ $12$ $-5.10$ $9$ $-4.80$ $7$ $-4.62$ $3\, 10^{-3}$ $14$ $-4.91$ $11$ $-4.79$ $8$ $-4.60$ $4\, 10^{-3}$ $15$ $-4.88$ $12$ $-4.70$ $10$ $-4.55$
 Solver $Alg_1$ $Alg_2$ $Alg_3$ $\Delta t$ Optimal order $Log|Res|$ Optimal order $Log|Res|$ Optimal order $Log|Res|$ $10^{-3}$ $6$ $-5.23$ $4$ $-5.2$ $3$ $-5.13$ $2\, 10^{-3}$ $12$ $-5.10$ $9$ $-4.80$ $7$ $-4.62$ $3\, 10^{-3}$ $14$ $-4.91$ $11$ $-4.79$ $8$ $-4.60$ $4\, 10^{-3}$ $15$ $-4.88$ $12$ $-4.70$ $10$ $-4.55$
Comparison between three solvers $Alg_1$, $Alg_2$ and $Alg_3$: Effect of truncation order
 Solver $Alg_1$ $Alg_2$ $Alg_3$ $p$ $IM$ $RHS$ $CPU(s)$ $IM$ $RHS$ $CPU(s)$ $IM$ $RHS$ $CPU(s)$ $2$ $a_{max}<1$ $a_{max}<1$ $a_{max}<1$ $3$ $a_{max}<1$ $a_{max}<1$ $12$ $600$ $26$ $4$ $a_{max}<1$ $13$ $800$ $42$ $11$ $800$ $30$ $5$ $a_{max}<1$ $8$ $1000$ $46$ $6$ $1000$ $34$ $6$ $15$ $1200$ $102$ $7$ $1200$ $54$ $5$ $1200$ $36$ $7$ $4$ $1400$ $126$ $2$ $1400$ $65$ $1$ $1400$ $40$
 Solver $Alg_1$ $Alg_2$ $Alg_3$ $p$ $IM$ $RHS$ $CPU(s)$ $IM$ $RHS$ $CPU(s)$ $IM$ $RHS$ $CPU(s)$ $2$ $a_{max}<1$ $a_{max}<1$ $a_{max}<1$ $3$ $a_{max}<1$ $a_{max}<1$ $12$ $600$ $26$ $4$ $a_{max}<1$ $13$ $800$ $42$ $11$ $800$ $30$ $5$ $a_{max}<1$ $8$ $1000$ $46$ $6$ $1000$ $34$ $6$ $15$ $1200$ $102$ $7$ $1200$ $54$ $5$ $1200$ $36$ $7$ $4$ $1400$ $126$ $2$ $1400$ $65$ $1$ $1400$ $40$
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## Introduction

Systems of identical particles constitute the basic building blocks of quantum information theory, being present in Bose-Einstein condensates1,2, quantum dots3,4,5,6, superconducting circuits7 and optical setups8,9. Completely characterizing the quantum features of these composite systems is thus a crucial requirement from both fundamental and technological viewpoint. Investigation of bipartite entanglement for identical particles started some time ago10,11,12,13,14,15 but, differently from the case of distinguishable particles, the subject has remained controversial10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27. The controversy mainly arises from the way identical particles are ordinarily treated in quantum mechanics by the standard particle-based approach, that is by making them artificially distinguishable with the attribution of nonobservable labels27,28. This practice has the consequence that the structure of the states inevitably is, with respect to labels, that of an entangled state. As a consequence, some viewpoints has been advanced12,14,16,17,18,19,20,21,22,23,24,25 differing both in interpretation and, in some cases, in the quantification of the part of entanglement attributable to particle identity. The last aspect can hardly be considered devoid of importance in view of entanglement being in general a resource for quantum information and communication12,14,18,29,30,31,32,33,34,35,36,37. Differences among the viewpoints clearly emerge in the simple paradigmatic case of identical particles independently prepared in far regions, such that they are expected not to have any correlations, and then allowed to merge spatially without any other change. In one viewpoint, as epitomized in a textbook27, the entanglement due to indistinguishability is present for far particles but nevertheless is physically no matter of concern because it cannot be exploited. In a widely held second viewpoint18,26, this entanglement is a merely formal artifact even when the particles are brought to overlap. In order to characterize it in the latter situation, a third viewpoint has been proposed which resorts to extraction procedures17. The consequent conclusions may be however subject to criticism on the ground that the extracted entanglement does not represent the one in the overlapping condition being instead produced by the extraction operation itself.

Among tools which are at the heart of quantum information and quantum computation, there is the Schmidt decomposition (SD) for bipartite systems of multilevel particles in pure states. It is of general application to entanglement characterization, theory of measurement, state purification, quantum erasure29,30 and also in black-hole physics38,39. Despite its wide utilization in systems of distinguishable particles, even the SD remains debated for identical particles10,16,26 where it is replaced by Slater decomposition. The associated Slater rank witnesses entanglement but its interpretation is different for bosons or fermions26. For distinguishable particles, the SD unveils the entanglement of the system by the von Neumann entropy of the reduced density matrix, whose eigenvalues are the squares of the Schmidt coefficients appearing in the decomposition30. Instead “the relationship between Schmidt coefficients and the eigenvalues of the reduced density matrix breaks down in the case of identical particles”26. Therefore, the ordinary notion of partial trace to get the reduced state has not been considered suitable for assessing the entanglement of identical particles16,18,26. Attempts to provide a SD for indistinguishable particles give wrong results, as nonzero entropy and thus the presence of entanglement for uncorrelated fermions10.

Using a recent non-standard particle-based approach40, here we develop a SD valid for both bosons and fermions, which allows the direct characterization of the physical entanglement of the particles. Differently from previous viewpoints, application of this method to the paradigmatic example exposed above allows, without resorting to extraction, to demonstrate that entanglement is zero for separated particles and nonzero when they overlap. Moreover, our SD overcomes the problems occurring in other proposals of SD within the second quantization10, being the Schmidt coefficients always the square roots of the eigenvalues of the reduced density matrix, exactly as for distinguishable particles. This SD gives the novel tool to straightforwardly characterize the entanglement for identical particles.

## Results

### Theory

We recall the notation of the intrinsically symmetric particle-based approach introduced in ref. 40. Hereafter, we mean by “symmetric states” (or “symmetric Hilbert space”) the symmetric or antisymmetric behavior of the system states depending on the bosonic or fermionic nature of the particles, respectively. The overall state of two identical particles, one in the state ϕ and one in ψ, is completely characterized by enumerating the one-particle states and represented as |ϕ, ψ〉. Two particles in |ϕ, ψ〉 are not independent and their overall state is a whole which cannot be written as a tensorial product of one-particle states, i.e. . However, a nonseparable external symmetric product of one-particle states (wedge product) can be introduced as . Analogously, we have (this wedge product will be crucial in demonstrating the theorem below). The probability amplitude of finding the two particles in if they are in |ϕ, ψ〉, is obtained by the symmetric two-particle scalar product defined in terms of one-particle amplitudes as40

where η is +1 for bosons and −1 for fermions. This probability amplitude immediately shows that the generic state |ϕ, ψ〉 is symmetric, i.e. . The state |ϕ, ψ〉 spans a linear symmetric two-particle Hilbert space . A symmetric inner product between state spaces of different dimensionality (one-particle projective measurement) can also be introduced as40

In it is possible to choose an orthonormal two-particle basis , |i〉 and |j〉 being single-particle states, where an arbitrary state of two identical particles can be expressed as . By Eq. (2), one then gets the reduced (single-particle) density matrix via partial trace as40 , where . We emphasize that now partial trace depends on the single-particle basis being local or nonlocal, as we shall show in the following. This behavior differs from the case of distinguishable particles where, since a single-particle basis always addresses a given particle, the partial trace is not affected by the local or nonlocal nature of the basis. We can now give the following theorem.

### Theorem 1

Within a symmetric two-particle Hilbert space , a pure state of two d-level identical particles |Ψ〉 can always be written in the Schmidt decomposition (SD)

The “Schmidt coefficients” are the square roots of the eigenvalues of the reduced density matrix and the states its eigenstates. The state belongs to the basis and the symmetric two-particle basis is the “Schmidt basis”.

Proof. We express the state |Ψ〉 in terms of the symmetric two-particle basis as , where the symmetric two-particle identity matrix has been inserted. By defining , the state can be further cast as . Generally, the states are not orthonormal. Nevertheless, as for distinguishable particles29, there exists a basis where they are orthogonal, i.e. . We thus write

where we have used the partial trace . When the states are the eigenstates of , i.e. , the states are orthogonal and satisfy . Denoting by the set of orthonormal states associated to , we have

Both and are eigenstates of ρ(1) with the same eigenvalue λi (as proven in Methods). Thus, given a set of eigenstates , each is one of the states within the set. For bosons, if the eigenvalues are nondegenerate then ; for fermions, Pauli exclusion principle dictates and the eigenvalues are always degenerate. Substituting of Eq. (5) in , the SD of Eq. (3) is finally demonstrated.

When the states are characterized by more than one observable, for instance when the single-particle basis is (|a〉 and |b〉 being two independent observables), one can be interested in studying the system for a fixed value of one of the observables. In such cases, the theorem above needs to be specialized. Let us take a two-particle state of the form , where u, v are arbitrary single-particle states. This means that the SD of is obtained by following the theorem above with the difference that the partial trace is now performed on the subspace of b (a) with the observable a (b) fixed. The corresponding reduced density matrix is indicated as (see Methods). The universality of SD just proven entails its application to identical particles in many scenarios of quantum information (entanglement characterization, purification, measurement theory) in analogy with distinguishable particles30. Knowledge of the Schmidt basis is essential to find the suitable set of measurements (Schmidt observables41) to acquire information on correlated identical particles in experimental contexts42,43,44.

The SD of Eq. (3) defines an entangled state in terms of nonseparability, whatever the overlap between the particles. As for distinguishable particles29, we define the positive integer “Schmidt number” s as the number of terms appearing in Eq. (3), that is the number of nonzero eigenvalues of . If s = 1, is pure and identifies a nonentangled state; if s >1, testifies an entangled state. The Schmidt number thus acts as entanglement witness. Analogous considerations hold for . In particular, the (symmetric) basis state with single-particle states |i〉, |j〉 containing only one observable results to be unentangled when , while it is maximally entangled when (see Methods). Being the Schmidt coefficients the square roots of the eigenvalues of the single-particle reduced state, they immediately lead to the von Neumann entropy

as a quantifier of entanglement for identical particles, exactly as happens for nonidentical particles29. Given any pure state , its SD is obtained in a recipe format as follows:

1. i

perform the trace of ρ on a chosen single-particle basis to get the reduced single-particle density matrix (or );

2. ii

calculate eigenvalues, λi, and eigenstates, , of ();

3. iii

construct the states and express the state in terms of the Schmidt basis .

### Applications

In the following, we apply this recipe to some states of interest (see Fig. 1). The first one is a situation already known40 which is here particularly useful to present how our method works. The other ones are new examples which evidence the usefulness of SD in finding novel entanglement features of identical particles.

#### Two qubits in two separated sites (Bell-like state).

We consider two identical particles (bosons or fermions) with orthogonal internal degrees of freedom (pseudospins) located in separated sites, described by

where ( real, with θ being the relative phase). The site M (Left (L) or Right (R)) and the pseudospin σ (↑, ↓) are independent observables. The two sites are nonoverlapping, behaving thus as “physical” labels. The state of Eq. (7) recalls Bell-like states26,45. It permits us to discuss the role of local and nonlocal measurement in the structure of the SD for identical particle states. When a single-particle property (e.g., the pseudospin) is measured in a localized region of space (e.g., L), the measurements itself and the corresponding partial trace are local40. According to the recipe above, this measurement corresponds to project on the local basis, i.e. on the subspace . The reduced single-particle density matrix is

It has eigenvalues , , and eigenstates , , , , which define the Schmidt basis. We notice that and that the particle statistics is intrinsically included by the presence of . The SD of is

with von Neumann entropy

This result coincides with the known von Neumann entropy for two distinguishable particles in a Bell-like state, giving maximal entanglement for a Bell state ()31.

When nonlocal (one-particle) measurements are performed simultaneously on both sites (L and R) where the particle has nonzero probability of being found, the trace is nonlocal. Operationally, it corresponds to perform the partial trace of ρ on the global single-particle basis, i.e. on . Following the recipe above by a global partial trace, we get a SD of the Bell-like state different from Eq. (9) (see Methods) leading to

The difference between and highlights the importance of measurement localization on the structure of the SD and in turn on the entanglement between two identical particles located in different sites. To further clarify this aspect, we consider the particular case α = 1 when the state becomes , which is unentangled40 since the particles are in separated sites and behave as uncorrelated distinguishable particles31. For this state, and . For systems of identical particles, local single-particle measurements supply the intrinsic entanglement40, whilst nonlocal measurements yield “measurement-induced entanglement”26,46. This feature must be contrasted with what happens for distinguishable particles, where single-particle measurements always address individual particles. We point out that these results overcome the issue existing with previous proposals of Schmidt decomposition and entanglement measure for identical particles which give nonzero entanglement for uncorrelated separated particles10,11.

#### Two qubits in the same site with arbitrary pseudospins

Entanglement is a measure of nonseparability of the state47. When the particles are in the same site, their internal states (pseudospins) establish such nonseparability. A recent experiment showed that the entanglement in a Cooper pair can be extracted by means of graphene quantum dots, so that it can be possibly used as a resource for quantum information in the solid state4. Moreover, it was recently observed48 that it is possible to prepare two maximally entangled ultra-cold atoms with opposite spin states by bringing them into the same optical tweezer (site). In general, one physically expects that situations may occur where particles are in the same site M with pseudospins in arbitrary directions. Such a condition is possible only for bosons, since for fermions the only allowed state by the Pauli exclusion principle is that with opposite pseudospins which is maximally entangled40. We hence study two identical boson qubits (e.g., photons) with one pseudospin along z-direction () and the other one along the direction , as displayed in Fig. 2 (this situation generalizes that of two bosons with opposite pseudospins treated previously40). By exploiting linearity and omitting the spatial index M, this state has the (unnormalized) form

where . Following the recipe above by performing the partial trace on the basis (see Methods), we obtain its (normalized) SD

where , and . Notice the dependence of SD on θ, which represents the pseudospin state overlap of the two particles. Entanglement of the two boson qubits is quantified by the von Neumann entropy and plotted in Fig. 3. It is maximum for θ = π (opposite pseudospins) and zero for θ = 0 (same pseudospins).

Two indentical qubits in the same site can thus be physically entangled40, a result which is indirectly confirmed by extraction procedures17 and is instead uncaught by other entanglement measures for identical particles, like the detection-level concurrence26.

#### Two identical qutrits in the same site

Systems of three-level particles (also called qutrits) are promising alternative candidates to be used in quantum processors instead of the standard two-level qubits49,50. We apply our method to two identical qutrits in the same site, each characterized by the basis (the spatial index is omitted for simplicity). Specifically we take this system, which is equivalent to that of two spin-1 bosons, in the previously analyzed state16.

By the partial trace on the above one-particle basis, we obtain the following reduced density matrix

from which one can calculate the eigenvalues and eigenvectors required for constructing the SD. Since their explicit expressions are cumbersome, we focus on a simple particular case which allows us to make the comparison to another method16 that provides different entanglement predictions for this state. In particular, we choose in the state of Eq. (14), which thus reduces to

By following the usual recipe (see Methods), we get its SD

where , , , . Expressing in the single-particle basis (i = 1, 2, 3) and exploiting the linearity of the symmetric Hilbert space40, we get . The von Neumann entropy of Eq. (6) is , which represents a maximally entangled state independently of ϕ. We provide a physical motivation to support this result. We notice that , where . The independence of ϕ is due to the fact that the amount of entanglement only rests on the scalar product and hence on the angle between the single-particle states , , as depicted in Fig. 4. Moreover, entanglement is maximum because , are orthogonal (as mentioned in Theory section and demonstrated in the Methods). This situation is analogous to the case of two identical qubits in the same site with pseudospin states in arbitrary directions treated before. Our result (ϕ-independent) contrasts with the previous one (ϕ-dependent)16.

## Discussion

We observe that a difference exists from an operational viewpoint between nonidentical and identical particles. For distinguishable particles, SD and its corresponding entanglement are known to be exploitable within a resource theory by local operations, addressing each individual particle, and classical communication (LOCC)29,30. Differently, indistinguishable particles are not individually addressable. Nevertheless, the SD here presented for identical particles identifies an entanglement still exploitable by LOCC, which then allows quantum information protocols like teleportation. As an instance, this can be achieved by resorting to extraction procedures which make the overlapping identical particles tunnel with certain probabilities into two separated spatial modes17. For particles in the same site, where an intrinsic entanglement can be defined40, it is straightforward to realize that the original Schmidt decomposition is reproduced, in a conditional fashion, into the two-particle state of the two accessible separated modes. The operational aspects will be treated elsewhere in detail, including the case of partially overlapping identical particles where the definition of entanglement is more subtle40.

Summarizing, we have supplied, within a non-standard approach to identical particles40, a universal SD of bipartite quantum systems, holding for particles of any nature. This result shows that the problems arising with other attempts of SD for identical particles within the second quantization, which are solved here in a natural way, are not therefore to be settled by arguments related to quantum information coding and processing10. In our approach the Schmidt number maintains its role of entanglement witness and the Schmidt coefficients can be faithfully used to calculate the von Neumann entropy. These aspects permit, differently from what has been claimed12,26,31, unambiguous quantification of entanglement for indistinguishable particles by ordinary notions, like the von Neumann entropy after partial trace.

We have first tested the reliability of the SD by using it to study the well-known condition of two identical qubits with opposite pseudospins in spatially separated sites. The SD and the corresponding entanglement entropy give the physically expected results, as zero entanglement for a product state and maximal entanglement for a Bell state, also showing that nonlocal measurements induce entanglement in the system. We have then applied the SD to analyze two boson qubits in the same site, finding how the amount of their entanglement depends on their pseudospin overlap: the entanglement increases as the two internal states tend to be orthogonal. This behavior generalizes previous results limited to orthogonal pseudospins40, which appear to confirm recent experimental observations of entanglement extraction in Cooper pairs4 and of entanglement generation between two cold atoms in the same optical tweezer48. We have finally studied a system of two identical qutrits, which are relevant for storing quantum information49,50. We have straightforwardly obtained their entanglement and provided a physical interpretation in the case when they are in the same site. Our result differs from that determined for the same system by an alternative approach16. The origin of this difference in the entanglement measurement remains to be understood, requiring experimental verification and comparison of both theoretical approaches.

Our result allows the natural generalization of the SD to arbitrary bipartitions of systems of identical particles. Our work enables the exploitation of this tool for characterizing composite quantum systems in theoretical and experimental relevant conditions where identical particles live in partially overlapping sites (e.g., electrons in quantum dots3,4,5,6, Bose-Einstein condensates1, solid-state qubits in circuit quantum electrodynamics7 and wave-guided and integrated photons8,9), which remain little explored. Our research demonstrates that, differently from what stated before, two identical particles prepared independently in spatially separated sites are not entangled and that, when these particles are brought to overlap with no other change, there is a physical entanglement which is to be attributed to quantum indistinguishability. This fact settles the ambiguity on the interpretation of identical particle entanglement and establishes that entanglement between identical particles is not a mere mathematical artefact, as has been argued26,18. As a further novel aspect, our approach evidences how the local and nonlocal nature of single-particle measurements, which define the partial trace operation, and the single-particle state overlap influence the structure of the SD and therefore the quantification of the entanglement. The Schmidt decomposition for identical particles here reported supply methods to exploit the resources of entanglement coming from particle identity for applications such as state teleportation, quantum metrology and quantum cryptography.

## Methods

### Eigenstates of the reduced density matrix

Here we demonstrate that the states are eigenstates of with eigenvalues , analogously to the eigenstates . We start by using Eq. (5) of the manuscript and reminding that to have

Since and , we obtain

At this point, inserting the two-particle identity matrix between and and using Eq. (2) of the main text to get , we find

where the last equality is due to Eq. (5) of the manuscript and to the symmetry property (). Hence, we conclude that

that is what we intended to demonstrate. Notice that the states belong to the basis of the eigenstates of the reduced density matrix.

### Relationship between the eigenstates and

According to Eq. (5) of the manuscript, one has

Expressing by the SD of Eq. (3) of the main text, we obtain

where we have used (see Equation (1) of the manuscript) and .

From the previous equation, it is immediately seen that for fermions, as expected, it is always , since two of them cannot occupy the same state (Pauli exclusion principle). The orthogonality of the eigenstates and implies that the eigenvalues of the reduced density matrix for a state of two fermions must be degenerate. For states of two bosons, instead, both cases of degenerate and non-degenerate eigenvalues can occur. In particular, if the eigenvalues of the reduced density matrix are non-degenerate, it immediately follows : the eigenstates and coincide. We stress that these properties are always true when the eigenvalues of the reduced density matrix are calculated within the complete single-particle basis (including all possible outcomes of the observables which define a single-particle state) or in the specific case when the single-particle state is described by an observable alone, which are the conditions assumed in proving the theorem of the manuscript. Wider scenarios arise when the reduced density matrix is instead calculated by fixing a given value of an observable.

### Partial trace on a given subspace of an observable

Let us consider a single-particle state and a two-particle state , where , , , , , are arbitrary states corresponding to two independent observables and (e.g., the site and the spin of the particle). We show a general criterion to perform the partial trace of on the subspace of an observable (e.g., ) by varying the other one (e.g., ). We first calculate the one-particle projective measurement (see Equation (2) of the manuscript)

The action of on the state can be thus defined as

The reduced single-particle density matrix performed on the subspace of the observable (that is, obtained by fixing and summing on ) reads

Once the reduced density matrix is so obtained and normalized, the entanglement can be quantified by von Neumann entropy, as usual.

### Entanglement of a two-particle basis state

Here we calculate the entanglement of a basis state within the complete single-particle basis, showing that it depends on the scalar product between , . We thus consider the two-identical particle state , where and are generic single-particle states. The reduced single-particle density matrix, performed on the basis , reads

By using Eq. (2) of the manuscript, we obtain

Recognizing the presence of the single-particle identity , the reduced density matrix reduces to

If and are orthogonal, i.e. , one obtains

Since the eigenvalues of are , entanglement as quantified by the von Neumann entropy is maximum (see Equation (6) of the main text). Moreover, the presence of entanglement (independently of its amount) is witnessed by the number of the nonvanishing eigenvalues, that is by the Schmidt number.

On the other side, when the states , coincide, that is , one obtains

Such a condition, allowed only for bosons (), leads to which is a pure state whose unique nonvanishing eigenvalue is . Entanglement of this state is thus zero (the von Neumann entropy vanishes), as already witnessed by the presence of only one nonvanishing eigenvalue in the reduced density matrix.

We then conclude that the entanglement of a two-particle basis state depends on the scalar product (and thus on the angle, from a geometrical viewpoint) between the single-particle states and . It is maximum when they are orthogonal and zero when they are the same. Furthermore, we have here confirmed that the Schmidt number is an entanglement witness, as for distinguishable particles29. We remark that these results are valid when the partial trace is performed within the complete single-particle basis in the specific case when the single-particle state is described by an observable alone. We have already seen (see section above and the case of two spatially separated particles of the manuscript) that for a single-particle state described by a number of observables, new scenarios surface for determining the entanglement of the identical particle system.

### Schmidt decomposition of the Bell-like state by a global partial trace

We give here the Schmidt decomposition of the Bell-like state of Eq. (7) of the manuscript, by following the recipe in the main text. We first perform the global partial trace of on the total single-particle space , and obtain

It has eigenvalues , , and eigenstates , , , , which define the Schmidt basis. The Schmidt decomposition of the Bell-like state thus results

and it permits to write the von Neumann entropy of Eq. (11) of the manuscript. Notice the difference between the SD given here and that reported in Eq. (9) of the main text.

### Schmidt decomposition of the two-boson state |Φ〉

We here provide the Schmidt decomposition of the two-boson state defined in Eq. (12) of the manuscript. The reduced density matrix , obtained by performing the partial trace of on the basis , is

where , , and . It is straightforward to find its eigenvalues

and the corresponding eigenstates

As we see, they only depend on the angle between the pseudospins (). Since we are dealing with two bosons in the same site, whose single-particle states are described by only an observable (the pseudospin), and the eigenvalues are nondegenerate, the single-particle states , defining the Schmidt basis are and . Therefore, the (normalized) Schmidt decomposition of the state , obtained by Eq. (3) of the manuscript, is given by

The corresponding entanglement is quantified by the von Neumann entropy .

### Schmidt decomposition of the state |Ψϕ〉 of two qutrits in the same site

We give the Schmidt decomposition of two identical qutrits in the same site, each characterized by the basis . This system is equivalent to that of two spin-1 bosons in the same hole, previously analyzed by an alternative method16. We consider the state

where the spatial index has been omitted for simplicity. By performing the partial trace of ρ onto the basis , we obtain the reduced density matrix

which has eigenvalues , and eigenstates , , , , which define the Schmidt basis. From Eq. (3) of the main text, the SD of the state is

format: plain


text:
# Are there any laws in Islam which it is better to die rather than violate the law?

Are there any laws in Islam which it is better to die rather than violate the law?

Does it matter whether violating the law would cause the person to die naturally (for example, eating pork while stranded with no other food) or whether the person is being forced to violate the law (for example, someone threatens someone with a gun and promises to kill him if he doesn't eat pork).

Eating pork is just the example I have used here. Are there any laws which should be broken in either of the two scenarios mentioned above?

Preservation of life is perhaps the most crucial objective of the Shariah. If one were to organize the aims of Islam and shariah into a pyramid, this would be sitting at the top.


text:
Math Help - Convergent sequences

1. Convergent sequences

let

2. Originally Posted by Godisgood
Let X_n and y_n be two convergent sequence in the metric space (X,d) with X_n converging to x and y_n converging to y. show that d(x_n, y_n) converges to d(x,y)

Thanks

$d(x_n,y_n) \le d(x_n,x)+d(x,y_n)\le d(x_n,x)+d(x,y)+d(y,y_n)$

This implies

$d(x_n,y_n)-d(x,y) \le d(x_n,x)+d(y,y_n)$

This should get you started. This is half of the inequality.

3. That is true only if d(*,*) is continous in [x,y]...

Kind regards

$\chi$ $\sigma$

4. Do you know that $\left| {d(x,y) - d(x_n ,y_n )} \right| \leqslant \left| {d(x,y) - d(x_n ,y)} \right| + \left| {d(x_n ,y) - d(x_n ,y_n )} \right|~~(\#1)$

There is a well-known theorem for metric spaces:
for any points $a,~b~\&, t \in S$ it is the case that $\left| {d(a,t) - d(b,t)} \right| \leqslant d(a,b)$.

Apply that to both terms on the right of $~(\#1).$


text:
# 7. Define a Markov Chain on S-0,1,2,3,... with transition probabilities Pi,i+1 with 0<p < 1/2. Prove...

###### Question:

7. Define a Markov Chain on S-0,1,2,3,... with transition probabilities Pi,i+1 with 0<p < 1/2. Prove that the Markov Chain is reversible.


text:
Animated Mandelbrot

This is a seemingly difficult question I think. I am (or at least will be) writing a dissertation of Fractal Geometry which is to be submitted electronically. For obvious reasons, I will be hoping to include an illustration of the Mandelbrot Fractal.

There are answers available for a 'simple' image of the fractal but I was hoping there would be a way to animate this image to zoom in (in theory endlessly) – a little like this

but not as detailed. Does anybody know how to do this?

My preamble is currently empty so I full list of necessary packages would be great. Thank you in advance.

If you really want to burn your rader's/reviewer's CPU, here you are:

\documentclass[tikz]{standalone}

\def\pspreparation#1#2#3{
20 div exch 20 div % y x
#3 div exch #3 div % x y
#2 add exch #1 add 2 copy true % y x y x true
}
\def\psiteration{ % y x Y X in?
{dup dup mul 2 index dup mul sub 3 index sub 3 1 roll 2 mul mul 3 index sub exch 2 copy dup mul exch dup mul add 3 le}{false}ifelse
}
\def\psdecoration{
5 1 roll pop pop pop pop % in?
{0 0 0}{1 1 1}ifelse % R G B
}
\def\pgfdeclaremandelbrot#1#2#3{
\pspreparation{#1}{#2}{#3}
\psiteration\psiteration\psiteration\psiteration\psiteration\psiteration\psiteration\psiteration
\psiteration\psiteration\psiteration\psiteration\psiteration\psiteration\psiteration\psiteration
\psdecoration
}
}
\def\pgfprintmandelbrot#1#2#3{
\pgfdeclaremandelbrot{#1}{#2}{#3}
}

\begin{document}
% x-shift, y-shift, scale
\pgfprintmandelbrot{ .6}{ 0}{1}
\pgfprintmandelbrot{ .4}{ 0}{1.2}
\pgfprintmandelbrot{ .2}{ 0}{1.44}
\pgfprintmandelbrot{ .0}{ 0}{1.728}
\pgfprintmandelbrot{-.2}{ 0}{2.0736}
\pgfprintmandelbrot{-.4}{ 0}{2.48832}
\pgfprintmandelbrot{-.4}{.1}{2.985984}
\pgfprintmandelbrot{-.4}{.2}{3.5831808}
\pgfprintmandelbrot{-.4}{.3}{4.29981696}
\pgfprintmandelbrot{-.4}{.4}{5.159780352}
\end{document}

GIF version: (Due to the ImageMagick fails to convert PDF files containing functional shading, it is going to be glitchy)

Fixed GIF version (using Preview on Mac)

• That... is... amazing! You are a genius! Thank you. – PercyF2519 Mar 12 '17 at 20:15
• It seems like my plot heading the opposite direction. That is in fact my prefered direction. See my profile picture. (Yes, it is done by TeX.) – Symbol 1 Mar 12 '17 at 20:22
• Why is it your favourite direction? I suppose it makes no real difference for visual displays. – PercyF2519 Mar 12 '17 at 20:24
• @PercyF2519 Because it makes me unique! But computationally, as you said, I am not :( – Symbol 1 Mar 12 '17 at 20:27


text:
# Inrush control transient at power on

I have a circuit with a not-so-small capacitor on the supply and I need to reduce the inrush surge; as a matter of fact the power supply breaks after a while (these cinese supplies are not fun)

I followed ON Semi AND9093/D (which is in fact very well made) and come up with the following circuit:

From the left comes a 12V supply, on the bottom there is a BJT enabling the switch when the times come and on the right after the beefy capacitor there a mixed load; at the moment is still disabled but there's some leakage anyway on the right side, if only for the capacitor one.

The circuit actually works quite well and limits the current as designed (5A, by the way):

However there is still a surprise: when I initially apply the supply there is still a surge while the load switch itself stabilizes:

I explain this as follow:

• At start up everything is discharged, C12 and C13 included;

• When I supply power Q1 is in conduction even if /HSGRAIL is not pulled down since the source is at supply level and the gate is still low since C12 is not charged yet (probably some leakage from somewere make it starting low, even the /HSGRAIL has some leakage of course)

• The gate initial charge to the supply is slowed down since C12 needs to be charged initially, too.

However if I lower R2 to raise the charge current, R3 need to go down to maintain the FET below Vth and then C12 would need to be bigger in proportion (the sad rule about RC time constant)

What would be the best way to solve this?

• Some magic topology trick?
• A push-pull gate drive (i.e. R2 and R3 replaced by transistors) so that it actually starts charged quickly? probably would need to make it discrete since most of the integrated ones stay high impedance until supply is stable;
• Switch to an N-channel switch and a charge pump driver? (a lot more expensive but if it's the only way)

EDIT: I forgot the oldest trick in the book:

• No gate-source capacitor.
• A 'suitably big' charge resistor (refined people would want to use a inrush control NTC)
• After some time bypass the resistor/NTC with the MOSFET to reduce losses (even when the cap is charged it will be need to be recharged)

22 ohm about 1W seems to do the trick, I'm doing tests

• Yes, C12 is the culprit because when it is discharged and you connect 12V, it will immediately turn on the P-FET until it is C12 is charged. Why it is between drain and gate, and why it isn't between source and gate? Jul 22, 2021 at 7:32
• I'm wondering the same, but the application note says it's better (because the miller plateu is bigger) and in fact in similar designs I usually see in that position. Even putting it on gate-source however would need time to charge it (maybe it would be faster but bigger to have the same control effect) Jul 22, 2021 at 7:36

I would recommend using a dedicated soft start load switch IC, for example an NCP45524. The advantages (to my mind):

• Low $$\R_{DS}\$$ N-Channel MOSFET, with a built in charge pump.
• Controlled soft start.
• Controlled discharge of the load when switched off.
• Thermal protection.
• Power good indicator.
• Reduced BOM complexity and space usage.


text:
# Find and correct the errors in the statement: x + 2x + 3x = 5x

81 views
Find and correct the errors in the statement:

x + 2x + 3x = 5x

by (24.8k points)
selected

L.H.S = x + 2x + 3x =1x + 2x + 3x = x(1 + 2 + 3)

= 6x ≠ R.H.S.

The correct statement is x + 2x + 3x = 6x


text:
SEARCH HOME
 Math Central Quandaries & Queries
 Question from Demetrie, a student: Dwayne's garden is triangle-shaped with two equal sides and a third side that is 4 ft more than the length of an equal side. If the perimeter is 49 ft, how long is the longest side?

Hi Demetrie,

In these algebra problems you sometimes need to decide which quantity is to be the "unknown". I am going to use the length of one of the equal length sides to be the unknown.

Let $L$ be the length of one of the two sides of equal length, in feet. Thus you have 2 sides of length $L$ feet each. The third side has a length $L + 4$ feet. What is the sum of the lengths of the three sides? This sum is the perimeter if 49 feet. Solve for $L$.

Penny

Math Central is supported by the University of Regina and the Imperial Oil Foundation.


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## Why this particular problem?

Let me briefly try to defend my choice of problem. I wanted to choose a genuine research problem in my own area of mathematics, rather than something with a completely elementary statement or, say, a recreational problem, just to show that I mean this as a serious attempt to do real mathematics and not just an amusing way of looking at things I don’t really care about. This means that in order to have a reasonable chance of making a substantial contribution, you probably have to be a fairly experienced combinatorialist. In particular, familiarity with Szemerédi’s regularity lemma is essential. So I’m not expecting a collaboration between thousands of people, but I can think of far more than three people who are suitably qualified in the above way.

Other criteria were that I didn’t want to choose a famous unsolved problem, or a problem where I had no idea whatever where to start. For a first attempt, it seemed a better idea to choose a problem that I’d love to solve, about which I already have some ideas, but in which I don’t (yet) have a significant emotional investment.

Does the problem split naturally into subtasks? That is, is it parallelizable? I’m actually not completely sure that that’s what I’m aiming for. A massively parallelizable project would be something more like the classification of finite simple groups, where one or two people directed the project and parcelled out lots of different tasks to lots of different people, who go off and work individually. But I’m interested in the question of whether it is possible for lots of people to solve one single problem rather than lots of people to solve one problem each.

However, my contention would be that any reasonably complex solution to a problem is somewhat parallelizable and becomes increasingly so as one thinks about it: when one solves a problem, one doesn’t first try to guess what Lemma 1.1 might be, but instead one tries to think of relevant mathematical statements that one has a chance of proving. And often when one gets stuck on one of these, one isn’t stuck on the main problem because there were still several unexplored avenues. If lots of people were working on a problem, then all these avenues could be explored at once — but they would have to be created first.

With this seemingly narrow project — to try to decide whether one particular approach can be made to work for one (interesting) special case of a theorem that has already been proved by different methods — there are already a few different things to think about. For example, there is the question of whether the graph $\Gamma$ has any useful properties that can be exploited, whether one can give it a useful and natural system of edge weights that makes it easier to handle, what one would actually want of a graph in order for a sparse regularity argument to be feasible, what sparse regularity statements there actually are out there, whether the averaging trick could somehow allow one to operate in a dense part of $\Gamma$, despite the objection in comment CC, whether there is some false statement that would need to be true for the approach to work, and so on.

I have two other ideas for projects of this kind if this one turns out to not to work very well, but I’ll keep those to myself for the time being.


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# sklearn.gaussian_process.kernels.Product¶

class sklearn.gaussian_process.kernels.Product(k1, k2)[source]

The Product kernel takes two kernels $$k_1$$ and $$k_2$$ and combines them via

$k_{prod}(X, Y) = k_1(X, Y) * k_2(X, Y)$

Note that the __mul__ magic method is overridden, so Product(RBF(), RBF()) is equivalent to using the * operator with RBF() * RBF().

Read more in the User Guide.

New in version 0.18.

Parameters:
k1Kernel

The first base-kernel of the product-kernel

k2Kernel

The second base-kernel of the product-kernel

Attributes:
bounds

Returns the log-transformed bounds on the theta.

hyperparameters

Returns a list of all hyperparameter.

n_dims

Returns the number of non-fixed hyperparameters of the kernel.

requires_vector_input

Returns whether the kernel is stationary.

theta

Returns the (flattened, log-transformed) non-fixed hyperparameters.

Examples

>>> from sklearn.datasets import make_friedman2
>>> from sklearn.gaussian_process import GaussianProcessRegressor
>>> from sklearn.gaussian_process.kernels import (RBF, Product,
...            ConstantKernel)
>>> X, y = make_friedman2(n_samples=500, noise=0, random_state=0)
>>> kernel = Product(ConstantKernel(2), RBF())
>>> gpr = GaussianProcessRegressor(kernel=kernel,
...         random_state=0).fit(X, y)
>>> gpr.score(X, y)
1.0
>>> kernel
1.41**2 * RBF(length_scale=1)

Methods

 __call__(X[, Y, eval_gradient]) Return the kernel k(X, Y) and optionally its gradient. clone_with_theta(theta) Returns a clone of self with given hyperparameters theta. Returns the diagonal of the kernel k(X, X). get_params([deep]) Get parameters of this kernel. Returns whether the kernel is stationary. set_params(**params) Set the parameters of this kernel.

Return the kernel k(X, Y) and optionally its gradient.

Parameters:
Xarray-like of shape (n_samples_X, n_features) or list of object

Left argument of the returned kernel k(X, Y)

Yarray-like of shape (n_samples_Y, n_features) or list of object, default=None

Right argument of the returned kernel k(X, Y). If None, k(X, X) is evaluated instead.

Determines whether the gradient with respect to the log of the kernel hyperparameter is computed.

Returns:
Kndarray of shape (n_samples_X, n_samples_Y)

Kernel k(X, Y)

K_gradientndarray of shape (n_samples_X, n_samples_X, n_dims), optional

The gradient of the kernel k(X, X) with respect to the log of the hyperparameter of the kernel. Only returned when eval_gradient is True.

property bounds

Returns the log-transformed bounds on the theta.

Returns:
boundsndarray of shape (n_dims, 2)

The log-transformed bounds on the kernel’s hyperparameters theta

clone_with_theta(theta)[source]

Returns a clone of self with given hyperparameters theta.

Parameters:
thetandarray of shape (n_dims,)

The hyperparameters

diag(X)[source]

Returns the diagonal of the kernel k(X, X).

The result of this method is identical to np.diag(self(X)); however, it can be evaluated more efficiently since only the diagonal is evaluated.

Parameters:
Xarray-like of shape (n_samples_X, n_features) or list of object

Argument to the kernel.

Returns:
K_diagndarray of shape (n_samples_X,)

Diagonal of kernel k(X, X)

get_params(deep=True)[source]

Get parameters of this kernel.

Parameters:
deepbool, default=True

If True, will return the parameters for this estimator and contained subobjects that are estimators.

Returns:
paramsdict

Parameter names mapped to their values.

property hyperparameters

Returns a list of all hyperparameter.

is_stationary()[source]

Returns whether the kernel is stationary.

property n_dims

Returns the number of non-fixed hyperparameters of the kernel.

property requires_vector_input

Returns whether the kernel is stationary.

set_params(**params)[source]

Set the parameters of this kernel.

The method works on simple kernels as well as on nested kernels. The latter have parameters of the form <component>__<parameter> so that it’s possible to update each component of a nested object.

Returns:
self
property theta

Returns the (flattened, log-transformed) non-fixed hyperparameters.

Note that theta are typically the log-transformed values of the kernel’s hyperparameters as this representation of the search space is more amenable for hyperparameter search, as hyperparameters like length-scales naturally live on a log-scale.

Returns:
thetandarray of shape (n_dims,)

The non-fixed, log-transformed hyperparameters of the kernel


text:
Home > Standard Error > Standard Error And Regression Coefficient

# Standard Error And Regression Coefficient

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text:
# Projective Geometric Algebra

In the conformal model, null vectors are instrumental in throwing off the shackles of coordinates. We cleverly built null vectors starting from a wacky inner product that induced negative squared norms.

Nothing. However, it results in a degenerate inner product, leading to a trickier algebra that historically deterred researchers. For example, $$\I$$ is no longer invertible, which makes duals less accessible.

Happily, other intrepid researchers overcame the obstacles, leading to projective geometric algebra (PGA). Simpler and more efficient than the conformal model, projective geometric algebra still handles rotations and translations beautifully.

Look familiar? Let $$f(X) = e^{X/2}$$ and:

$V = f(12 \z\infty) f(\y\x \gamma) f(8\x\infty) f(\z\y \beta) f(2\y\infty) f(\x\z \alpha)$

Again, the map $$p \mapsto V p V^{-1}$$ as $$\alpha, \beta, \gamma$$ range over all angles produces the above animation. But this time we started from a 4-dimensional vector space, leading to a 16-dimensional geometric algebra. Thus the number of entries in a multivector is at most the number of entries in a traditional workhorse of computer graphics, a 4x4 matrix.

There are drawbacks. We can no longer wedge 4 points to describe a circle. We lose succinct inversion and dilation. Perhaps for proving advanced geometry theorems we may be better off with the conformal model. But for many applications projective geometric algebra may be ideal.

## PGA Tour

We copy over definitions from older articles:

{-# LANGUAGE CPP #-}
import Data.Bool
import Data.List
import Data.Map (Map)
import qualified Data.Map as M
import Data.Maybe
#ifdef __HASTE__
import Data.IORef
import Haste.DOM
import Haste.Events
import Haste.Graphics.Canvas
import Haste.Graphics.AnimationFrame
#endif

type Multivector = Map String Double

m ! s = M.findWithDefault 0 s m

a ~== b = abs (a - b) < 10**(-6)

wedge :: Multivector -> Multivector -> Multivector
wedge m n = M.filter (not . (~== 0)) $M.fromListWith (+)$ catMaybes $f <$> M.assocs m <*> M.assocs n where
f a@(s, _) b@(t, _)
| length u == length s + length t = Just c
| otherwise                       = Nothing
where c@(u, _) = basisMul a b

dot :: Multivector -> Multivector -> Multivector
dot m n = M.filter (not . (~== 0)) $M.fromListWith (+)$ catMaybes $f <$> M.assocs m <*> M.assocs n where
f a@(s, _) b@(t, _)
| length u == length t - length s = Just c
| otherwise                       = Nothing
where c@(u, _) = basisMul a b

mul :: Multivector -> Multivector -> Multivector
mul x y = M.filter (not . (~== 0)) $M.fromListWith (+)$ basisMul <$> M.assocs x <*> M.assocs y  The bluntest approach might be to define one null vector for each dimension in our space. Points will certainly be represented by null vectors, but this takes degeneracy too far. In particular, if we want dot products to give useful information, then we need $$n$$ non-null basis vectors. Therefore we start with the standard inner product of $$\mathbb{R}^n$$, then add a null vector $$\e_0$$ giving us an $$(n+1)$$-dimensional space known as $$\mathbb{R}_{n,0,1}$$. In our code, we write 0 for $$\e_0$$ and handle this case specially, defaulting to the standard behaviour for all other basis vectors. basisMul (s, m) (t, n) = gnome [] (s ++ t) (m * n) where gnome pre (a:b:rest) c | a < b = gnome (pre ++ [a]) (b:rest) c | a == b = case a of '0' -> ([], 0) _ -> back pre rest c | a > b = back pre (b:a:rest) (-c) where back [] rest c = gnome [] rest c back pre rest c = gnome (init pre) (last pre:rest) c gnome pre rest c = (pre ++ rest, c)  Next, we move to the dual space $$\mathbb{R}^*_{n,0,1}$$ to spread $$\e_0$$ around. Recall the dual of a $$k$$-blade is an $$(n-k)$$-blade representing its orthogonal complement. For $$\mathbb{R}^*_{2,0,1}$$, if $$\e_1$$ and $$\e_2$$ are non-null basis vectors (which show up as 1 and 2 in our code), then we represent 1-vectors with the duals of $$\e_0, \e_1, \e_2$$. Already we are hampered by degeneracy. We usually multiply by $$\I$$ to compute a dual, but this fails for deriving $$\E_0$$ from $$\e_0$$, since $$\e_0$$ is a null vector. Furthermore, without this portal gun to teleport us between a space and its dual, we are forced to think "backwards" all the time: wedge products describe intersections and meet products describe spans. (If an orthogonal complement grows, then the original subspace shrinks, and vice versa.) It so happens we can temporarily pretend to have a standard inner product and multiply $$\e_0, \e_1, \e_2$$ by $$\I$$ to get their duals: $\E_0 = \e_1 \e_2, \E_1 = \e_2 \e_0, \E_2 = \e_0 \e_1$ The $$\E_1$$ and $$\E_2$$ represent the $$x$$ and $$y$$ directions, and $$\E_0$$ is the origin. All but $$\E_0$$ are null. We represent euclidean points and ideal points as follows: one :: Multivector one = [("", 1)] canon = mul one point2 x y = canon [("20", x), ("01", y), ("12", 1)] ideal2 x y = canon [("20", x), ("01", y)]  With these representations, for a finite point $$\P$$ we have $$\P^2 = -1$$, and for an ideal point $$V$$ we have $$V^2 = 0$$: square x = mul x x prop_finitePointSquaredIsMinusOne = square (point2 123 456) == [("",-1)] prop_idealPointSquaredIsZero = square (ideal2 123 456) == []  As we’re in a projective space, $$\lambda \P$$ also represents $$\P$$ for any nonzero scalar $$\lambda$$; a finite point is normalized when $$\P^2 = -1$$. Ideal points always square to zero. Like before, we avoid geometrically meaningless squares thanks to null vectors. The square of a (normalized) point is a constant, so is independent of the origin. We test our code by reproducing the multiplication table of the geometric algebra basis vectors from one of Gunn’s papers: basis2 :: [Multivector] basis2 = one : map (flip M.singleton 1) (words "0 1 2 12 20 01 012") pseudo2 = [("012", 1)] :: Multivector table1 :: [[Multivector]] table1 = [[mul r c | c <- basis2] | r <- basis2] crudeTable1 = putStr$ unlines $map (unwords . map (show . M.toList)) table1  [("",1.0)] [("0",1.0)] [("1",1.0)] [("2",1.0)] [("12",1.0)] [("02",-1.0)] [("01",1.0)] [("012",1.0)] [("0",1.0)] [] [("01",1.0)] [("02",1.0)] [("012",1.0)] [] [] [] [("1",1.0)] [("01",-1.0)] [("",1.0)] [("12",1.0)] [("2",1.0)] [("012",1.0)] [("0",-1.0)] [("02",-1.0)] [("2",1.0)] [("02",-1.0)] [("12",-1.0)] [("",1.0)] [("1",-1.0)] [("0",1.0)] [("012",1.0)] [("01",1.0)] [("12",1.0)] [("012",1.0)] [("2",-1.0)] [("1",1.0)] [("",-1.0)] [("01",-1.0)] [("02",-1.0)] [("0",-1.0)] [("02",-1.0)] [] [("012",1.0)] [("0",-1.0)] [("01",1.0)] [] [] [] [("01",1.0)] [] [("0",1.0)] [("012",1.0)] [("02",1.0)] [] [] [] [("012",1.0)] [] [("02",-1.0)] [("01",1.0)] [("0",-1.0)] [] [] [] The line $$a x + b y + c = 0$$ is represented by: line2 a b c = [("1", a), ("2", b), ("0", c)] :: Multivector  A euclidean line $$\a$$ is normalized when $$a^2 = 1$$. We provide similar routines for 3D geometry: point3 x y z = canon [("032", x), ("013", y), ("021", z), ("123", 1)] ideal3 x y z = canon [("032", x), ("013", y), ("021", z)] plane3 a b c d = [("1", a), ("2", b), ("3", c), ("0", d)] :: Multivector pseudo3 = [("0123", 1)] :: Multivector  Our demo needs to retrieve the coordinates of a euclidean point for the 3D case: -- Assumes input is a euclidean point. coords3 :: Multivector -> [Double] coords3 m = [-get "023", get "013", -get "012"] where d = m!"123" get s = m!s / d  ## Squared away Define the squared norm of a euclidean $$k$$-vector $$\vv{X}$$ by: $|| \vv{X} ||^2 = \vv{X}^2$ (This formula does not apply to all $$k$$-vectors, but rather only those representing points, lines, planes, and so on.) The square of an ideal element is always zero, so we also define the square of the ideal norm $$||\vv{X}||^2_\infty$$ to be the sum of the squares of the coefficients, and say an ideal $$k$$-vector $$\vv{X}$$ is normalized when $$||\vv{X}||^2_\infty = 1$$. This can be formulated without coordinates: $||\vv{X}||^2_\infty = ||\vv{X}\vee\P||^2$ where $$\P$$ is any normalized finite point. However, our implementation simply reads off the coefficients: scalarAdd = M.insertWith (+) "" scalarMul a v = (*a) <$> v

normSquared :: Multivector -> Double
normSquared v = square v ! ""

idealNormSquared :: Multivector -> Double
idealNormSquared = sum . map (^2) . M.elems

normalize :: Multivector -> Multivector
normalize v
| d ~== 0 = scalarMul (sqrt (1/idealNormSquared v)) v
| otherwise = scalarMul (sqrt (-1/d)) v
where d = normSquared v

## The meet product

Recall in the conformal model we computed wedge products to find the span of subspaces, such as the line defined by two points, or the plane defined by three points. Because PGA uses the dual space, we need meet products instead.

Usually we take advantage of $$A \vee B = (A^* \wedge B^*)^*$$. But without an easy dual operation, we resort to a shuffle product to compute the meet, a sort of geometric version of the inclusion-exclusion principle.

We need a helper function that returns all partitions of a sequence into two subsequences, along with the sign of the permutation obtained by concatening the subsequences.

signedPartitions sz xs = f sz xs where
f n xs
| n > length xs = []
| n == 0 = [(1, [], xs)]
| (x:xt) <- xs = [(s, x:as, bs) | (s, as, bs) <- f (n - 1) xt]
++ [(parity n s,as, x:bs) | (s, as, bs) <- f n xt]
parity n | odd n = negate
| otherwise = id

prop_signedPartitionsExample = signedPartitions 2 [1..5] ==
[(1,[1,2],[3,4,5])
,(-1,[1,3],[2,4,5])
,(1,[1,4],[2,3,5])
,(-1,[1,5],[2,3,4])
,(1,[2,3],[1,4,5])
,(-1,[2,4],[1,3,5])
,(1,[2,5],[1,3,4])
,(1,[3,4],[1,2,5])
,(-1,[3,5],[1,2,4])
,(1,[4,5],[1,2,3])
]

To compute meets with a shuffle, we need the dimension of the space, which in a cleaner library might be held somewhere by each Multivector value. (Such a library could also prevent mixing up multivectors from different spaces.) Instead, we require the caller to supply the dimension.

vee :: Int -> Multivector -> Multivector -> Multivector
vee dim m n = M.filter (not . (~== 0)) $M.fromListWith (+)$ concat $f <$> M.assocs m <*> M.assocs n where
f a@(s, m) b@(t, n)
| dim > length s + length t = []
| otherwise = [(bs, fromIntegral sgn*m*n) |
(sgn, as, bs) <- signedPartitions (dim - length t) s,
intersect as t == []]

vee2 = vee 3
vee3 = vee 4

## Products in 2D

Multiplication by $$\I$$ is commutative, and computes the orthogonal complement in the Euclidean 2D space. (Not the geometric algebra itself, otherwise we would have easy access to the dual!)

For a euclidean line $$\a$$, the product $$\a\I$$ is the ideal point in the direction perpendicular to the line.

For a euclidean point $$\P$$, the product $$\P\I$$ is $$\e_0$$, the ideal line.

prop_linePerp = line2 1 2 3 mul pseudo2 == ideal2 1 2
prop_pointPerp = point2 42 9000 mul pseudo2 == [("0",-1)]

The product of two euclidean lines is:

$\a\b = \a \cdot \b + \a \wedge \b$

The dot product $$a \cdot \b$$ is the cosine of the angle $$\theta$$ between them.

Let $$\P$$ denote their normalized point of intersection. If $$\P$$ is finite, that is, if the lines intersect, then $$\a \wedge \b = (\sin \theta) \P$$. If $$\P$$ is ideal, that is, the lines are parallel, then $$\a \wedge \b = d_{ab} \P$$ where $$d_{ab}$$ denotes the distance between the lines.

The product of two euclidean points is:

$\P \Q = -1 + d_{PQ} \vv{V}$

where $$d_{PQ}$$ is the distance between them and $$\vv{V}$$ is an ideal line representing the direction perpendicular to the line joining them. We define

$\P \times \Q = d_{PQ} \vv{V}$

hence $$d_{PQ} = || \P \times \Q ||_\infty$$.

The product of a euclidean line and a euclidean point is:

$\a \P = \a \cdot \P + \a \wedge \P = \a^\perp_\P + d_{\a\P}\I$

where $$\a^\perp_\P = \a \cdot \P$$ is the line through $$\P$$ perpendicular to $$\a$$ and $$d_{\a\P}$$ is the distance between the point and the line.

It turns out we also have meaningful products when one or both of the inputs are ideal.

## Transformations

There are more geometric interpretations to explore and we have yet to touch the 3-dimensional case. But let’s fast-forward to the part we’ve all been waiting for: rotors!

The reflection of $$\vv{X}$$ in the $$(n-1)$$-hyperplane $$\a$$ is:

$\a\vv{X}\a$

This formula reflects points, lines, planes, and so on.

We could finish here, as all Euclidean isometries can be generated from reflections. Composing reflections through $$\a$$ then $$\b$$ is simply:

$\b\a\vv{X}\a\b$

If $$\vv{R} = \b \a$$ is normalized then we call it a rotor, and transform a subspace $$\vv{X}$$ by computing $$\vv{R} \vv{X} \tilde{\vv{R}}$$, where $$\tilde{\vv{R}}$$ is the reversal of $$\vv{R}$$, meaning we multiply the factors in reverse order. Calculation shows $$\vv{R}\tilde{\vv{R}} = 1$$.

Rotations have an exponential form:

$e^{\frac{\theta}{2} \P}$

where $$\P$$ is the normalized point (in 2D) or axis (in 3D) at the center of a rotation by the angle $$\theta$$. For euclidean points, $$\P^2 = -1$$ so $$\P$$ takes the role played by $$\vv{i}$$ in other models.

For example, a rotation around the $$y$$-axis by 120 degrees is given by:

approxEq v w = all (~== 0) $M.elems$ M.unionWith (-) v w

prop_rotateExample =
axis == [("13",-1)] && rotor approxEq [("",0.5),("13",-sqrt(3)/2)]
where
rotor = scalarAdd (cos (a/2)) $scalarMul (sin (a/2)) axis axis = normalize$ point3 0 0 0 vee3 point3 0 1 0
a = 2*pi/3

Observe we computed the meet product of two points to find the multivector describing the $$y$$-axis. It turns out to have a simple form, though in general lines in 3D are less straightforward in PGA:

prop_lineExample =
line == [("01",-23),("02",-14),("03",17),("12",-7),("13",-3),("23",-7)]
where
line = point3 1 2 3 vee3 point3 6 5 4

In 2D, translation by $$d$$ units can be viewed as a rotation of size $$d$$ with the center at an ideal point $$\P$$ in the perpendicular direction. By considering the Taylor series expansion, and recalling $$\P^2 = 0$$, we find:

$e^{\frac{d}{2} \P} = 1 + \frac{d}{2} \P$

For a 3D translation, the axis is an ideal line, which Gunn states can be found by computing $$(\E_0 \vee \vv{V})\I$$ where $$\vv{V}$$ is the direction of travel. It seems we find the line joining the origin and $$\vv{V}$$ then take its orthogonal complement to yield the axis.

$e^{\frac{d}{2} (\E_0 \vee \vv{V})\I}$

For example, the following rotor translates 12 units along the $$z$$-axis:

prop_translateExample =
zdir == [("03",-1)] && rotor == [("",1),("03",-6)]
where
zdir = mul (M.singleton "123" 1 vee3 ideal3 0 0 1) pseudo3
rotor = scalarAdd 1 $scalarMul (12/2) zdir  Rotations and translations involving the coordinate axes have a simple form so are great for demos. moon a b c p = foldl1 mul$ rs ++ [p] ++ map inv (reverse rs) where
inv = M.mapWithKey f where
f [] v = v
f _  v = -v
rs =
[ tra "03" 12
, rot "21" c
, tra "01" 8
, rot "32" b
, tra "02" 2
, rot "13" a
]
rot axis ang = [("", cos (ang/2)), (axis, sin (ang/2))] :: Multivector
tra dir d = [("", 1), (dir, d/2)] :: Multivector

Above, our definition for $$V$$ uses notation from our adventure in the conformal model: the symbols $$\infty, \x, \y, \z$$ correspond to $$\e_0,\e_1,\e_2,\e_3$$ which are 0 1 2 3 in our code.

## Into orbit

Lastly, we slap together some code to animate a cube on this webpage.

#ifdef __HASTE__
-- Cube faces and their colours.
cubeFC = zip
[[0, 1, 3, 2], [4, 6, 7, 5],
[0, 4, 5, 1], [2, 3, 7, 6],
[1, 5, 7, 3], [0, 2, 6, 4]]
-- http://the-rubiks-cube.deviantart.com/journal/Using-Official-Rubik-s-Cube-Colors-268760351
[RGB 0 0x51 0xba, RGB 0 0x9e 0x60,
RGB 0xff 0xd5 0, RGB 0xff 0xff 0xff,
RGB 0xff 0x58 0, RGB 0xc4 0x1e 0x3a]

paint canvas tRaw = sequence_ $renderOnTop canvas . drawFace <$> filter vis cubeFC
where
ms = round tRaw :: Int
ang s = fromIntegral (mod ms (s*1000)) * 2 * pi / fromIntegral (s*1000)
cubeVertices = (\[a,b,c] -> point3 a b c) <$> replicateM 3 [-1, 1] ws = coords3 . moon (ang 2) (ang 5) (ang 23) <$> cubeVertices
drawFace (face, rgb) = color rgb . fill . path $project . (ws!!) <$> face
project :: [Double] -> (Double, Double)
project [x, y, z] = (x/z*160 + 160, y/z*160 + 160)
-- Cull faces pointing away from us.
vis (face, _) = d!"" <= 0 where
(u:v:_) = zipWith (zipWith (-)) t ts
d = foldl1' mul [toGA u, toGA v, M.singleton "zyx" 1, toGA h]
ts@(h:t) = (ws!!) <$> face toGA = M.fromList . zip (pure <$> "xyz")

main = withElems ["canvas"] $\[cElem] -> do Just canvas <- fromElem cElem paused <- newIORef False let anim t = do render canvas$ fill $rect (0, 0) (320, 320) paint canvas t b <- readIORef paused unless b$ void $requestAnimationFrame anim pause = do b <- readIORef paused writeIORef paused$ not b
when b $void$ requestAnimationFrame anim
_ <- cElem onEvent MouseDown \$ const pause
requestAnimationFrame anim
#endif

Ben Lynn blynn@cs.stanford.edu 💡


text:
# Solve the that that that rebus

It's a hard rebus to do and the rebus is: 3 "that"s

                                                                               that
that
that

• Probably the author has a stutter. – rhsquared Jun 5 '19 at 8:45
• It's a list of things Meat Loaf won't do for love. – hexomino Jun 5 '19 at 9:18

My guess is...

"That's all right."

• This is actually the correct answer, as the OP confirmed in a comment yesterday on the other answer :-) – Rand al'Thor Jun 7 '19 at 19:57
• My answer includes "all" (since the three are all to the right) which is a bit different, but I see your point. I didn't realize OP had accepted an answer since there was no green check mark, – Headblender Jun 7 '19 at 20:20
• Yep, I upvoted your answer, and hope it would be accepted if the OP returns. – Rand al'Thor Jun 7 '19 at 20:22

I thought there was a phrase

It’s all that

but couldn’t find a link at short notice. I did find the following show, though:

And now that we’ve come to the end of this answer,

That’s all, folks.

PS: from the horse’s mouth,

That’s right!

• Sorry its thats right – user60673 Jun 6 '19 at 3:50
• @user60673 Apologies, I don't understand that comment. – Lawrence Jun 6 '19 at 5:09
• I believe the OP is telling you the answer to the rebus! – Gordon K Jun 6 '19 at 14:56
• @GordonK Oh! Got it now, thanks :) ! But isn't the original that-that-that text in the middle of a horizontal scroll-box? When I scroll all the way to the right, the block of text appears to move to the left. – Lawrence Jun 6 '19 at 14:58
• Since OP unfortunately decided to give up the answer in a comment, and it isn't actually in this answer, perhaps you could edit your answer to provide the intended solution. – Rubio Jun 6 '19 at 20:53


text:
2. Economics
3. 1 all of the assumptions regarding consumer preferences is correct...

# Question: 1 all of the assumptions regarding consumer preferences is correct...

###### Question details

1. All of the assumptions regarding consumer preferences is correct except for:
a) Having more of a good is better for the consumer.
b) Consumer preferences are transitive.
c) Consumer preferences are complete.
d) Consumer preferences may be irrational.