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# Reprogrammable plasmonic topological insulators with ultrafast control

## Abstract

Topological photonics has revolutionized our understanding of light propagation, providing a robust way to manipulate light. So far, most of studies in this field are focused on designing a static photonic structure. Developing a dynamic photonic topological platform to switch multiple topological functionalities at ultrafast speed is still a great challenge. Here we theoretically propose and experimentally demonstrate a reprogrammable plasmonic topological insulator, where the topological propagation route can be dynamically changed at nanosecond-level switching time, leading to an experimental demonstration of ultrafast multi-channel optical analog-digital converter. Due to the innovative use of electric switches to implement the programmability of plasmonic topological insulator, each unit cell can be encoded by dynamically controlling its digital plasmonic states while keeping its geometry and material parameters unchanged. Our reprogrammable topological plasmonic platform is fabricated by the printed circuit board technology, making it much more compatible with integrated photoelectric systems. Furthermore, due to its flexible programmability, many photonic topological functionalities can be integrated into this versatile topological platform.

## Introduction

The discovery of topological states of matter in condensed matter physics has inspired the search for analogous effects in classical and bosonic systems, especially fruitful in photonics. The beginnings of topological photonics can be traced to the efforts to emulate the quantum Hall effect (QHE), where the time-reversal symmetry of the system is broken by exploiting magneto-optic photonic crystals under external static magnetic fields1,2,3,4,5,6. However, magneto-optical effects in most magnetic materials are weak and in many cases, it can be cumbersome and inconvenient to implement a magnetic device. To overcome these limitations, nonmagnetic photonic topological systems are developed subsequently, in which the time-reversal symmetry is preserved, to mimic the quantum spin-Hall effect (QSHE) or quantum valley-Hall effect (QVHE)7,8,9,10,11. The unique topological features of light in topological photonics, such as unidirectional and backscattering-immune propagation against defects and sharp bends, offer a robust way to control the behavior of light. The study of topological photonics has revolutionized our understanding of light propagation, and holds great promise for practical photonic applications, especially for the development of robust linear and nonlinear photonic devices12,13,14,15,16.

Currently, the studies of photonic topological insulators (PTIs) are mainly focused on the development of different static photonic systems to realize a specific photonic topological phenomenon or functionality17,18,19, and hence their controllability and reconfigurability are limited. In practical applications, however, multiple photonic topological functionalities are expected to be achieved in a single but reconfigurable PTI, so that the time and costs associated with the design and fabrication process can be reduced. To this end, several key works on reconfigurable topological insulators20,21,22,23,24,25 have been reported recently, in which the reconfigurability is mostly implemented by changing either the geometry or material parameters. Unfortunately, these features in PTIs cannot be easily altered once the photonic structure has been fabricated. Moreover, the mechanical, thermal or optical method used to realize the reconfigurability is not convenient for use in compact photoelectric integrated systems, and the reconfiguring speed is limited. Recently, the emergence of programmable metasurfaces opens a digital way to dynamically control the metasurface functionality, which has spurred wide interest in metamaterial community26,27,28,29,30,31. However, limited by the reflection-type binary state, current programmable metasurfaces are mainly used to tailor the reflection and radiation of electromagnetic (EM) waves, and the study that aims to control the propagation route of guiding waves has been hardly reported.

In this work, taking advantage of both the flexible programmability of digital metasurfaces and the robust EM propagation of PTIs, we propose and experimentally demonstrate an ultrafast reprogrammable plasmonic topological insulator (RPTI), which is comprised of metallic elements arranged in a hexagonal lattice structure. To achieve an electrically driven programmability, each metallic element has six arms whose EM response can be electrically controlled via a positive-intrinsic-negative (PIN) diode, thus a topological bandgap can be created by breaking the space-inversion symmetry of the unit cell to emulate the QVHE. Moreover, each unit cell of the RPTI can be encoded independently to enable a dynamic control of topological light propagation routes.

Compared to existing controllable PTIs, our RPTI has two unique features. First, its programmability is achieved by electrically encoding the binary states of the PIN diode, leading to an ultrafast dynamic control of different topological propagation routes. Second, our RPTI is fabricated by a widely-used printed circuit board (PCB) technology, and thus it can be seamlessly integrated with the commonly used PCB-based photoelectric integrated circuits. These unique features are crucial for the development of versatile and intelligent topological photoelectric devices for future practical applications.

## Results

### The system

The proposed RPTI (Fig. 1a) consists of programmable unit cells (Fig. 1b) arranged in a hexagonal lattice, with each unit cell being in one of its four digital states “0”, “1”, “2”, and “3” (see Fig. 1c). This system configuration enables us to encode one RPTI into several distinct plasmonic crystal domains, as shown in Fig. 1a. Applying different coding sequences to each programmable unit cell at different times (t1, t2, and t3) using a field-programmable gate array (FPGA), we can dynamically reconfigure the size and pattern of each plasmonic crystal domain to realize an on-demand manipulation of the domain-wall interface along which the microwave propagates. Moreover, this reconfigurable topological propagation route can be observed directly in experiments using a scanning near-field microwave microscopy (SNMM). Specifically, the programmable unit cell in Fig. 1b consists of a dielectric spacer sandwiched between a controllable printed metallic element at the top and, at the bottom, a metallic ground plate perforated by six holes forming a hexagon. Unlike previous studies32,33, our disk has six inner arms and six outer arms, where each inner arm is bridged with one outer arm via a PIN diode. Furthermore, to ensure that the direct current (DC) bias voltage of the inner arms is zero, the six inner arms are connected to the ground plate by a metallic via hole. The six outer arms are connected to a FPGA control network by six metallic via holes. As a result, applying different DC voltages to each outer arm, one can dynamically switch “on” and “off” the corresponding PIN diode to emulate the binary states “1” and “0”, respectively. The geometry parameters of our RPTI studied in this work are $$a=15/\sqrt{3}$$ mm, t = 3 mm and r = 7.5 mm (see Fig. 1b). In addition, the radius of the inner metallic circular disk is 1.5 mm, and the gap distance between the inner and outer arms is 0.3 mm.

In principle, the reprogrammable unit cell in Fig. 1b is a 6-bit digital element possessing 26 = 64 digital states. To simply our discussion, the odd (even) outer arms are designed to be in the same binary state as detailed in Supplementary Fig. 1, so that the 6-bit digital element can be reduced to a 2-bit digital element possessing 22 = 4 digital states. In the context of programmable metasurfaces31, they could be naturally encoded as 0 = “00”, 1 = “01”, 2 = “10” and 3 = “11”, as listed in Fig. 1c. Here, the binary codes “1” and “0” mean that the PIN diode is switched “on” and “off”, respectively. In the following, we denote each digital state as a digital unit in Fig. 1c, though in practice they are the same programmable unit cell. Our RPTI is designed in the microwave regime, where metallic loss can be suppressed and spoof surface plasmons (SSP) exist34. As demonstrated in Supplementary Figs. 2-5, the field of SSP is concentrated at the outer arm, if the corresponding PIN diode is switched on. However, if the diode is switched off, the field of SSP is located at the inner arm, and the corresponding outer arm is “invisible” to EM waves35. Based on this feature, we can engineer and dynamical control the band diagram of the RPTI, as demonstrated in Fig. 1d. In particular, due to the C6 symmetry of the unit cell, there is a Dirac point in the band diagram of digital unit-0, in which all diodes are switched off. Similarly, Dirac points also appear in the band diagram of digital unit-3, in which all diodes are switch on. For the digital unit-1, the diodes on its odd (even) arms are switched off (on), so the C6 symmetry is reduced to C3 symmetry. As a result, the C6-symmetry-protected Dirac points are gapped out to form bandgaps at K and K valley points, as illustrated in the band diagram of digital unit-1.

The emergence of valley-Hall states within the bandgap, which underpins the connection between the observed phenomena and applications on one side and the topological invariants characterizing the system on the other, can be explained as follows: as digital unit-1 and unit-2 are related by a π/3 in-plane rotation in real space, the first Brillouin zone (FBZ) of digital unit-2 in momentum space can also be obtained from that of digital unit-1 via a π/3 rotation (Fig. 1e). As a result, the K(K′) valleys of unit-1 are transformed to K′(K) valleys of unit-2. Since the valley Chern numbers of K and K′ are ±1/233,36,37, for a domain-wall interface between a unit-1-type domain and a unit-2-type domain, the difference of the valley Chern number across the interface for K and K is ±1. According to the principle of bulk-edge correspondence36,37, a valley Hall edge mode band would emerge in the valley bandgap. In Fig. 1f, we further show that this nontrivial valley bandgap can be broadened by increasing the length s of the outer arm. By switching the “on” and “off” states of the PIN diode, the binary expression of topological phase transition between unit-1 and unit-2 domains is encoded as the digital expression, which leads to a digital and dynamically controllable topological propagation interface. In condensed matter physics, manipulating the valley Hall states in a dynamical way is challenging and as such we believe that our work could inspire renewed research efforts in programmable valleytronics38 in condensed matter physics.

### Projected bands of domain-wall interfaces

To demonstrate the topological nature of nontrivial domain-wall interface modes, we encode the unit cells of RPTI so as to construct two valley-Hall domain-wall interfaces with zigzag shape, namely the interfaces A and B in Fig. 2a, which separate two domains filled with digital unit-1 and unit-2, respectively. As demonstrated in Fig. 1f, the size of the valley bandgap is tunable with respect to the length s of the outer arm. Here the projected band diagrams corresponding to different lengths s are given in Fig. 2b. In each case, there are two valley-Hall interface modes. The blue (red) solid line indicates the nontrivial interface mode of the interface A (B), and the gray-shaded regions represent the projected bulk modes. From physical point of view, the valley bandgap should be as small as possible so as to differentiate it from a glide-plane symmetric interface39. However, in order to achieve a device with improved functionality, one should aim to achieve a large bulk topological bandgap that can well preserve the topological properties of the valley-Hall states. After a careful consideration of the tradeoffs between achieving a good valley-Hall phase and a broad topological spectral range, we found out that the optimum value of the length s is 1.8 mm.

To study the property of unidirectional propagation of topological interface modes, a straight RPTI waveguide with the B-type interface is encoded in Fig. 2c, where a chiral source consisting of six electric dipoles with ±π/3 phase difference between adjacent dipoles is employed to excite a right-circularly polarized (RCP) or left-circularly polarized (LCP) wave. As shown in Fig. 2b, if the source has frequency of 6.7 GHz, two distinct waveguide modes with opposite group-velocity, corresponding to points 1 and 2, are generated. Furthermore, from the valley-chirality-locking property of the topological interface modes, the chirality of the interface mode at point 1 is also opposite to that at point 2. Consequently, a unidirectional propagation of microwave can be achieved by selectively exciting an RCP or LCP phase vortex at the source point in Fig. 2c, which is numerically demonstrated in Fig. 2d, e.

### Reconfigurable topological light propagation

In addition to the unidirectional propagation, the feature of immunity against sharp bends is also experimentally studied here. As illustrated in Fig. 3a, a fabricated RPTI is dynamically controlled using an FPGA control network. To verify the backscattering-immune propagation against sharp bends, different topological and non-topological propagation routes are programmed in our RPTI by properly encoding each of its unit cells via the FPGA controller, and the corresponding near-field distributions of |Ez| are measured by a SNMM experimental setup (see Fig. 3b).

To be more specific, a straight-waveguide propagation route is studied first with the measured near-field distribution of |Ez| at 7.2 GHz (~0.5 GHz frequency shift with respect to simulated results) presented in Fig. 3c. The corresponding coding pattern used to implement the straight propagation route is indicated in inset. Moreover, the simulated and measured transmissions of the straight topological propagation route are compared in Fig. 3e, where a good agreement is observed. In addition to the straight propagation route, we further study the case of a 120-degree-bend propagation route. As shown in Fig. 3d, the propagation remains immune to the sharp bend, and the corresponding transmission spectra are given in Fig. 3f. It shows that the transmission of a 120-degree-bend topological interface only decreases slightly with respect to that of a straight topological interface. In addition to the robustness against the sharp bending of propagation routes, it is also relevant to study the effect of defects and disorder, which could generally arise from manufacturing imperfections and operational failure of devices. Our study shows that the proposed topological interface is rather robust even in the presence of high levels of disorder (for details see Supplementary Fig. 7).

### Ultrafast multi-channel optical analog-digital converter

Due to the flexible programmability, many existing photonic topological functionalities can be readily implemented in our universal RPTI platform. More importantly, our RPTI can also be used to achieve some dynamic functionalities that cannot be implemented in static PTIs or low-speed reconfigurable PTIs. Here, a proof-of-principle experiment of a topologically protected multi-channel optical analog-digital converter with ultrafast control is demonstrated in Fig. 4. To simplify our demonstration, four waveguide ports are designed in our RPTI as illustrated in Fig. 4a–c, and a continuous wave (CW) with frequency 7.2 GHz is pumped at the input port 1. By encoding different topological propagation routes, we can dynamically switch the output port from ports 2 to port 3 or port 4, thus the input CW analog signal can be directly discretized into different digital signals (DS) at the output ports. To demonstrate this remarkable feature, the input CW signal is modulated as three ASCII sequences (the corresponding glyph sequences are “Hi!”, “SEU”, and “UCL”), and the three output digital signals are measured experimentally, as shown in Fig. 4d. The measured results show that the switching time in this system is as short as 50 ns, namely more than seven orders of magnitude faster than that of previous work20. Moreover, since only one topological propagation channel exists at each time step, the interference among the output ports is negligible, even when the output ports are closed to each other. This low cross-talk feature is crucial in practical high-fidelity digital communications.

In principle, the reprogrammable plasmonic topological insulator demonstrated here can be viewed as a programmable photonic circuit. In the current state-of-the-art photonic technology, the dynamic programmable functionality of most photonic integrated circuits (PICs) is mainly achieved by locally switching analog optical gates40, which in turn are usually controlled mechanically, thermally or electrically. As a result, the number and physical location of output ports are fixed once the PIC is fabricated. Since the dynamic switching functionality of our PTI is implemented by tuning its global topology, the tunable degree of freedom of the proposed PIC is much higher than that of current programmable photonic circuits. This translates in a more flexible and agile design of light propagation routes. However, facilitated by advanced fabrication methods (such as MEMS, CMOS etc.), the current programmable PIC is higher in integration and lower in energy consumption than our PTI. Thus, in the future, it will be a promising and important work to further reduce the energy consumption or improve the integration of the proposed PTI by using advanced fabrication methods.

## Discussion

To summarize, we have theoretically proposed and experimentally demonstrated a reprogrammable plasmonic topological insulator with ultrafast switching speeds at nanosecond scale. By investigating the band diagrams of the proposed RPTI, we shown that a topological bandgap emulating QVHE can be created by digitally encoding its unit cells via switching electrically the PIN diodes “on” and “off”. As each unit cell can be independently programmed, it was experimentally demonstrated that valley-Hall topological light propagation routes in our RPTI can be dynamically reconfigured. Moreover, we experimentally implemented a topologically protected multi-channel optical analog-digital converter to illustrate the ultrafast control feature of our RPTI, and demonstrated that the switching time in our case is as short as 50 ns, which is more than seven orders of magnitude faster than that of previous work. Our work brings the current studies of photonic topological insulators to a digital and intelligent era, which could open an avenues towards the development of software-defined photoelectric elements in high-speed communications and computation-based intelligent devices with built-in topological protection.

## Methods

### Numerical simulations

The numerical simulations were performed using the finite-element method based software COMSOL Multiphyiscs. In Fig. 1b, the metallic material (yellow) is copper and the dielectric substrate (semitransparent gray) is a widely-used commercial PCB board (F4B), whose relative permittivity is 2.65 and loss tangent is 0.001. The PIN diodes (bold pink) are modeled by lumped elements31,41. The “off” state is modeled by an inductor-capacitor series circuit, where the inductance L = 0.4 nH and the capacitance C = 40 fF. The band diagrams in Fig. 1d were evaluated at the outer-arm length s = 1.7 mm. The “on” state is modeled by a resistor-inductor series circuit, where the resistance R = 2.2 Ω and the inductance L = 0.32 nH. The projected band diagrams in Fig. 2a, b were evaluated using a supercell containing 10 unit cells on either side of the domain wall, and periodic boundary conditions were imposed at left and right sides.

### Sample fabrication

Our RPTI sample was fabricated in multiple-layer lamination process of PCB technique. In the layer stack management, as presented in Supplementary Fig. 1b, L1 and L2 are printed on the top and bottom facets of a F4B substrate (3 mm thickness), whereas L3 and bias-voltage-line layers (Supplementary Fig. 6b) are printed on the top and bottom facets of a FR4 substrate (0.5 mm thickness), whose relative permittivity is 4.3 and loss tangent is 0.025. A prepreg layer (FR4 with 0.2 mm thickness) for bonding the F4B and FR4 substrates is used between L2 and L3 layers. The thickness of the printed copper film in each layer is 35 µm. The overall size of our RPTI is ~465 × 221 mm2, containing 488 digital units with 2928 PIN diodes. Commercially available Aluminum Gallium Arsenide PIN diodes (MADP-000907-14020) are employed in the fabrication.

### Experimental measurements

A scanning near-field microwave microscopy is employed to measure the near electric fields and transmissions. Our SNMM is composed of vector network analyser (Agilent N5230C), low-noise amplifier, and several phase-stable coaxial cables. Two coaxial probes that act as the emitter and receiver are employed in measurements. One probe is soldered on the edge of the RPTI to generate a plasmonic mode, whereas the other one is fixed on a scanning support to scan the near-field distributions. During the measurements, the RPTI is placed on an acrylic supporter, as demonstrated in Supplementary Fig. 6a. Since each digital unit requires two voltage-control signals to realize the 2-bit states (Supplementary Fig. 1), our RPTI has 976 independent channels for voltage control. In order to dynamically steer the coding pattern of our RPTI, we use three field programmable gate arrays with a total of 1200 digital-voltage channels. The PIN diodes are connected to FPGA through white flexible printed circuit (FPC) cables, as shown in Supplementary Fig. 6c. In the measurement of the multi-channel optical analog-digital converter, a high-frequency oscilloscope (Agilent DSO9013A) and a signal generator (Agilent E8267D) are used to measure the wave forms in the time domain.

## Data availability

The data in Figs. 14 that support the findings of this study are available from the corresponding author upon reasonable request.

## Code availability

The code that supports the findings of this study are available from the corresponding author upon reasonable request.

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## Acknowledgements

This work was supported by European Research Council (ERC), Grant Agreement no. ERC-2014-CoG-648328; National Key Research and Development Program of China (Grant nos. 2017YFA0700201, 2017YFA0700202, and 2017YFA0700203); and the 111 Project (Grant no. 111-2-05).

## Author information

Authors

### Contributions

J.W.Y., N.C.P. and T.J.C. conceived the idea, J.W.Y. performed the theoretical analysis and numerical simulations, Z.L. helped with the theoretical analysis, Q.M. and Q.X. performed the experiments. N.C.P. and T.J.C. supervised the project. All authors contributed to the manuscript preparation.

### Corresponding authors

Correspondence to Nicolae C. Panoiu or Tie Jun Cui.

## Ethics declarations

### Competing interests

The authors declare no competing interests.

Peer review information Nature Communications thanks Alexander Khanikaev and the other anonymous reviewer(s) for their contribution to the peer review of this work. Peer reviewer reports are available.

Publisher’s note Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

## Rights and permissions

Reprints and Permissions

You, J.W., Ma, Q., Lan, Z. et al. Reprogrammable plasmonic topological insulators with ultrafast control. Nat Commun 12, 5468 (2021). https://doi.org/10.1038/s41467-021-25835-6

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# K* production in Heavy Ion Collisions at RHIC

K* production in Heavy Ion Collisions at RHIC - Descarga este documento en PDF. Documentación en PDF para descargar gratis. Disponible también para leer online.

Descargar gratis o leer online en formato PDF el libro: K* production in Heavy Ion Collisions at RHIC
Study of resonances with their short life-times provides useful tools to probe the properties of hot and dense matter produced in relativistic heavy ion collisions. The high density and-or high temperature of the medium can modify resonance properties such as mass and width. Therefore, measurement of these properties can reveal important information about the evolution dynamics in heavy ion collisions. We report the measurements of $K^{*}$ transverse momentum$p T$ spectra at mid-rapidity via its hadronic decay channel up to intermediate $p {T}$ of 2.9 GeV-c using the STAR detector in Au+Au and Cu+Cu collisions at $\sqrt{s {\mathrm{NN}}}$= 62.4 GeV and 200 GeV. These results are compared to previously reported $K^{*}$ results from Au+Au collisions at RHIC. Integrated yield ratios of $K^{*}-K$ and $K^{*}-\phi$ are used to understand the rescattering and regeneration effects on $K^{*}$ production.

Autor: Sadhana Dash; for the STAR Collaboration

Fuente: https://archive.org/

format: plain


text:
# Angles from gyroscope readings using quaternions

I am using gyroscope only to get real time angles as I move the IMU using a micro controller.

I am able to get angles at a pretty decent accuracy(2 to 3 degree error). I am using quaternions for obtaining angles.

The angles are with respect to the initial position of the IMU.

I rotate the IMU around individual axis i.e around any one Axis at a time, I get good accuracy. But when I rotate around all axis at once the problem starts as stated below.

Problem:

The problem here is when I give a pitch and then yaw all seems fine. Now at this position of pitch and yaw, I give a roll, the other two angles also change.

The gyroscope gives raw angular rate in DPS, which is converted to radian/sec for processing. The quaternions are calculated from raw angular rate.

The code to convert quaternions to euler angles is as follows,

//Local variables for clarity
fqw = gsangleparam.fquaternion[0];
fqx = gsangleparam.fquaternion[1];
fqy = gsangleparam.fquaternion[2];
fqz = gsangleparam.fquaternion[3];

//Calculate Angles*/
gsangles.yaw = atan2(2 * (fqx*fqy + fqw*fqz), (fqw*fqw + fqx*fqx - fqy*fqy -    fqz*fqz)) * 180/PI;

gsangles.pitch = asin(-2 * (fqx*fqz - fqw*fqy)) * 180/PI;

gsangles.roll = atan2(2 * (fqy*fqz + fqw*fqx), (fqw*fqw - fqx*fqx - fqy*fqy  + fqz*fqz)) * 180/PI;

The Problem of roll is present only when initial position(and also 180 deg counterpart) is as follows:

X - Facing left

Y - Facing towards us

Z - Facing UP

The problem is, When I give a pitch(around X here),yaw(around Z) and then roll(around Y here) or any sequence of roll pitch and yaw, then yaw changes.

In fact whenever I rotate about Y axis(roll here), Yaw(around Z) changes.

There is no problem when the initial position is(and also 180 deg counterpart) as follows,

X - Facing away from us

Y - Facing towards us

Z - Facing UP

There is no problem. i.e When I give a pitch(around Y here),yaw(around Z) and then roll(around X here) or any sequence of roll pitch and yaw, there is no problem.

• Can you provide the model number and/or datasheet for the gyro you're using? Everything I've said is speculative; the datasheet should give an absolute answer. – Chuck Dec 13 '16 at 14:14
• I use MPU 6500 from Invensense. The Sensitivity is 1000 DPS. – Nithin G A Dec 14 '16 at 8:31
• Any inputs from your side?? – Nithin G A Dec 15 '16 at 8:46
• How are you defining the x and y axes relative to the text on the MPU-6500 package? Also, how are you defining your rotations? About the body axes or about global axes? – Chuck Dec 15 '16 at 15:34
• For me the package is the reference, the data sheet mentions the X and Y axis directions with respect to the package. – Nithin G A Dec 16 '16 at 7:47

I'll link you to some math but try to give a conceptual explanation, too.

Consider a case where you want to start with the IMU upright and pointed forward and want to end with the IMU upside-down (new +z = old -z) but still pointed forward (new +x = old +x).

How do you get from the starting position to the end position?

1. Easiest way is to rotate about +x an angle of 180 degrees. (Roll)
2. You could rotate about +y an angle of 180 degrees and then rotate about the new +z axis an angle of 180 degrees (Pitch then Yaw)
3. You could rotate about +z an angle of 180 degrees and then rotate about the new +y axis an angle of 180 degrees (Yaw then Pitch)

There are other ways you could do it, too - you could rotate about successive axes in 90 degree increments, a bunch of little increments, etc. The point is, you need to define what your order of rotations are to get from one particular starting orientation to get to a particular ending orientation for your rotation angles to make sense.

The problem that you're seeing is that your gyroscope is likely setup to assume a roll first, followed by a pitch, followed by a yaw. That's (again, probably - check your datasheet) the order of rotations the gyro is giving you, but you are giving it a a pitch first, then yaw, which are the last two rotations in the sequence, then giving it the first rotation - roll.

You're giving it the first motion last, which means that it has to go "back" and update the pitch/yaw angles to correct for the fact that you didn't rotate it the way it wants to report the angles to you.

Try doing a roll first, followed by a pitch then a yaw and see if your angles hold steady with that order of rotations. Check your datasheet for specifics, but I think most systems are setup for roll/pitch/yaw (rpy) order.

• Please check the edits in the question. I am not facing a problem in sequence of rotations, I think it is related to the initial position. Please do check the edit and share your thoughts. – Nithin G A Dec 13 '16 at 11:26
• After a few experiments I found that the Initial Position is the key and also based on that we need to select the sequence of rotation. Eg. If the Initial Position is ZXY, Z up and X left and Y towards us(Right Hand Cordinate system), We need to use ZXY rotation sequence. If Initial position is ZYX,(Z up, Y left and X away from us), then we have to use ZYX sequence. – Nithin G A Dec 13 '16 at 13:10
• Your inputs guided me to do some experiments and come up with these observations. Thanks a lot Chuck – Nithin G A Dec 13 '16 at 13:12


text:
# What values of l are permitted for an electron with n = 3?

Dec 31, 2016

$l = 0 , 1 , 2$. See below.

#### Explanation:

There are four quantum numbers: the principle quantum number, $n$, the angular momentum quantum number, $l$, the magnetic quantum number, ${m}_{l}$, and the electron spin quantum number, ${m}_{s}$. For this question we are concerned only with the first two.

The principle quantum number, $n$, describes the energy and distance from the nucleus, and represents the shell.

For example, the $3 d$ subshell is in the $n = 3$ shell, the $2 s$ subshell is in the $n = 2$ shell, etc.

The angular momentum quantum number, $l$, describes the shape of the subshell and its orbitals, where $l = 0 , 1 , 2 , 3. . .$ corresponds to $s , p , d ,$ and $f$ subshells (containing $s , p , d , f$ orbitals), respectively. Each shell has up to $n - 1$ types of subshells/orbitals.

Therefore, the $n = 3$ shell has subshells of $l = 0 , 1 , 2$, which means the $n = 3$ shell contains $s$, $p$, and $d$ subshells (each containing their respective orbitals).


text:
# OpenStack VMs are unable to communicate with Outer World

Hello!

I'm trying to implement OpenStack solution in my environment ( bunch of Mac Minis) with one NIC card limitation. So far, I was able to setup a Controller and two Compute Nodes and created several VMs in compute nodes.

My host (Controller/Compute) are in 10.x.x.x series and I've setup the br100 private network for VMs in 192.168.100.x series per document. So, both Host and VMs have different gateway.

Problem : My Hosts (Controller/Compute) can communicate (ping/SSH) to their VMs but other server from same network (10.x.x.x) are unable to communicate to VMs. Vice versa, VMs are unable to communicate to other Hosts of the 10.x.x.x network except controller & compute nodes and also to the outer world.

What could be possible solution for this? Any help would be really appreciated. Please let me know, what information/files should I provide to diagnose it further.

edit retag close merge delete

Sort by » oldest newest most voted

what the network type? flat, vlan, local, which one did you created?

more

Thanks Xiang Hui, that solved my question.

more

Can we do any kind of NATing or iptables rules in OpenStack like VMWare provides . By the way, I'm using FlatDHCPManager mode.

more

Hi Xiang,

I've used the FlatDHCP (nova.conf -> network_manager=nova.network.manager.FlatDHCPManager). I've created my network with below command:

nova network-create private --fixed-range-v4=192.168.100.1/24 --bridge-interface=br100

My Controller & Compute Hosts are in 10 series gateway and Virtual Instances are in 192.168 series. So my VM can ping its gateway and its Host but not other hosts of same 10 series network and vice versa.

more

Hi Amogh,

Sorry, I have misunderstood your question before.

So your Controller & Compute Hosts which in 10 series can ping its vms which in 192.168 series, but other 10 series server cann't , right? and you want other servers can ping these vms each other.

There's two way: 1. If the other servers in 10 series are in the same switch with your Controller & Compute Hosts, you can just simply assign these other servers an 192.168 series ip to the ethernet card , then you can ping the vms of our Controller & Compute Hosts.

2. assign a floating ip to the vms, of course, the ip is 10 series, which means, the vms have two ips, one is in 192.168.x, another is 10.x, and I don't know if nova-network can support the floating ip fuction, but quantum(neutron) have supported this floating ip feature by using quantum-l3-agent.

more

nova-network support floating ip, and floating ip will help you solve this issue.

more

Thanks everyone. I got it solved by making my virtual network in Host network itself.

more

# Get to know Ask OpenStack

Resources for moderators


text:
## GMS problems with attachments in forwarded mails

There are two ways for forwarding messages in mailsystems: Forward messages
as inline text or forward messages as attached files. I GroupWise we can do
this.
The is a problem if the original message had an attachment like a pdf-file
for example. If somebody forward this to me as attached file, I can't access
to the pdf-attachment, when I use the GroupWise Mobile Server. In my
smartphone I see only
-----Begin Embedded Message-----
the text of the original message
-----End Embedded Message Subject: xxx-----
and the original attachment was absent.
With the GroupWise client or Webaccess the attachment this present. I this a
bug or must I configure anything?


text:
#### 11th Standard Chemistry English Medium Free Online Test 1 Mark Questions 2020 - Part Four

11th Standard

Reg.No. :
•
•
•
•
•
•

Chemistry

Time : 00:25:00 Hrs
Total Marks : 25

25 x 1 = 25
1. Presence of which cation makes the water hard in nature?

(a)

Ca& Mg

(b)

Na&Mg

(c)

Ca&Na

(d)

Mg&F

2. Which of the following is produced by bombardment of neutrons with lithium?

(a)

Deuterium

(b)

Protium

(c)

Tritium

(d)

Beryllium

3. Hydrogen combines with carbon monoxide in the presence of copper catalyst will synthesise-------

(a)

Ethanol

(b)

Methane

(c)

Methanol

(d)

Methanal

4. Rubidium belongs to______group of metals

(a)

transition

(b)

inner transition

(c)

alkali

(d)

alkaline earth

5. Which of the following oxides is the most basic in nature?

(a)

Na2O

(b)

BeO

(c)

Li2O

(d)

H2O

6. Consider the following statements
1. Gypsum is calcium sulphate hemihydrate
2. Retrograde solubility is striking feature of gypsum
3. Alabaster is a variety of gypsum
Which of the following statement(s) given above is/ are incorrect?

(a)

1& 3

(b)

only

(c)

2 & 3

(d)

1,2 & 3

7. Hydroxides of beryllium are _______ in nature.

(a)

neutral

(b)

basic

(c)

acidic

(d)

amphoteric

8. The partial pressure of dry gas is

(a)

greater than that of wet gas

(b)

lesser than that of wet gas

(c)

equal to that of wet gas

(d)

none of these

9. Vanderwaal's constant a and b are related with  ______________ respectively

(a)

attractive force and volume of molecules.

(b)

repulsive force and volume of molecules

(c)

attractive force and bond energy of molecules.

(d)

shape and repulsive force of molecules

10. Equal volumes of He, O2 and SO2 are taken in a closed container. The ratio of the partial pressures of gases He, O2 and SO2 would be:

(a)

1 : 2 : 8

(b)

8 : 16 : 1

(c)

1 : 4 : 16

(d)

16 : 2 :1

11. The density of neon will be highest at:

(a)

STP

(b)

0°C, 2 atm

(c)

273°C, 1 atm

(d)

273°C, 2 atm

12. Which of the following gases will have the lowest rate of diffusion?

(a)

H2

(b)

N2

(c)

F2

(d)

O2

13. Pressure of a gas is equal to __________.

(a)

$\frac{F}{a}$

(b)

F x a

(c)

$\frac{a}{F}$

(d)

F - a

14. For a cyclic process the volume of ΔU is

(a)

maximum

(b)

minimum

(c)

zero

(d)

does not change

15. The entropy change can be calculated by using the expression, $\Delta$S = $\frac { { q }_{ rev } }{ T }$. When water freezes in a glass beaker, choose the correct statement amongst the following:

(a)

$\Delta$S (system) decreases but $\Delta$S (surroundings) remains same.

(b)

$\Delta$S (system) decreases but $\Delta$S (surroundings) decreases

(c)

$\Delta$S (system) decreases but $\Delta$S (surroundings) increases.

(d)

$\Delta$S (system) decreases and $\Delta$S (surroundings) decreases.

16. In which of the following reaction, standard entropy change ($\Delta$S°) is positive and standard Gibb's free energy change ($\Delta$G°) decreases sharply with increasing temperature?

(a)

C(graphite) + $\frac { 1 }{ 2 }$O2(g) $\rightarrow$ CO(g)

(b)

CO(g)+$\frac { 1 }{ 2 }$O2(g) $\rightarrow$ CO2 (g)

(c)

Mg (s) + $\frac { 1 }{ 2 }$O2(g) $\rightarrow$ Mg(s)

(d)

$\frac { 1 }{ 2 }$C(graphite) + $\frac { 1 }{ 2 }$O2(g) $\rightarrow$ $\frac { 1 }{ 2 }$CO2 (g)

17. Two moles of N2 and two moles of H2 are taken in a closed vessel of 5 litre capacity and suitable conditions are provided for the reaction. When the equilibrium is reached, it is found that a half mole of N2 is used up. The equilibrium concentration of NH3 is

(a)

0.2

(b)

0.4

(c)

0.3.

(d)

0.1

18. Equimolar solution of non-electrolyte in the same solvent have

(a)

same boiling point and same freezing point

(b)

different boiling point. and different freezing point

(c)

same boiling point but different freezing point

(d)

same freezing point but different boiling point

19. Statement I: A solution of potassium chloride in water deviates from ideal behavior.
Statement II : The solute dissociates to give K+ and Cl- ion which form strong ion dipole interaction with water molecules

(a)

Statement I& II are correct and II is the correct explanation of I

(b)

Statement! & IIare correct but II is not correct explanation of I

(c)

Statement I is correct but statement II is wrong.

(d)

Statement I is wrong but statement II is correct

20. Which one of the following has linear shape?

(a)

PbCl2

(b)

SnBr2

(c)

BeCl2

(d)

CCl2F2

21. In which of the following homolytic bond fision takes place?

(a)

Alkaline hydrolysis of ethyl chloride

(b)

Addition of HBr to double bond

(c)

Photochlorination of methane

(d)

Nitration of benzene

22. The most stable alkene is

(a)

CH2=CH-CH2-CH3

(b)

$C{ H }_{ 2 }=\underset { \overset { | }{ C{ H }_{ 3 } } }{ CH } =C{ H }_{ 2 }$

(c)

CH2=CH-CH=CH2

(d)

CH3-CH=CH-CH3

23. The difference in potential energy between eclipsed and staggered form of ethane is.

(a)

4 kJ/mol

(b)

12.55 kJ/mol

(c)

2 kJ/mol

(d)

44 kJ/mol

24. Hunsdiecker reaction is used to prepare alkyl chloride and alkyl bromide starting from

(a)

diazonium salt

(b)

silver salts of carboxylic acid

(c)

sodium salt of carboxylic acid

(d)

alcohol

25. Which of the following is a secondary haloalkane?

(a)

Bromoethane

(b)

2-Chloropropane

(c)

2-Iodo-2-methylpropane

(d)

1-Chloropropane


text:
# Why should Molniya orbit period be a half-synodic, not a half-sidereal day?

In the Wikipedia article for Molniya orbit the introduction mentions that the period should be 0.5 sidereal day, so that its path across the sky every ~12 hours as observed from a given point on Earth remains constant.

But later in the properties section it refines that statement, saying that it should actually be 0.5 synodic day instead, and goes on to discuss the effect of the earth's oblateness upon the precession of the right ascension of the ascending node $$\Omega$$ and the argument of perigee $$\omega$$.

Since a geosynchronous orbit needs a period of 1.0 sidereal day to remain fixed in the sky, why is the Molniya referenced to a synodic day instead of a sidereal day?

Is there a simple way to understand this? Some kind of "Aha!" one-liner?

• Could it just be an error in the article? The Earth rotates, by definition, once every sidereal day relative to the vernal equinox. If the ground track of the satellite does not shift relative to the surface, its orbit should match that period as well. Sep 9 '16 at 23:49
• @JulianHzg There is something a bit confusing for sure. It seems the discussion is about adjusting the period to compensate for precession of some of the orbital elements, but I can't follow the logic. Also, I thought the difference between a sidereal and a synodic day is about 3m 56s, not the 35 seconds mentioned there.
– uhoh
Sep 10 '16 at 2:30

The wording is quite confusing. First off, the article frivolously uses the term 'orbital period' without ever qualifying which they mean: sidereal, synodic, draconic or anomalistic?

That changes the result, depending on how you understand the orbital period: Is it the time between the satellite reaching periapsis on two consecutive passes, or is it the time its true anomaly repeats?

First, a bit about the orbit. The Molinya orbit is to be used with constellations of three satellites; its purpose is to provide observation of two "interesting" regions of Earth in high latitudes; areas on two sides of Earth. This all from altitudes much lower than GEO (both lower distance for contact and observation, and lower launch costs), and thanks to the constellation configuration, 24/7 observation period is assured, at good angle to the surface of the "interesting" regions.

In particular, Soviets used it to observe Siberia and northern US.

The Molinya orbit has a very high eccentricity. The reason is to keep the satellites of the constellation for above the interesting area of Earth for the longest time (when they move slowly near the apoapsis) and let them dip into periapsis briefly when the interesting area moves out of focus (to be taken over by two next satellites of the constellation), only to return "into position" shortly later.

First, let's neglect orbital precession of the satellite. The optimal orbital period for this configuration is one sidereal day - the three orbits of the three satellites remaining in the same orientation relative to distant stars, and while the terminator line of Earth moves as Earth circles the Sun (and synodic day shifts relative to sidereal), the whole configuration remains constant relative to "distant stars", each satellite reaching apoapsis 1/6 sidereal day from the previous, and landing above the same location every sidereal day (and one on the opposite side of Earth, but same hemisphere, half a sidereal day later).

Therefore, in this configuration - taking the orbit without precession - it's accurate to say its period is half a sidereal day. That's also the number you get When you write down the orbital elements assuming Earth is a perfect sphere (as is commonly done) - so within this simplification, 0.5 "sidereal day" is accurate.

Now, Earth not being a perfect sphere, perturbs the orbit, causing its precession. The orbit was chosen such, that the perturbation per day is nearly exactly the angular shift between sidereal and synodic day, per day. As result, the orbit's argument of periapsis travels around Earth at one rotation per year, remaining in constant orientation relative to the Sun, and resulting in the orbit being synchronous with Earth surface at given time of synodic day.

It has little practical consequence, as both all three satellites retain their synchronicity and observation of the designated locations just the same, and the ground stations would have little trouble adapting to synodic period. Except the J2 term of the gravitational field of the Earth does inevitably cause some precession of any inclined, eccentric orbit - removing it would cost a lot of fuel, so instead the inherent error is transformed into a feature, the orbit chosen such, that it's synchronous to convenient, common sidereal time.

The angular precession per day is just a bit under 1 degree (360 degrees/365 days), and while the satellite completes the orbit the "sidereal" number of times - reaches the apoapsis 364 times per year - the apoapsis completes one full circle, each day both argument of periapsis and true anomaly of the satellite shifting by that 1 degree, adding one more orbital rotation to the number and resulting in the "synodic day period" result.

• Thanks! I'll take some time to read this - it is extremely helpful!! At the end of Properties it says "The rotational period of the Earth relative to the node (i.e. the synodic day) will therefore be only 86,129 seconds, 35 seconds less than the sidereal day which is 86,164 seconds." Isn't a synodic day (on average) 24 hours - 236 seconds longer than a sidereal day, not 35 seconds shorter? Or is this a different kind of synodic day?
– uhoh
Sep 13 '16 at 11:00
• @uhoh: Ah. Very silly liberal application of the term "sidereal day". In this context, "sidereal day" is applied in the satellite orbit's frame of reference - forget the Sun, treat the orbit as an invariable. "central body" relative to which Earth rotates. I honestly have no clue what purpose is this paragraph serving. [also - the gist of the answer is the second paragraph.]
– SF.
Sep 13 '16 at 11:27
• A sidereal day - 23h 56m 4s is indeed 86,164 seconds like it says. It's "The rotational period of the Earth relative to the node (i.e. the synodic day) will therefore be only 86,129 seconds, " part that bothers me. Oh - did you just mean to say "synodic day" and not "sidereal day* in your comment?
– uhoh
Sep 13 '16 at 11:55
• @uhoh: damn, now I'm totally confused. Yes, that's what I meant, and after you pointed it out, the paragraph completely ceased to make sense to me.
– SF.
Sep 13 '16 at 13:53
• ya, I know - I've been looking at the article now and again for weeks. OK - how about you change your answer to "...though may not be incorrect."and we'll keep at it. I can think about your answer in a half day from now.
– uhoh
Sep 13 '16 at 17:20


text:
# How does Batch Normalization not lead to the model blowing up? [duplicate]

I was reading the Batch Norm paper and in this paragraph,

We could consider whitening activations at every training step or at some interval, either by modifying the network directly or by changing the parameters of the optimization algorithm to depend on the network activation values (Wiesler et al., 2014; Raiko et al., 2012; Povey et al., 2014; Desjardins & Kavukcuoglu). However, if these modifications are interspersed with the optimization steps, then the gradient descent step may attempt to update the parameters in a way that requires the normalization to be updated, which reduces the effect of the gradient step. For example, consider a layer with the input $u$ that adds the learned bias $b$, and normalizes the result by subtracting the mean of the activation computed over the training data: $\hat{x} = x − E[x]$ where $x = u + b$, $\mathcal{X} = \{x_{1 \ldots N} \}$ is the set of values of $x$ over he training set, and $E[x] = \frac{1}{N} \sum_{i=1}^N x_i$ . If a gradient descent step ignores the dependence of $E[x]$ on $b$, then it will update $b \leftarrow b + \Delta b$, where $\Delta b \propto −\partial l/ \partial \hat{x}$. Then $u + (b + \Delta b) − E[u + (b + \Delta b)] = u + b − E[u + b]$. Thus, the combination of the update to $b$ and subsequent change in normalization led to no change in the output of the layer nor, consequently, the loss. As the training continues, $b$ will grow indefinitely while the loss remains fixed. This problem can get worse if the normalization not only centers but also scales the activations. We have observed this empirically in initial experiments, where the model blows up when the normalization parameters are computed outside the gradient descent step.

What's the point of having $b$ anyway? (Later in the paper, it says that b is removed during batch normalization, so what's the aim of the math shown in this picture and how is this* math different from the one that we do during actual batch normalization).

This happens later in the paper,

Note that, since we normalize Wu+b, the bias b can be ignored since its effect will be canceled by the subsequent mean subtraction (the role of the bias is subsumed by β in Alg. 1). Thus, z = g(Wu + b) is replaced with z = g(BN(Wu))

The explanation comes in the next paragraph. Instead of normalizing by the mean of activations in a single batch, they normalize with respect to the whole training data ($\mathcal{X}$). The way they achieve this is by using moving averages that get updated for each mini-batch during training.
The reason why the bias term becomes redundant is due to learned scale and shift variables $\gamma^{(k)}$, and $\beta^{(k)}$. Without these, the normalization would squash everything which could limit what a non-linearity activation function can represent. They give the example of the sigmoid activation function which would be constraint to the linear part only and the tails would be squashed.


text:
# How do you find the domain of p(t)=(t+9)/(t+3)?

Aug 22, 2017

$t \in \mathbb{R} , t \ne - 3$
$\text{solve "t+3=0rArrt=-3larrcolor(red)" excluded value}$
$\text{domain is } t \in \mathbb{R} , t \ne - 3$


text:
# Simplify the radical $\sqrt{x-\sqrt{x+\sqrt{x-...}}}$

I need help simplifying the radical $$y=\sqrt{x-\sqrt{x+\sqrt{x-...}}}$$ The above expression can be rewritten as $$y=\sqrt{x-\sqrt{x+y}}$$ Squaring on both sides, I get $$y^2=x-\sqrt{x+y}$$ Rearranging terms and squaring again yields $$x^2+y^4-2xy^2=x+y$$ At this point, deriving an expression for $$y$$, completely independent of $$x$$ does not seem possible. This is the only approach to solving radicals which I'm aware of. Any hints to simplify this expression further/simplify it with a different approach will be appreciated.

EDIT: Solving the above quartic expression for $$y$$ on Wolfram Alpha, I got 4 possible solutions

• It is possible, but you have to solve a quartic, which is tedious (but doable in principle). Jul 28, 2020 at 4:45
• @Deepak I'm not yet familiar with a method of solving quartic expressions. Could you provide a reference for solving quartics? Jul 28, 2020 at 4:47
• I believe the answer is $\frac{\sqrt{4x+3}-1}{2}$ Jul 28, 2020 at 4:52
• Here is the link from where i learnt,youtube.com/watch?v=BO1T7ebJlO8 Jul 28, 2020 at 4:58
• There is a tag that you could usefully add of "nested-radicals" Jul 28, 2020 at 9:31

Perhaps it is more instructive to consider instead the following: let $$y = \sqrt{x - \sqrt{x + \sqrt{x - \sqrt{x + \cdots}}}}, \\ z = \sqrt{x + \sqrt{x - \sqrt{x + \sqrt{x - \cdots}}}},$$ so that if $$y$$ and $$z$$ exist, they satisfy the system $$y = \sqrt{x - z}, \\ z = \sqrt{x + y},$$ or $$y^2 = x - z, \\ z^2 = x + y.$$ Consequently $$0 = z^2 - y^2 - y - z = (z-y-1)(y+z).$$ It follows that either $$z = -y$$ or $$z = 1 + y$$. The first case is impossible for $$x \in \mathbb R$$ since by convention we take the positive square root, so both $$y, z > 0$$. In the second case, we can substitute back into the first equation to obtain $$y^2 = x - (1+y)$$, hence $$y = \frac{-1 + \sqrt{4x-3}}{2},$$ where again, we discard the negative root.

So far, what we have shown is that if such a nested radical for $$y$$ converges, it must converge to this value. It is not at all obvious from the above whether a given choice of $$x$$ results in a real-valued $$y$$, for any meaningful definition of $$y$$ must be as the limit of the sequence $$y = \lim_{n \to \infty} y_n, \\ y_n = \underbrace{\sqrt{x - \sqrt{x + \sqrt{x - \cdots \pm \sqrt{x}}}}}_{n \text{ radicals}},$$ and although the choice $$x = 1$$ appears at first glance permissible, we quickly run into problems; $$y_3 = \sqrt{1 - \sqrt{1 + \sqrt{1}}} \ne \mathbb R$$. In particular, we need $$x$$ to satisfy the relationship $$x \ge \sqrt{x + \sqrt{x}},$$ which leads to the cubic $$x^3 - 2x^2 + x - 1$$ with unique real root $$x = \frac{1}{3} \left(2+\sqrt[3]{\frac{25-3 \sqrt{69}}{2}}+\sqrt[3]{\frac{25+3 \sqrt{69}}{2}}\right) \approx 1.7548776662466927600\ldots.$$ However, any such $$x$$ meeting this condition will lead to a convergent sequence. The idea is to show that $$|y_{n+2} - y| < |y_n - y|$$ for all $$n \ge 1$$; then since $$\lim y_n$$ has at most one unique limiting value as established above, the result follows.

Consider the final relation you have obtained as a quadratic equation in $$x$$,i.e: $$x^2-(2y^2+1)x+y^4-y=0$$ Solving the above gives you $$x=y^2+y+1 \text{ or } x=y^2-y$$ Individually solve these the quadratics in $$y$$ to obtain the four solutions you got from Wolfram Alpha.

• Both answers are good, but this one is more natural +1 Jul 28, 2020 at 5:02
• @Arjun Thanks a lot! Jul 28, 2020 at 5:03

Note that :

$$(x-y^2)^2 = x+y \implies (x-y^2)^2 - y^2 = x+y-y^2 \implies (x-y^2-y)(x-y^2+y) = x-y^2+y \\ \implies \boxed{(x-y^2+y)(x-y^2-y-1) = 0}$$

So either one is correct.

Note : The problem is that one is still not sure when the radical above converges i.e. what is the set of all $$x$$ for which $$\sqrt{x + \sqrt{x-\sqrt{x+...}}}$$ forms a convergent sequence.

• I was solved to suppose an integral, where this radical was the integrand- the limit of integration was 1 to 3, so I suppose only positive expressions have to be assumed? Jul 28, 2020 at 5:08
• Yes, of course. You would run into square root problems if any intermediate $a_n$ became negative. Of course, if this is one of those IIT type questions where you just need an answer, then you need not bother and can go ahead with whatever expressions you get for $y$ in terms of $x$. Jul 28, 2020 at 5:09
• No, this is actually a subjective problem where I need an explicit method to solve. Jul 28, 2020 at 5:14
• Ok, so my question is : do you need to prove explicit convergence of the radical in an interval? See, if you don't justify the convergence, you can do what you like, right? You might know all those proofs with sequences like $1+1+1+...$ in videos ; you don't want to land in a situation like that. Jul 28, 2020 at 5:17
• Yes, I think you should do that, if the selection is enough then I think we can close the conversation. +1 to your question! Jul 28, 2020 at 5:22


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# Structure of the repository¶

But how does KeOps handle symbolic formulas on the GPU? How can its routines outperform the CUDA backends of Deep Learning frameworks by such a wide margin? To answer these questions, we need to dive into the mixed C++/Python/Matlab codebase of the KeOps package, whose structure may be summarized as follows:

• The pykeops/ folder, with common/, numpy/ and torch/ subfolders contains our Python wrappers and relies on the fantastic PyBind11 library.

• The keopslab/ folder provides a collection of entry points for Matlab scripts.

• The keops/ folder contains our C++ files and the associated compilation scripts. The generic KeOps engine that we are now about to discuss is implemented in the core/ subfolder which contains:

• The link_autodiff.cpp and link_autodiff.cu “main” C++ files, which define the methods that binding libraries may use to create high-level modules.

• The pack/ subfolder, which defines abstract types for lists and tuples within the C++ templating system. Using advanced concepts that were introduced with the C++11 revision, this file allows us to drive the nvcc compiler with declarative “variadic templating” and generate routines that manipulate an arbitrary number of parameters, $$i$$- and $$j$$-variables.

• The autodiff/ subfolder, which defines the primitives of the KeOps symbolic syntax: variables, abstract unary and binary operations, gradients.

• The mapreduce/GpuConv*_.cu CUDA files, which implement our massively parallel Map-Reduce schemes. These files contain the core logic of the KeOps library.

• The mapreduce/CpuConv*_.cpp C++ files, which implement simple Map-Reduce schemes using standard “for” loops. They may be used to test the correctness of our parallel implementations and provide a fall-back mode to users who do not have access to GPU chips on their machines.

• The reductions/ subfolder, which implements the supported $$\operatorname{Reduction}$$ operations: sum, arg-min, log-sum-exp, etc.

• The formulas/ subfolder, which implement the atomic operations that users may combine to define vector-valued formulas $$F$$.

As evidenced here, the KeOps engine is heavily reliant on modern features of the C++ language: every time Genred encounters a new kind of generic operation (up to the values of $$\mathrm{M}$$, $$\mathrm{N}$$ and the data arrays which are free to change between every call), the string that specifies a generic formula is parsed by the compiler and a new “.dll” or “.so” shared object is generated before being executed on the relevant Python or Matlab tensors.


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# Book:Joseph George Coffin/Vector Analysis: An Introduction to Vector-methods and their Various Applications to Physics and Mathematics/Second Edition

## Joseph George Coffin: Vector Analysis: An Introduction to Vector-methods and their Various Applications to Physics and Mathematics (2nd Edition)

Published $\text {1911}$, John Wiley & Sons.