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# Design and analysis of Maxwell fisheye lens based beamformer

## Abstract

Antenna arrays and multi-antenna systems are essential in beyond 5G wireless networks for providing wireless connectivity, especially in the context of Internet-of-Everything. To facilitate this requirement, beamforming technology is emerging as a key enabling solution for adaptive on-demand wireless coverage. Despite digital beamforming being the primary choice for adaptive wireless coverage, a set of applications rely on pure analogue beamforming approaches, e.g., in point-to-multi point and physical-layer secure communication links. In this work, we present a novel scalable analogue beamforming hardware architecture that is capable of adaptive 2.5-dimensional beam steering and beam shaping to fulfil the coverage requirements. Beamformer hardware comprises of a finite size Maxwell fisheye lens used as a scalable feed network solution for a semi-circular array of monopole antennas. This unique hardware architecture enables a flexibility of using 2 to 8 antenna elements. Beamformer development stages are presented while experimental beam steering and beam shaping results show good agreement with the estimated performance.

## Introduction

The vision for Beyond 5G (B5G) networks paves the path towards ultra-reliable low-latency (URLLC) communications. Recent times have seen a sharp rise in the number of connected devices, especially in the context of internet-of-everything (IoE)1. Wireless communication technologies have found widespread use in different applications that impact agriculture, transport and healthcare to name a few. Therefore, new and innovative techniques have been proposed for providing seamless coverage and improved link quality to provide high data rates. In the new communication infrastructure, multi antenna systems have been utilized to provide better antenna gain, enabling enhanced communication link quality2. Additionally, providing seamless connectivity becomes more challenging for applications in mobile platforms, e.g. unmanned aerial vehicle (UAV), vehicular and satellite networks where electrically scanned arrays are beneficial. To overcome these challenges, beamforming techniques have been utilized to enhance the network coverage and overcome the limitations due to interference and path loss. Similarly, beam steering allows the transmissions to be directed at a particular angle(s), ensuring seamless connectivity for mobile devices3.

Three well-known beamforming techniques are analog, digital and hybrid beamforming. Generally a single radio frequency (RF)-chain is used to connect all the antenna elements to a transmitter/receiver module in analog beamforming architecture, while in the digital beamforming, a dedicated RF-chain is required for each antenna element. The hybrid beamforming uses a two stage structure, employing both the analog and digital beamforming, thereby enhancing the array gain (analogue beamforming) and mitigating interference (digital beamforming)4. All three beamforming types require multi-antenna systems that helps to improve the signal-to-noise-ratio (SNR) when used at a receiver side. Interference can also be mitigated through generating highly directional beams along the desired direction when used at a transmitter side. This work is motivated by advantages offered by the analogue beamforming, either on its own or as a part a hybrid architecture, and here we explored the utilization of Maxwell fisheye lens to propose a unique scalable analogue beamformer solution.

Maxwell fisheye belongs to a class of gradient-index materials, where the refractive index changes with the geometry of the material5. Other gradient-index materials including Luneberg Lens6, Eaton Lens7 and Fresnel lens8 utilizes the energy focusing capability of the lens structure at the antenna end, while this work uses energy focusing capability of a Maxwell Fisheye lens at a feed network level. Maxwell fisheye lens is an inhomogeneous optical system, where rays emerging from one point within the lens follow a circular arc-shaped path about the origin (center) of the lens9. Maxwell fisheye lens possesses a characteristic that gives impetus to its use in analogue beamforming, i.e., for each point on the Maxwell fisheye lens, there is an image, where the image is a conjugate of the source. The optical length between two conjugate points on the circular arc-shaped trajectory is the same. Therefore, the capability of translating source signal to image signal by the Maxwell fisheye lens motivates its use for developing a self-scalable multi-antenna system feed network that does not require a corporate feed network with a dedicated number of ports to excite spatially distributed antenna elements operating as an array. Specifically, the contributions of this work are as follows: First, we propose a novel beamformer design architecture using Maxwell fisheye lens whose capability of on demand beam shaping and beam steering is demonstrated for the first time; Second, we demonstrate that our proposed approach does not require a fixed corporate feed network and is scalable to handle between 2 to 8 antenna array elements; Third, we validate the beam shaping and beam steering using proof-of-concept prototype. The remainder of the paper is organized as follows. In “Maxwell fisheye lens-based beamformer” section, we discuss the proposed beamformer design, present the operation of the Maxwell fisheye lens and its corresponding theoretical model. Prototype development is explained in “Prototype development method” section, followed by results and discussion in “Results and discussion” section. While “Conclusions and future work” section concludes the paper and presents future directions to this work.

## Maxwell fisheye lens-based beamformer

### Beamformer design

The proposed Maxwell fisheye lens based beamforming semi-circular array design is shown in Fig. 1a. The figure shows the transceiver coaxial input emanating from the a metallic sheet acting as a RF ground plane. On the bottom side of this RF ground plane is a planar Maxwell fish-eye lens, where probe of the transceiver coaxial input is exciting the lens. The RF ground plane is bounded by a copper mirror, which is necessary to confine the microwave signal within the lens structure. The circular lens structure can be characterised into two halves, where half of the circle represents the input-side and the remaining half represents the monopole side. The dielectric profile is surrounded by the input and output side probes. At the monopole side half of the Maxwell fisheye lens, coax probes (similar to that of the input-side) are connected to monopole antennas, which are vertically placed at the top side of the RF ground plane. Monopole antennas connected to the output probes of the Maxwell fisheye lens creates a semi-circular monopole antenna array, which is backed by a metallic reflector placed at a distance of $$\lambda /2$$. The lens structure and monopoles are designed to operate at 10 GHz. Relative locations of the monopole antenna elements are provided in Table 1. The source and drain locations used in this work coincide with the location identification via time reversed scattering transformation approach shown in10,11, while general form of geometrical solution of Maxwell fisheye lens can be found in12.

### Maxwell fisheye lens operation

To understand the operation of Maxwell fisheye lens and how it is used to develop a beamformer, let us consider a three dimensional Cartesian coordinate system shown in Fig. 1a when the trajectory of the rays within a 3 dimensional (3D) Maxwell fisheye lens is projected onto xz- plane13. Mathematically, ray trajectory can be represented as $$r(t)=x(t), y(t), z(t)$$, where t signifies the parametric variation of the ray. As shown previously in14, the rays from a 3D system projects onto a plane without any loss of generality, and this principle can be used to translate a 3D Maxwell fisheye lens on a finite plane.

The Rexolite lens profile in Fig. 1a is centered at the origin of the Cartesian coordinate system, while the base of the lens as well as the RF ground plane are at $$z=0$$. By mapping the ray trajectories from spherical points on the xz-plane15; the differential of the path length on the sphere corresponds to the differential of the optical length on the xz-plane. This allowed us to map the geodesic points on multiple circles that produces the sphere to a Maxwell fisheye lens onto the refractive index profile in xz-plane, which can be written as16

\begin{aligned} x=\left( \frac{1- sin\theta }{cos\theta }\right) ^2 cos (2\phi ) \text { and } z= \left( \frac{1- sin\theta }{cos\theta }\right) ^2 sin (2\phi ) \end{aligned}
(1)

In this stereographic projection, the line segments dx and dy can be represented using spherical coordinates, hence the cylindrical lens profile can be represented as

\begin{aligned} n^2(x,y,0)[dx^2 + dy^2]=n_0^2[d\theta ^2 + sin^2\theta d\phi ^2] \end{aligned}
(2)

where, at the equator of the reference sphere index $$n = n_0$$. Using this principle, the Maxwell fisheye lens profile can be represented by14,17:

\begin{aligned} n=\sqrt{\varepsilon }_r=\frac{2n_o}{1+\frac{r^2}{R^2}}, \quad \mathrm {when} \ r \in [0,\infty )] \end{aligned}
(3)

where R represents the radius of the reference sphere and $$2n_o$$ is the refractive index at the center of the lens. The traditional methods used to create a refractive index profile for fisheye focusing include thin plates18 and holy parallel plates19. In this paper, we use mode theory of parallel-plate waveguide20 for realizing the fisheye principle. Our design comprises of ideally conducting parallel plates, where the field propagating parallel to the plates has a general form of propagation constant given by

\begin{aligned} k_m=\sqrt{k^2-m^2\left( \frac{\pi }{d}\right) ^2} \end{aligned}
(4)

where $$k=\omega \sqrt{\varepsilon _r}/c.$$ and $$m \in {1,2,...}$$. The propagation will take place when $$k_m$$ is real. For $$k<\pi \sqrt{\varepsilon _r}/d$$, where $$k_m$$ is imaginary and the propagating waves undergo an exponential decay in the direction of propagation. This leads to development of parallel plate waveguide structure that can act as a high pass filter with a cut-off frequency given by

\begin{aligned} \omega _c=\frac{m \pi c}{d\sqrt{\varepsilon (\omega )}}, \quad \mathrm {when} \ m \in \{1,2,...\} \end{aligned}
(5)

where m represents the mode number. The wave will propagate at a phase velocity $$v=\omega /k_m$$ for $$\omega >\omega _c$$ and energy will be transported at a group velocity $$v_g=d\omega /dk_m$$. The group velocity can be controlled by creating a graded index profile through Eq. (3). The parallel plate waveguide is partially filled with a dielectric material of permittivity $$\varepsilon _r$$ with a varying thickness across the lens, which leads to an effective dielectric $$\acute{\varepsilon _r}$$ given by

\begin{aligned} \acute{\varepsilon _r}=1-t(r)\left( \frac{1+\varepsilon _r}{d}\right) . \end{aligned}
(6)

In Eq. (6), r denotes the parallel plate’s radius, while t is dielectric material thickness. The effective permittivity $$\acute{\varepsilon _r}$$ can be controlled by the method explained in21. Equations (3), (5) and (6) are used to formulate the dielectric profile between the parallel plates shown in Fig. 1a, where coax probes are use to excite a signal at 10 GHz within the parallel plate region.

There is a physical limitation regarding the design of the lens which is related to the near zero index area of the lens profile resulting due to the approach given in Eqs. (3) and (6), which is the requirement of an infinitely large lens to realize the perfect fisheye operation. We mitigate this by including a cylindrical reflecting metal sheet to limit the wave propagation within a finite area as postulated in22. Figure 1b demonstrates the function of the mirror where multiple circular ray trajectories from the source are reflected by the mirror and form an image point, while this operation occurs within a confined space. In practical terms, the signal excited at the source point will have an extension at the image as shown in Fig. 1c. It shows that the wave leaving a Maxwell fisheye lens follows a circular path and converges on an image point. This illustrates the focusing capability of the lens, here $$85\%$$ of the inserted field at the source port can be extracted from the drain probe at operation frequency of 10 GHz. In this case, the electromagnetic field energy loss is related to the imperfect imaging at the drain located at the image point of the lens, which is discussed in11,23.

### Beamformer operation

When a 10 GHz signal is simultaneously excited at the transceiver coax inputs, monopole antennas are excited, hence forming a beam along the $$-z$$-direction. Figure 1d presents the beamforming results validated via full-wave electromagnetic simulation tool CST Microwave Studio, and compared them with that of a mathematical semi-circular antenna array factor. To achieve beam shaping with maximum directivity, phase alignment at the array excitation is required, thus phase shifting depicted in Table 2 is applied at each of the 8 input ports of the beamformer. The results demonstrate that directivity of the Maxwell fisheye lens based antenna array and an ideal circular array with monopoles are comparable, validating the utility of the fish-eye lens structure. Moreover, we also observe that the directivity patterns have significantly high back lobes, prompting a need of metallic reflector shown in Fig. 1a, which is added to redirect the radiated energy towards $$-z$$-direction. When beam steering is performed in an ideal circular array, the maximum directivity level remains the same, as shown in the Fig. 1d. Although, there is a slight degradation ($$<1\hbox {dB}$$) in the maximum directivity when the steering angle of $$25^\circ$$ is applied to the Maxwell fisheye lens based beamformer, that doesn’t effect the beam steering capability. Additionally, it can be observed that the half power beamwidth for an ideal semicircular array stays consistent when beam steering is performed. Finally, for a Maxwell fisheye lens based antenna array we can observe a side lobe at steering angle of $$50^\circ$$, however with 6.8dB difference from the maximum directivity the overall beamfomring response is comparable with the ideal semicircular antenna array.

## Prototype development method

Figure 2a shows the dimensions of the Maxwell fisheye lens developed for the beamforming performance evaluation, where the design configuration of the prototype is depicted. The schematic shows the phase aligned transmission lines which are connecting the external signal input through SMA connectors to the inputs of the Maxwell fisheye lens. The monopole antenna array elements are at the output of the Maxwell fisheye lens backed by a metallic reflector when looking at the top side, whereas the Rexolite lens profile is shown when looking at the bottom side of the beamformer in Fig. 2a. Length of the monopole antenna and gap from the metallic reflector is also provided. The Maxwell fisheye lens has two perfect electrically conducting (PEC) plates spaced $$d=5$$mm apart. Note that the bottom plate from the Maxwell fisheye lens is removed for the purposes of showing the Rexolite lens and probes assembly. The substrate between the parallel plates is Rexolite with $$\varepsilon _r = 2.53$$, dispersion factor=0.00066 and the coefficient of linear thermal expansion $$=3.8 \times 10^{-5} 1/^oF$$ inch. These properties make Rexolite a suitable candidate for lens development. A copper mirror surrounds the Rexolite lens, where the diameter of the circular band is 100 mm. The lens is excited through input coaxial probes with a height (h) of 4.5mm to provide impedance matching to the 50 $$\Omega$$ line. Accordingly, the output probes have the same height as the input probes, and the dimension of the input and output coaxial probes are also provided. The lens profile is made from a cylindrical Rexolite stock using Triumph Duplex milling machine. To ensure the best Maxwell fisheye lens performance, we made sure that the surface roughness across the lens profile is $$<\lambda /8$$ at 10 GHz, while this was realised through polishing the Rexolite lens surface with a Tech-Gen precision finisher. The prototype lens structure has overall diameter of 100mm, whereas Rexolite lens has 60mm diameter. To conduct experiments, we have developed phased aligned module in which all the transmission lines incur the same phase shift. It is important to mention that these phase align transmission lines are used to connect the lens structure to the edge of the board and are not a part of the proposed beamformer. Any length of transmission should work the same way provided that the phase incurred by the propagating wave at 10 GHz is the same.

As in Fig. 1a, in the prototype shown in Fig. 2a, the monopole side of the Maxwell fisheye lens beamformer is where the metal backed monopole antennas are connected to the output of the Maxwell fisheye lens beamformer. Figure 2b show the fabricated prototype used to measure the beamformer performance. Figure 2c shows that the signals travel in the phased align transmission lines independently and the mutual coupling between signals in parallel transmission lines is minimized deliberately. It is important to note that the overall transmission line length is kept as small as possible to avoid any additional signal insertion loss from the edge of the substrate to the input of the Maxwell fisheye lens. Figure 2c also presents a side-view of the transition between the microstrip transmission line, coaxial input, Rexolite lens and coaxial output, where signal propagating through the transmission line is delivered to the input of the monopole antenna through the Maxwell fisheye lens.

## Results and discussion

From the simulations, we observed an impedance mismatch when the reflective mirror is placed close to the probe in the Maxwell fisheye lens. This impacts the device operation and reduces its efficiency. The reason being the classical substrate-filled parallel plate waveguide excitation impedance principle24 does not directly apply to parallel plate waveguide with growing/decaying substrate profile, and an additional step of impedance matching optimization is needed. When the probe location and depth is optimized for maximum device efficiency in full-wave electromagnetic simulator, the source excitation forms an image at the drain position. This is done by optimizing close-to-perfect phase alignment of the propagating waves emerging from source side, and terminating at the drain point. In particular, where the Rexolite material is not present, the outer edge of the lens acts purely as a parallel plate waveguide, in which a portion of the signal propagates at uniform velocity.

The solution of wave equation is characterized by m in Eq. (5), while the $$TE_m$$ and $$TM_m$$ modes are defined on discrete wavelengths, having specific cut-off frequencies. It is pertinent to note that the wave equation solution with no magnetic-fields along the direction of propagation, i.e., $$TM_m$$ has a special case of having mode $$TM_0$$ with no cutoff frequency. Hence, the lens structure supports the $$TM_0$$ mode along the direction of propagation, where this mode doesn’t possess an electric or magnetic field, leading to a quasi-TEM mode. The mode of propagation in the lens structure close to metallic mirror is different from that of within the Rexolite filled parallel plate waveguide (evident from Fig. 1c).

Moreover, the transitions between sections of the lens and other parts of the beamformer prototype contribute differently to the signal propagation, which are: (i) the part of the lens where air transition to the Rexolite substrate; (ii) transition between dielectric filled plates to the coax probe; (iii) transition between microstrip transmission line and source probe, and (iv) the transition between drain probe and monopole antenna array. All of these transitions have different wave propagation characteristics. During prototype development stage, these transitions were separately optimized for maximum power transfer and low return loss operation before combining together in the form of Maxwell fisheye lens-based beamformer shown in Fig. 2a,b.

Figure 3 shows the beam shaping and the directivity of the propagating waves when different number of ports of the prototype are excited. The 2 port excitation signifies signal input at two central ports, and so on. It can be observed that the directivity increases as we increase the number of ports from 2 to 8, i.e., a directivity of approximated 6 dBi is observed when 2 ports are excited. The directivity increases to approximately 11 dBi when 8 ports are excited. Moreover, if 2 ports are excited then a similar level of directivity is achieved between $$\pm 35^\circ$$. When 4 ports are excited, a beam with higher directivity is achievable with a smooth slope between $$\pm 50^\circ$$. For 6 port excitation, a similar sharp beam shaping is achievable, while the slope can be observed at around $$\pm 35^\circ$$. It is important to note that the sidelobe level for 4, 6 and 8 port excitation is higher as compared to sidelobe level at 2 port excitation. However, the difference between main lobe and 1st side lobe is around 8 dB. Figure 3a represents the beam steering of Maxwell fisheye lens beamformer,when an ideal phase ramp is applied to all the ports. Although the beam shapes are non-ideal, these are useful for this feasibility study about the behavior of beams steering using this disruptive concept as shown in the Fig. 3b. A phase ramp of $$15^\circ$$, $$30^\circ$$, $$45^\circ$$ and $$60^\circ$$ is applied separately to form the beam 2, 3, 4, and 5, respectively. It can be observed that when beam 1 is steered towards beam 2 direction, the maximum directivity is slightly reduced from 10.2 dB to 10.1 dB, while the maximum directivity becomes 10.7 dB for beam 3. Furthermore, we observe that beam shifting impacts the beam shape. However, this difference is not significant for this study and the overall device operation is not compromised.

Figure 4 depicts the three dimensional directivity view of the beam shaping in response to the excitation of 2,4,6 and 8 ports. The beam becomes narrower as the number of ports excited increases from 2 to 6. While the wide equalized directivity beam can be used to realize broadcast mode, the sharp beam with 6-port excitation can be used for applications demanding high directivity. When all 8 ports of the beamformer were excited, we did not observe any additional benefit for boresight transmission as the beamshaping function is similar to that of the 6 port excitation. Hence it is better to use 6 port excitation to achieve similar operation, while the remaining two ports are interchangeably used for beam steering. The patterns in Figs. 4 and  5 shows contours drawn after every 2dB power difference. The half power beamwidth (HPBW) for the 4 port excitation is narrower when compared to 6 ports excitation, while the maximum area of equal power is wider in 2 port excitation as previously shown in Fig. 3a. In a situation where a narrow beam is required, we can use the 4 port excitation while for a wider beam 2 port excitation can be utilized. The 6 port excitation provides a beam shaping that is intermediary between 2 and 4 port excitation. In all the cases, the sidelobe level is below 8 dB from the main lobe. One important point to note here is that 2, 4 and 6 port excitation can be used for beam steering without using an external phase shifter, whilst the latter is required for beam steering using 8 port excitation and the results are depicted in Fig. 5 which signifies the beam steering in response to the phase ramp applied in a similar fashion as described in Fig. 3a. Beam 1 is generated through an 8 port excitation. The rest of the beams 2,3 and 4 are generated as a result of applying the phase ramp also with 8 ports. It can be observed that the beam steering is achieved but the mean maximum radiation area is disturbed. However, the performance of beam steering is reliable even at higher steering angles, when observed at the 3 dB contour. For example, if we observe beam 5 with higher beam steering, the mean maximum radiation area is still in the desired angular direction i.e. $$50^\circ$$. This aspect reveals the benefits and flexibility of utilizing the circular array structure.

The prototype shown in Fig. 2b is used to evaluate beam shaping and beam steering capabilities of the Maxwell fisheye lens beamformer. The prototype was placed in a $$10\ \mathrm {m}\times 5\ \mathrm {m}\times 5\ \mathrm {m}$$ far-field anechoic chamber facility at Queen’s University Belfast, where each port was separately excited by a 10 GHz signal. Co-polarized component in far-field was recorded along the xz-plane (same as in Fig. 2a), for all the 8 input SMA ports of the prototype. The far-field measurement data for each port excitation was combined in a post processing step and the normalised results are compared with that of full-wave electromagnetic simulations, shown in Fig. 6. The beam shaping depicted in Fig. 6a,c is close to the simulated predictions. Beam steering depicted in Fig. 6b shows a slight difference in simulated and measured 3 dB beamwidth, while the beam shape and side lobe level is maintained as predicted. The beam shaping for two port excitation resulting in equalized radiations fields along azimuth direction is verified with the help of measurements results shown in Fig. 6c. The measurement results clearly validate the performance of the Maxwell fisheye lens based proposed beamformer hardware.

## Conclusions and future work

In this work, we present a novel scalable analogue beamforming hardware architecture that helps in realizing adaptive beam steering and beam shaping to fulfil the coverage requirements. The proposed beamforming hardware comprises of a finite size Maxwell fisheye lens used as a feed network for a semi-circular array of monopole antennas. The hardware architecture enables on-demand beamforming with a flexibility of using 2 to 8 antenna elements. The unique aspect of the proposed approach is that it does not require dedication of the number of ports in a feed network and is scalable to handle between 2 to 8 antenna array elements. The simulation and measured results are presented to validate the performance of the beamformer in terms of beam shaping and beam steering. As a future direction to this work, we plan to investigate means to achieve better impedance matching between transitions. Moreover, the proposed design can be expanded to act as a high performance wireless repeater due to the simplicity of the feed network.

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## Acknowledgements

Authors would like to thank W. McKeown and C. Colligan for their assistance in the lens machining. This work was partially supported by the Engineering and Physical Sciences Research Council (EPSRC) under Grant EP/P000673/1 and Grant EP/S007954/1.

## Author information

Authors

### Contributions

M.A.B.A. and V.F. conceived the idea. M.A.B.A. designed and simulated the proposed beamformer and performed the measurements. R.I.A., G.G.M. and M.A.B.A. interpreted the results. R.I.A., M.A.B.B. and G.G.M. wrote the paper. V.F. supervised the research. All authors reviewed the manuscript.

### Corresponding author

Correspondence to Muhammad Ali Babar Abbasi.

## Ethics declarations

### Competing interests

The authors declare no competing interests.

### Publisher's note

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

## Rights and permissions

Reprints and Permissions

Abbasi, M.A.B., Ansari, R.I., Machado, G.G. et al. Design and analysis of Maxwell fisheye lens based beamformer. Sci Rep 11, 22739 (2021). https://doi.org/10.1038/s41598-021-02058-9

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• DOI: https://doi.org/10.1038/s41598-021-02058-9


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# Hill Climbing Algorithm - Optimum Step Size

I am implementing a standard hill climbing algorithm to optimise hyper-parameters for a predictive model. The hill climbing algorithm is being applied as part of a two-stage approach:

1. Apply grid search with large values applied to the hyper-parameter to find a 'best' starting point
2. Apply hill climbing algorithm in this space with a large number of different, random start points to find a local optimum

The large values that are passed in the first step, the grid search are

1*10^seq(-4, 5, by=1)
[1] 1e-04 1e-03 1e-02 1e-01 1e+00 1e+01 1e+02 1e+03 1e+04 1e+05

So I am struggling to choose an optimum step size that isn't too large that is skips the peak or too small that it takes too long to converge. I don't think a single value for the step size is appropriate for all values passed in the grid search since, for example the difference between 1e-04 and 1e-03 is vastly different to 1e03 and 1e04. So I want the step size to be proportionate to the grid search start point. I know the search space I'm looking at is

grid_search_optimum/10 to grid_search_optimum*10

My question, therefore, is what is an accepted value for the step size in relation to the search space? I haven't found any opinions in literature around this and, in general, the only advice is to choose a step size that is "sufficiently small". Any advice or pointers to relevant papers would be greatly appreciated!

Usually it worked for me that if the search space was know then annealing rate (divide the size size with number of iteration)helped to decrease/increase the step size gradually to get to local max/min but the draw back is it might get stuck in local and might need some "momentum" to go on, another draw back it it might be very slow.however it doesn't seems like the case with your question that you bother about it.

• Hey BlackCurrant, thanks for the quick post! Do you have any reference for the step size being search size divided by the number of iterations? The paper proposing a self-adaptive step size is very interesting but wouldn't be appropriate in this case. – A_Murphy May 7 at 13:01
• I have implemented annealing rate in context of gradient decent algorithms , if you are able to extend the concept here- cs231n.github.io/neural-networks-3/#anneal few articles- towardsdatascience.com/… jeremyjordan.me/nn-learning-rate one in context of deep learning- auai.org/uai2015/proceedings/papers/73.pdf – BlackCurrant May 7 at 13:12
• Sorry about the delay I was spending time looking through the attached! However, I am struggling to see how to implement an adaptive step size for hill climbing, how would the gradient be calculated in this situation? – A_Murphy May 8 at 16:35
• Here is the article with code handy, i shall have to dug to find mine, basically you just divide the learning rate in each iteration by iteration count. This is simple break down- machinelearningmastery.com/… – BlackCurrant May 8 at 16:58
• That makes perfect sense thanks! The only thing left unclear is the start value for the step size. In my problem the search space can vary massively so what size should the first step be in terms of the search space? I'm happy to anneal it thereafter as a good solution – A_Murphy May 8 at 17:17

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# LAX to Anaheim - how much slower is the Super Shuttle than a cab?

I am flying to LAX and need to get to the Anaheim Hilton (essentially the convention centre). Although it's only 29 miles, I am told it will take 90 minutes and cost $100 by cab. That's quite a cab fare, so I am thinking of the Super Shuttle. Problem is, every time I try the shuttle, it seems we spend 30-45 minutes driving around the airport, and then go to everyone else's hotels (and homes!) first, adding what feels like another hour to the trip. Since I'll be landing at 10pm California time, which will feel like 1am, I'm really not up for that. But perhaps the freeway part of the journey is the bigger part, and it won't be that bad? Does anyone have experience with cabs, shuttles etc specifically from LAX to Anaheim, and suggestions for which I should choose? - Great question! – Michael Pryor Jun 25 '11 at 17:30 Are you going to the BlizzCon? – RoflcoptrException Sep 17 '11 at 8:22 Orbiting LAX in a SuperShuttle until we'd gained enough passengers to escape is one of the most frustrating starts to a trip I have yet encountered. I guess the reverse trip isn't as bad, but I got a cab (not as far as Anaheim, though) – e100 Jan 11 '13 at 13:58 ## 4 Answers Some thoughts: • That 29 miles sounds like it came from someone in PR—Google maps puts it at over 35 miles and 60-80 minutes with traffic. Back when I lived there, I'd have estimated that drive at about 90 minutes without traffic. • Do you know what your schedule will be like during the trip? Unless you know you're going no further than a few blocks away from the hotel/convention center during the whole trip, you'll almost always want to rent a car in Southern California. Keep in mind that you can't catch taxis on the street, and public transportation is very limited. • Are you traveling by yourself, or with others? If it's just you, it's hard to justify paying$120 for a cab (each way) versus $16 for a shuttle (each way). However, the cab fare is the same no matter how many people they let you cram in. • I agree with Joel that John Wayne (SNA) and Long Beach (LGB) are better bets, but it's worth keeping in mind that the shuttle fare is: •$16 to/from LAX
• $10 to/from SNA •$35 to/from LGB

Which makes SNA look like the best bet. I have no idea why LGB is that much more when the cab fare would be under $70. - +1. This left me wondering: why can't you catch taxis on the street in Southern California? Are they not allowed to stop, or are there so few of them or what? – Jonik Sep 17 '11 at 7:31 @Jonik - I think it's a chicken & egg issue: taxi riders don't expect to see taxis on the street, so taxi drivers don't expect to pick up fares on the street. Also, So Cal is very spread out; it would take a massive number of taxis to cover the area, and the taxi-taking population isn't there to support them. – Dori Sep 17 '11 at 7:42 Yep, that's the nature of SuperShuttle... it takes twice as long and drives you batty, but it does save some money. Do you really have to fly to LAX? There are two commercial airports closer than LAX to Anaheim -- John Wayne and Long Beach -- both of which are pleasant, quieter airports. - I'm flying from YYZ - there are no direct flights to SNA etc. I figured the cost and time of changing flights would outweigh the cost and time of the cab. – Kate Gregory Jun 26 '11 at 1:23 @Kate Split flights are occasionally cheaper from YYZ, it depends on the date and a number of other factors. I would check, just in case. – Matthew Read Jun 27 '11 at 3:45 As it happened I did not go to Anaheim from LAX but to Carlsbad. However I have just taken the reverse trip. I booked a 5:05 shuttle from the Anaheim Hilton. We pulled out about 5:15 but proceeded to go back and forth from hotel to hotel and in fact drove past the Hilton for the last time about 6pm. We pulled onto airport property at 6:40, much to the consternation of the fellow with the 7:30 flight who'd been told it was a 45 minute trip, and since we went to his terminal first, I ended up walking into my terminal about 7:00 for let's call it a two hour trip. So yeah, my history of the shuttle - that it adds a minimum of an hour to the trip - holds true even here in California where the intercity trip might be thought to outweigh the fiddling around time. I knew what I was up for which is why I booked a 5pm shuttle. - Just did it last night. Depends how many people are travelling for us it was 4 so we decided on a taxi. Bad idea it cost$110 from LAX to Anaheim, I should've taken the \$18 pp in a shuttle. Hope that helps!

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# Wilsonian versus old-fashioned renormalization points of view

+ 5 like - 0 dislike
2483 views

Some participants point out that Wilsonian point of view is much better than all old-fashioned renormalization ideologies. So I would like to learn what makes you feel more comfortable with Wilsonian POV? What makes you accept it without hesitation? What is its advantage?

Here's a good explanation of the Wilsonian point of view by @LubosMotl on TRF. Kenneth Wilson, RIP.

Thank you, dimension10, I read it when Lubosh wrote it. Still, there might be "personal" differences in different people. I would like to learn as much as possible about the "convincing part".

I accept it simply because it is logically more coherent and natural than the "subtracting infinity" picture, and it reproduces perturbative renormalizability criteria in a natural way. It doesn't tell us what the underlying short distance theory really looks like, but it tells us it's ok not knowing what the underlying theory is.

Thanks, Jia. By the way, I am not going to argue at all; I just want to listen to argumentations/insights that make people convinced in advantages of the Wilsonian POV on QFT.

A wise decision!

I am sorry, I cannot (may not) vote up, guys, so I write a "thanks" comment when I want to vote up your answers/comments.

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The ''old-fashioned renormalization" presumably refers to renormalization by subtracting infinities. This is just a loose way of talking, and not really done on the formal level. Indeed, there is no subtraction of infinities once one introduces a cutoff $\Lambda$ - one only subtracts huge finite numbers, and takes the limit after the subtracted numbers are of physical size. This is just like calculating a limit $\lim_{\Lambda\to\infty} \sqrt{\Lambda^2+\Lambda}-\sqrt{\Lambda^2-\Lambda}$ by expanding the expression into powers of $\Lambda^{-1}$ (including negative powers, i.e., a term linear in $\Lambda$, which cancels) before taking the limit, thereby getting in place of $\infty-\infty$ the finite result $1$. The procedure is completely unambiguous.

The magic and uneasiness enters the QFT version of this simple and undisputed toy problem in calculus because in QFT the parameters of the Lagrangian are an (often diverging) function of the cutoff $\Lambda$, and the quest is for understanding how this comes about.

Wilson's picture of integrating out high energy degrees of freedom (cf. http://motls.blogspot.in/2013/06/kenneth-wilson-rip.html) gives an intuitive understanding of what happens - it naturally explains why at a different cutoff scale one should expect to get different values of the parameters of the resulting effective theory, and hence allows an interpretation of the bare parameters in the Lagrangian without cutoff as parameters of an effective theory. It also suggest a systematic way of obtaining a differential equation for the cutoff-dependence of the bare parameters - the Wilson renormalization group. In fact, the latter is only a semigroup, as one loses information in the process of integrating out high energy degrees of freedom, and in general cannot go backwards. cf. http://www.physicsoverflow.org/6902/under-conditions-renormalization-group-equations-reversible.

Wilson's picture is physical only in the quantum field theory of thermodynamics and solid state physics, where the cutoff is an observational scale and hence has a physical meaning. In relativistic QFT, the cutoff is a computational device as it destroys Poincare invariance and locality, which are both valid rigorously in a QFT. But the analogy is fairly compelling when one interprets the cutoff as resulting from an unknown but neglected submicroscopic structure (e.g., a grand unified theory, effects of quantum gravity, or a string theory). Integrating out these submicroscopic degrees of freedom would indeed result in a nonlocal cutoff in the resulting effective field theory.

On the other hand, one expects that at least asymptotically free theories such as QCD are well-determined even without an existing substructure. (The Yang-Mills millennium problem asks to settle a related problem, cf. the discussion in http://www.physicsoverflow.org/21784/yang-mills-millenium-question-and-dynin-formalism). In this case, Wilson's picture is just that - a picture to motivate the cutoff-dependence of the coupling that leads to a finite limit when the cutoff is removed.

However, asymptotically free theories show an additional phenomenon called dimensional transmutation. For simplicity, I discuss massless QCD, which depends on just a single bare parameter $g$. Renormalization requires [notation as in Chapter 17 of Peskin/Schroeder]  that the bare coupling constant in QCD is a function $g(\Lambda,\alpha_s,M)$ of three arguments, the cutoff $\Lambda$, the dimensionless strong fine structure constant $\alpha_s$ (the QCD analogue of the electromagnetic fine structure constant), and an arbitrary renormalization scale $M$. Wilson's renormalization semigroup shows how  $g(\Lambda,\alpha_s,M)$ changes as $\Lambda$ is reduced, but because we know - or at present rather assume - that the exact, local theory at $\Lambda=\infty$ exists, we can look for the limit $\Lambda\to\infty$ of the family of Wilson theories when $\Lambda$ increases beyond limits.. This is believed to give a well-defined QFT, the theory we call QCD. It still has a dependence on two parameters $\alpha_s$ and $M$, although there is only a 1-parameter family of  QFTs (since we have a limit of a 1-parameter family of bare theories with cutoff). Therefore there must be a definite functional relation between $\alpha_s$ and $M$, usually expressed by writing $\alpha_s$ as a function of the mass scale $M$. [Since massless QCD is scale invariant, one may scale the fields such that $\alpha_s=1$ and gets ''the mass scale $M$ of QCD'' - to be fitted to experiments. However, then quark masses must also be taken into account, which complicates the story.) The dependence  $\alpha_s(M)$ (and of other observables) is called the running coupling constant, and there is a second Stueckelberg renormalization group (this time a true group) that expresses how quantities depend on $M$. In causal perturbation theory, where there is no cutoff and hence no Wilson renormalization semigroup, the Stueckelberg renormalization group is still present.

Note, however, that the Wilson renormalization semigroup and the Stueckelberg renormalization group have nothing in common except for some formal analogy. In particular, in relativistic QFT, Wilson's point of view is an interpretation aid, no essential ingredient (since the true theory has no cutoff), while the Stueckelberg renormalization group is a property of the renormalized equations, and choosing $M$ of the order of the experimental energy is important to keep the approximation errors in truncated computations small. (In exact computations to all orders, the value of $M$ would be immaterial.)

answered Aug 17, 2014 by (14,019 points)
edited Aug 17, 2014

Thank you, Arnold, too!

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# Are there situations where improper priors can be avoided via a prior on a subset of the real line and a transformation?

There are many situations where improper priors are "permissable" (Berger, 2009). In many cases, these improper priors are improper because they are "flat" on the real line. A well known example is inference on a normal mean, where Jeffery's prior turns out to be $$\pi(\mu)= 1$$.

$$[a,b] \subset \mathbb{R}$$ and $$\mathbb{R}$$ are in bijection with each other. Because it's easy to write down such bijections explicitly, I'm wondering it is ever possible to map the problem of inference on the real line with an improper prior to a problem of inference on $$[a,b]$$ with a proper one.

My intuition is that the answer is no, but improper priors are still so strange for me I'm searching for ways to grasp them.

• @Xi'an This comment should be an answer. – Robin Ryder Feb 18 at 20:08

When one considers a (differentiable) bijection $$\psi:(a,b)\to\mathbb R$$, and an $$\sigma$$-finite measure $$\pi$$ over $$\mathbb R$$, the measure $$\nu$$ defined by $$\nu(A)=\pi(\Psi(A))$$over measurable sets $$A\subset(a,b)$$ has a density with respect to the Lebesgue measure $$\lambda$$ equal to $$\frac{\text{d}\nu}{\text{d}\lambda}(x)=\frac{\text{d}\pi}{\text{d}\lambda}(\Psi(x))\times\left|\frac{\text{d}\Psi}{\text{d}x}(x)\right|$$ which means that the density of the transformed measure is no longer constant. In any case, $$\nu\{(a,b)\}=\pi(\Psi\{(a,b)\})=\pi(\mathbb R)=\infty$$


text:
Sergei Yakovenko's blog: on Math and Teaching

Monday, January 30, 2017

Lecture 11 (Jan 16, 2017)

Filed under: Calculus on manifolds course,lecture — Sergei Yakovenko @ 3:54
Tags: , ,

Geodesics

Definitions of geodesic curves on a Riemmanian manifold. Differential equations of the second order. Local existence of solutions. Geodesic map. Geodesic spheres, orthogonality. Local minimality of geodesic curves. Metric and geodesic completeness of Riemannian manifolds.

Survey of adjacent areas. Behavior of the nearby geodesics. Jacobi field. On surfaces: conjugated points, the role of the Gauss curvature, global properties of manifolds with positive and negative curvature (comparison). Hyperbolic plane $\mathbb H$, geodesics on it. Realization of the hyperbolic geometry on a surface in $\mathbb R^3_{++-}$. Impossibility of global embedding of $\mathbb H$ into $\mathbb R^3_{+++}$.

Introduction to the Riemannian geometry

1. Flat structure of the Euclidean space and coordinate-wise derivation of vector fields.
2. Axiomatic definition of the covariant derivative and its role in defining the parallel transport along curves on manifolds. Connexion.
3. Covariant derivative $\overline\nabla\text{ on }\mathbb R^n$ and its properties (symmetry, flatness, compatibility with the scalar product).
4. Smooth submanifolds of $\mathbb R^n$. The induced  Riemannian metric and connection. Gauss equation.
5. Weingarten operator on hypersurfaces and its properties. Gauss map.
6. Curvatures of normal 2-sections (the inverse radius of the osculating circles). Principal, Gauss and mean curvatures.
7. Curvature tensor: a miracle of a 2-nd order differential operator that turned out to be a tensor (“0-th order” differential operator).
8. Symmetries of the curvature and Ricci tensors.
9. Uniqueness of the symmetric connexion compatible with a Riemannian metric. Intrinsic nature of the Gauss curvature.

The lecture notes are available here.

Meromorphic flat connexions on holomorphic manifolds: Integrability, monodromy, classification

1. Pfaffian systems and their integrability
2. From local to global solutions: monodromy
3. Geometric language: covariant derivative and its curvature
4. Meromorphic functions, meromorphic forms
5. Example: multidimensional Euler system
6. Regular singularities
7. Flat connexions vs. isomonodromic deformations

Recommended reading: D. Novikov & S.Y., Lectures on meromorphic flat connexions, sect. 1-2.

Blog at WordPress.com.


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27 Jan 2011, 06:33
There are three people Sam, Rick and Husam. What is the probability that they don't have birthday on a same day (Sunday, MOnday---Saturday)?
I am sorry I don't have OA for this question.Bunuel, Pkit, shrouded1, icandy, IanStewart or any other members please help me!
Solutions:
I attempted it from two approaches but got two different answers
Approach 1:
7/7 * 6/7 * 5/ 7

Approach 2:
If we consider Sunday-Mon- Tue-Wed-Th-Fri- Sat as seven different seats and we can find in how many ways three people can make an arrangement on these seats.
7! / { 3! * (7-3)!} = 7! / { 3! * 4!}

Please let me know which approach is correct. Why one is wrong and another is right? or Why both of them are wrong?
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27 Jan 2011, 07:59
bhandariavi wrote:
There are three people Sam, Rick and Husam. What is the probability that they don't have birthday on a same day (Sunday, MOnday---Saturday)?

I am sorry I don't have OA for this question.Bunuel, Pkit, shrouded1, icandy, IanStewart or any other members please help me!
Solutions:
I attempted it from two approaches but got two different answers
Approach 1:
7/7 * 6/7 * 5/ 7

Approach 2:
If we consider Sunday-Mon- Tue-Wed-Th-Fri- Sat as seven different seats and we can find in how many ways three people can make an arrangement on these seats.
7! / { 3! * (7-3)!} = 7! / { 3! * 4!}

Please let me know which approach is correct. Why one is wrong and another is right? or Why both of them are wrong?

The question is quite ambiguous.

There are three people Sam, Rick and Husam. What is the probability that they don't have birthday on a same day (Sunday, Monday, ..., Saturday)?

I think that the question simply asks the probability of an event that not all 3 men have their birthdays on a same day (meaning that all 3 have birthday on Sunday is not OK, but if 2 of them have their birthday on Sunday and the third one on Monday is OK):

$$P=1-\frac{7}{7^3}=\frac{48}{49}$$. So in this case neither of approaches is right.

But if the question were asking about the probability that all 3 men have their birthdays on different days (for example Sam-Sunday, Rick-Monday, Husam-Tuesday), then:

First one can have his birthday on any day - 7/7, the probability that the second guy won't have his birthday on the same day will be 6/7 and the probability that the third guy won't have his birthday on these two days will be 5/7, so $$P=\frac{7}{7}*\frac{6}{7}*\frac{5}{7}=\frac{30}{49}$$.

Or if you want to solve with combination then: probability=# of favorable outcomes/total # of outcomes --> total # of outcomes is 7^3 as each has 7 options for a birthday. Now, # of favorable outcomes will be: $$C^3_7*3!$$, where $$C^3_7$$ is ways to choose 3 different days out of 7 and 3! is # of different ways to assign these days to 3 people (the nominator also could be represented as $$P^3_7$$). So, $$P=\frac{C^3_7*3!}{7^3}=\frac{30}{49}$$.

So in this case your first approach would be correct and second would not. By the way second approach doesn't give a probability at all (probability must be between 0 and 1 inclusive), and it gives # of selections of 3 different items (days in our case) out of 7.

Hope it's clear.
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28 Jan 2011, 04:37
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bhandariavi wrote:
There are three people Sam, Rick and Husam. What is the probability that they don't have birthday on a same day (Sunday, MOnday---Saturday)?
I am sorry I don't have OA for this question.Bunuel, Pkit, shrouded1, icandy, IanStewart or any other members please help me!
Solutions:
I attempted it from two approaches but got two different answers
Approach 1:
7/7 * 6/7 * 5/ 7

Approach 2:
If we consider Sunday-Mon- Tue-Wed-Th-Fri- Sat as seven different seats and we can find in how many ways three people can make an arrangement on these seats.
7! / { 3! * (7-3)!} = 7! / { 3! * 4!}

Please let me know which approach is correct. Why one is wrong and another is right? or Why both of them are wrong?

I am not sure if I am correct, but let's try...

Suppose, all their birthdays are on a Sunday...

therefore, probability is = 1/7 * 1/7 * 1/7

For 7 such days, it is = 7 * 1/(7*7*7) = 1/49

Now, assuming only Sam and Rick's birthday is on a Sunday...

Therefore, probability is = 1/7 *1/7 * 6/7

For 7 days, it is = 7* 1/49 *6/7 = 6/49.

There can be 3 such combo ( S & R, R & H, S & H).

Therefore, net = 6/49 * 3 = 18/49

Therefore, total probability of same birthdays = 1/49 + 18/49 = 19/49

Therefore, required answer is = 1 - 19/49 = 30 /49
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28 Jan 2011, 07:28
Thank you Bunuel and Samidh. It is crystal clear now
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28 Jan 2011, 20:34
HI Bunuel

Could you please explain this bit to me ?

P = 1 - 7/7^3. So in this case neither of approaches is right.

I'm keen to know how the numerator is 7.

Regards,
Subhash
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28 Jan 2011, 22:41
" Suppose, all their birthdays are on a Sunday...

therefore, probability is = 1/7 * 1/7 * 1/7

For 7 such days, it is = 7 * 1/(7*7*7) = 1/49

Now, assuming only Sam and Rick's birthday is on a Sunday...

Therefore, probability is = 1/7 *1/7 * 6/7

For 7 days, it is = 7* 1/49 *6/7 = 6/49.

There can be 3 such combo ( S & R, R & H, S & H).

Therefore, net = 6/49 * 3 = 18/49

Therefore, total probability of same birthdays = 1/49 + 18/49 = 19/49

Therefore, required answer is = 1 - 19/49 = 30 /49 "

i think the approach is ok. that includes both the possiblity of all 3 having b'day on same day and only 2 having b'day on same day.

whats the OA ?
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29 Jan 2011, 03:00
subhashghosh wrote:
HI Bunuel

Could you please explain this bit to me ?

P = 1 - 7/7^3. So in this case neither of approaches is right.

I'm keen to know how the numerator is 7.

Regards,
Subhash

What is the probability that all 3 men have birthday on a specific day, for example on Sunday? P=1/7*1/7*1/7=1/7^3. Now, as there are 7 days then the probability that all 3 have birthday on a same day is 7/7^3 and the probability they don't have birthday on a same day is P=1-7/7^3=48/49.

garimavyas wrote:
" Suppose, all their birthdays are on a Sunday...

therefore, probability is = 1/7 * 1/7 * 1/7

For 7 such days, it is = 7 * 1/(7*7*7) = 1/49

Now, assuming only Sam and Rick's birthday is on a Sunday...

Therefore, probability is = 1/7 *1/7 * 6/7

For 7 days, it is = 7* 1/49 *6/7 = 6/49.

There can be 3 such combo ( S & R, R & H, S & H).

Therefore, net = 6/49 * 3 = 18/49

Therefore, total probability of same birthdays = 1/49 + 18/49 = 19/49

Therefore, required answer is = 1 - 19/49 = 30 /49 "

i think the approach is ok. that includes both the possiblity of all 3 having b'day on same day and only 2 having b'day on same day.

whats the OA ?

Again: 30/49 would be the answer if the question were asking about the probability that all 3 men have birthday on different days (for example Sam-Sunday, Rick-Monday, Husam-Tuesday) BUT if the question simply asks the probability of an event that not all 3 men have their birthdays on a same day (meaning that all 3 have birthday on Sunday is not OK, but if 2 of them have their birthday on Sunday and the third one on Monday is OK) then the answer is P=1-7/7^3=48/49.
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29 Jan 2011, 08:23
Yeah, understood now. Thanks a lot.
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Publicité ▼

## définition - Magma (algebra)

voir la définition de Wikipedia

Wikipedia

# Magma (algebra)

In abstract algebra, a magma (or groupoid; not to be confused with groupoids in category theory) is a basic kind of algebraic structure. Specifically, a magma consists of a set $M$ equipped with a single binary operation $M \times M \rightarrow M$. A binary operation is closed by definition, but no other axioms are imposed on the operation.

The term magma for this kind of structure was introduced by Nicolas Bourbaki. The term groupoid is an older, but still commonly used alternative which was introduced by Øystein Ore.

## Definition

A magma is a set $M$ matched with an operation "$\cdot$" that sends any two elements $a,b \in M$ to another element $a \cdot b$. The symbol "$\cdot$" is a general placeholder for a properly defined operation. To qualify as a magma, the set and operation $(M,\cdot)$ must satisfy the following requirement (known as the magma axiom):

For all $a$, $b$ in $M$, the result of the operation $a \cdot b$ is also in $M$.

And in mathematical notation:

$\forall a,b \in M : a \cdot b \in M$

### Etymology

In French, the word "magma" has multiple common meanings, one of them being "jumble". It is likely that the French Bourbaki group referred to sets with well-defined binary operations as magmas with the "jumble" definition in mind.

## Types of magmas

Magmas are not often studied as such; instead there are several different kinds of magmas, depending on what axioms one might require of the operation. Commonly studied types of magmas include

Note that both divisibility and invertibility imply
the existence of the cancellation property.

## Morphism of magmas

A morphism of magmas is a function $f\colon M\to N$ mapping magma $M$ to magma $N$, that preserves the binary operation:

$f(x \; *_M \;y) = f(x) \; *_N\; f(y)$

where $*_M$ and $*_N$ denote the binary operation on $M$ and $N$ respectively.

## Combinatorics and parentheses

For the general, non-associative case, the magma operation may be repeatedly iterated. To denote pairings, parentheses are used. The resulting string consists of symbols denoting elements of the magma, and balanced sets of parenthesis. The set of all possible strings of balanced parenthesis is called the Dyck language. The total number of different ways of writing $n$ applications of the magma operator is given by the Catalan number $C_n$. Thus, for example, $C_2=2$, which is just the statement that $(ab)c$ and $a(bc)$ are the only two ways of pairing three elements of a magma with two operations.

A shorthand is often used to reduce the number of parentheses. This is accomplished by using juxtaposition in place of the operation. For example, if the magma operation is $*$, then $xy * z$ abbreviates $(x * y) * z$. Further abbreviations are possible by inserting spaces, for example by writing $xy * z * wv$ in place of $((x * y) * z) * (w * v)$. Of course, for more complex expressions the use of parenthesis turns out to be inevitable. A way to avoid completely the use of parentheses is prefix notation.

## Free magma

A free magma $M_X$ on a set $X$ is the "most general possible" magma generated by the set $X$ (that is there are no relations or axioms imposed on the generators; see free object). It can be described, in terms familiar in computer science, as the magma of binary trees with leaves labeled by elements of $X$. The operation is that of joining trees at the root. It therefore has a foundational role in syntax.

A free magma has the universal property such that, if $f\colon X\to N$ is a function from the set $X$ to any magma $N$, then there is a unique extension of $f$ to a morphism of magmas $f^\prime$

$f^\prime\colon M_X \to N.$

## Classification by properties

 Totality Associativity Identity Inverses Group-like structures Required not required not required not required not required Required not required not required Required not required Required not required Required not required not required Required not required Required Required not required Required Required not required not required not required Required Required Required not required not required Required Required Required Required not required Required Required Required Required Required not required Required Required Required not required not required Required Required not required not required not required Required not required not required not required Note: A quasigroup with associativity (equivalently, a semigroup with inverses) already has an identity element, and is therefore a group.

A magma (S, *) is called

• unital if it has an identity element,
• medial if it satisfies the identity xy * uz = xu * yz (i.e. (x * y) * (u * z) = (x * u) * (y * z) for all x, y, u, z in S),
• left semimedial if it satisfies the identity xx * yz = xy * xz,
• right semimedial if it satisfies the identity yz * xx = yx * zx,
• semimedial if it is both left and right semimedial,
• left distributive if it satisfies the identity x * yz = xy * xz,
• right distributive if it satisfies the identity yz * x = yx * zx,
• autodistributive if it is both left and right distributive,
• commutative if it satisfies the identity xy = yx,
• idempotent if it satisfies the identity xx = x,
• unipotent if it satisfies the identity xx = yy,
• zeropotent if it satisfies the identity xx * y = yy * x = xx,
• alternative if it satisfies the identities xx * y = x * xy and x * yy = xy * y,
• power-associative if the submagma generated by any element is associative,
• left-cancellative if for all x, y, and z, xy = xz implies y = z
• right-cancellative if for all x, y, and z, yx = zx implies y = z
• cancellative if it is both right-cancellative and left-cancellative
• a semigroup if it satisfies the identity x * yz = xy * z (associativity),
• a semigroup with left zeros if there are elements x for which the identity x = xy holds,
• a semigroup with right zeros if there are elements x for which the identity x = yx holds,
• a semigroup with zero multiplication or a null semigroup if it satisfies the identity xy = uv, for all x,y,u and v
• a left unar if it satisfies the identity xy = xz,
• a right unar if it satisfies the identity yx = zx,
• trimedial if any triple of its (not necessarily distinct) elements generates a medial submagma,
• entropic if it is a homomorphic image of a medial cancellation magma.

If $*$ is instead a partial operation, then S is called a partial magma.

See n-ary group.

## References

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SERVING THE QUANTITATIVE FINANCE COMMUNITY

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FaridMoussaoui
Posts: 400
Joined: June 20th, 2008, 10:05 am
Location: Genève, Genf, Ginevra, Geneva

### Re: Are Artificial Intelligence methods (AKA Neural Networks) for PDEs about to rediscover the wheel ?

$\circ$ is the standard composition operator.

The reviewer was asleep for the Hessian. Lower $#$ follows Vellani notations.
Last edited by FaridMoussaoui on February 13th, 2019, 9:07 am, edited 1 time in total.

JohnLeM
Topic Author
Posts: 298
Joined: September 16th, 2008, 7:15 pm

### Re: Are Artificial Intelligence methods (AKA Neural Networks) for PDEs about to rediscover the wheel ?

I think there's a typo on page 3; you say $\nabla^2$ is the Jacobian but it is the Laplacian operator (or Hessian which become singular/negative definite).

BTW is reference [6] available yet?

I don't get what $A \circ S$ does/is: it seem to be used a number of times without a sharp definition.

// Haven't got my head yet around article [1] and how to pull-back a measure..It uses upper case # while the article uses lower case #. Are they the same #?
$\nabla^2h = \nabla S$, that is a Jacobian, seen as the Hessian of a convex function, that is important when one want to define S as a one-to-one map.
[6] is written, we should have released it since years, we are all busy :/ I'll try to give a talk about it ASAP.
$A \circ S$ composition
article [1] is magical. It tells you that any map $S$ can be decomposed as $S = (\nabla h) \circ T$, h convex, T Lebesgue measure preserving. It is the analogous for functions of the polar decomposition for matrix $M = A U$.
Not really, but quite close : if $S_\# \mu = \nu$, then $S^\# \nu = \mu$, that is nothing but a variable change $\int \varphi \circ S d\mu = \int \varphi d\nu$(AFAIR)

Cuchulainn
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### Re: Are Artificial Intelligence methods (AKA Neural Networks) for PDEs about to rediscover the wheel ?

I'm kind of allergic to magic It's very measure-theoretic which is computationally not obvious?

BTW if you write $(\nabla S)$ then it is the gradient,(vector) not Jacobi (matrix)??

Regarding mapping from n-d space to the unit cube $(0,1)^n$ I'm assuming we are talking about the independent variable?
Last edited by Cuchulainn on February 13th, 2019, 10:17 am, edited 8 times in total.

JohnLeM
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### Re: Are Artificial Intelligence methods (AKA Neural Networks) for PDEs about to rediscover the wheel ?

I'm kind of allergic to magic

BTW if you write $(\nabla S)$ then it is the gradient,(vector) not Jacobi (matrix)??
I just wanted to express my deep admiration for this work of Brenier.
$(\nabla S)$ is a field of vectors if S is a scalar function. It is a Jacobian (i.e. field of matrices) if $S : R^D \mapsto \ R^D$ is a map ?

Cuchulainn
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### Re: Are Artificial Intelligence methods (AKA Neural Networks) for PDEs about to rediscover the wheel ?

I don't agree with the notation! I do agree with the conclusion.

Jacobian

https://nl.wikipedia.org/wiki/Jacobi-matrix

Or am I missing something?
Cal it $J(S)$ otherwise everyone thinks it's $grad$. It confuses me no end,
Last edited by Cuchulainn on February 13th, 2019, 10:30 am, edited 1 time in total.

JohnLeM
Topic Author
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Joined: September 16th, 2008, 7:15 pm

### Re: Are Artificial Intelligence methods (AKA Neural Networks) for PDEs about to rediscover the wheel ?

I don't agree with the notation.

Well, this is the same notation : $\nabla f := (\partial_1 f, \cdots, \partial_D f)$ for a scalar function $f : R^D \mapsto R$. $\nabla S := (\partial_i S_j )_{i,j=1,\cdots,D}$ for the Jacobian of the map $S:=(S_1,\ldots,S_D)$. It seems correct: it is the gradient applied to each component. For instance, I can think to the Hessian operator as $\nabla^2 = \nabla \nabla^T$, and the Laplacian one as $\Delta = \nabla^T \nabla$ (with a slight notation abuse)
Last edited by JohnLeM on February 13th, 2019, 10:46 am, edited 1 time in total.

Cuchulainn
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### Re: Are Artificial Intelligence methods (AKA Neural Networks) for PDEs about to rediscover the wheel ?

OK, but the define it in the article! IMO you are still using notation I have never seen in this context.

The gradient is a special case of Jacobian and writing the latter using forrmer's notation is scary IMO.

JohnLeM
Topic Author
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### Re: Are Artificial Intelligence methods (AKA Neural Networks) for PDEs about to rediscover the wheel ?

Regarding mapping from n-d space to the unit cube $(0,1)^n$ I'm assuming we are talking about the independent variable?
I am not sure to understand your question. Yes all is mapped into a fixed set in this paper, for instance $(0,1)^D$, that is a localization principle. In other words, instead of solving the Fokker planck equation (1), describing a probability density $\mu(t,x),x\in R^D$, we solved equivalently the equation satisfied by $S(t,x) : (0,1)^D \mapsto R^D$, transporting the Lebesgue measure of the unit cube (not. $dx$) into $\mu(t,x)$, i.e. $S(t,\cdot)_\# dx = d\mu(t,\cdot)$.

Cuchulainn
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### Re: Are Artificial Intelligence methods (AKA Neural Networks) for PDEs about to rediscover the wheel ?

Regarding mapping from n-d space to the unit cube $(0,1)^n$ I'm assuming we are talking about the independent variable?
I am not sure to understand your question. Yes all is mapped into a fixed set in this paper, for instance $(0,1)^D$, that is a localization principle. In other words, instead of solving the Fokker planck equation (1), describing a probability density $\mu(t,x),x\in R^D$, we solved equivalently the equation satisfied by $S(t,x) : (0,1)^D \mapsto R^D$, transporting the Lebesgue measure of the unit cube (not. $dx$) into $\mu(t,x)$, i.e. $S(t,\cdot)_\# dx = d\mu(t,\cdot)$.
I mean e.g. solving the heat equation u_t = u_xx on the real line. Define new independent variable y = x/(1+x)  to get a pde u_t = a(y)(a(y)u_y)_y on (0,1) where a(y) = 1-y*y.

I am not familiar with the term "localization principle". Maybe that's my confusion. Is it a term from physics?

Is it possible for motivation to give an example (even 1d/2d) to show how equations (5) and (6) work?

JohnLeM
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### Re: Are Artificial Intelligence methods (AKA Neural Networks) for PDEs about to rediscover the wheel ?

I mean e.g. solving the heat equation u_t = u_xx on the real line. Define new independent variable y = x/(1+x)  to get a pde u_t = a(y)(a(y)u_y)_y on (0,1) where a(y) = 1-y*y.
If you consider a fixed-in-time change of variable, you are done. This is precisely an error that the AI community does today. Recall that Fokker-Planck equations are considered with Dirac measures as initial conditions. Try to solve numerically the heat equation in one dimension  $u_t = u_{xx}, u(0,x) = \delta_{x_0}(x)$, where $x_0$ can be anywhere, with your variable change $y = x/(1+x)$ to see this.

FaridMoussaoui
Posts: 400
Joined: June 20th, 2008, 10:05 am
Location: Genève, Genf, Ginevra, Geneva

### Re: Are Artificial Intelligence methods (AKA Neural Networks) for PDEs about to rediscover the wheel ?

The change of variable is for the probabilty (transition) density (as he is solving the Fokker Planck equation).
For example for the GMB process $dX = X r dt + X \sigma dW$, we have $X \in R_{+}$.
We look for a map to localise from $R_{+}$ to $(0~1)$.
The map is defined in terms of the cumulative probability function which is in $(0~1)$

PS: sorry, this is duplicate to Jean-Marc answer. I was typing when he posted his answer.

JohnLeM
Topic Author
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Joined: September 16th, 2008, 7:15 pm

### Re: Are Artificial Intelligence methods (AKA Neural Networks) for PDEs about to rediscover the wheel ?

I am not familiar with the term "localization principle". Maybe that's my confusion. Is it a term from physics?

Is it possible for motivation to give an example (even 1d/2d) to show how equations (5) and (6) work?
I am not familiar too with this term, but it describes quite well what we wanted to do.

For one dimensional, there are plenty of example in my old paper.
For two dimensional, the CRAS paper contains one numerical example for Heston. Here is another example for log normal process.

JohnLeM
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Joined: September 16th, 2008, 7:15 pm

### Re: Are Artificial Intelligence methods (AKA Neural Networks) for PDEs about to rediscover the wheel ?

The change of variable is for the probabilty (transition) density (as he is solving the Fokker Planck equation).
For example for the GMB process $dX = X r dt + X \sigma dW$, we have $X \in R_{+}$.
We look for a map to localise from $R_{+}$ to $(0~1)$.
The map is defined in terms of the cumulative probability function which is in $(0~1)$

PS: sorry, this is duplicate to Jean-Marc answer. I was typing when he posted his answer.
No problem Farid, it is better to have several point of views. It is almost what you are saying : in one-dimension, the map is a quantile, and its inverse map is a cumulative. In several dimensions, the correct generalization of a quantile is a map $S = \nabla h : (0,1)^D \mapsto R^D$, h convex (Brenier again).

JohnLeM
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### Re: Are Artificial Intelligence methods (AKA Neural Networks) for PDEs about to rediscover the wheel ?

The article alone is not enough to implement the algorithm (in my point of view). As usual, you have to look to the references.
Farid, as I wrote, I think this is going to be really difficult to implement alone. A suggestion could be : find a public research group that could help to release a version for academical purposes, so that anybody could toy with it. Any suggestions ?

Cuchulainn
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### Re: Are Artificial Intelligence methods (AKA Neural Networks) for PDEs about to rediscover the wheel ?

Any suggestions ?
Yes


text:
## General Chemistry 10th Edition

$1.19\times 10^{-10}$
We know that $H_3O^+=\frac{K_w}{OH^-}$ We plug in the known values to obtain: $H_3O^+=\frac{1.00\times 10^{-14}}{8.4\times 10^{-5}}=1.19\times 10^{-10}$


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Tag Info

5

I'm not sure how deep of a question you are asking. The dog that did not bark is from a Sherlock Holmes murder mystery. The dog at the house did not bark at the intruder, so Holmes believed the dog knew the intruder. Therefore, the lack of evidence like barking, was itself the evidence. In the Chochrane paper, the introduction mentions that the lack of ...

3

Since you're asking on a quant finance forum, the mathematical approach would be Decide on a model that the stock price follows, and Compute the expected value of the price, conditional on the most recent price. A famous model, made ubiquitous by Black, Scholes and Merton, is a geometric Brownian motion. Under this model, the stock price $S_T$ at time ...

3

Why not just use Geometric Mean Returns? Each time you buy/sell an ETF calculate the holding period return as a percentage and plug into the formula. The answer is a percentage that you can use to calculate the approximate money appreciation (or loss) against your "fixed notional"

2

Actually prices dont make sense as they are correlated with previous samples (prices), returns are not. Better will be difference between prices, but then you dont have reference point and comparability between assets, so eventually you need returns. At the end that is what you are interested in I think as profit is usually measured in return.

1

Whenever you are looking to estimate total return, you would use adjusted closing prices. If you are strictly looking for the future stock price, you would use unadjusted closing price. I assume, though, that you are looking to predict the value of holding a stock during a given period, so you would want to use adjusted prices. The only time I've used actual ...

1

The average would be called the mid-price, not the best in my opinion, but that depends on your modeling. Another strategy is to weight the bid and offer prices according to size, also called the micro-price or bid-offer weighted price. This has the advantage of moving your calculated price closer to where it is traded as volume is depleted from whatever ...

1

You don't have to have a large percentage of the participants able to take physical delivery, just a small percentage. For every contract there are plenty of people who will take it in to inventory or sell out of it if the spread between the spot and future is wide enough. So there's a band that's created. For example, if Henry Hub Gas is 2.9 and the ...

1

Although, I think this question is a bit off-topic for a Quant S.E., I'll try to answer it with my background sitting at a commodities desk at a bank. Since most of the future contracts are never settled physically (therefore, no actual trades occour), why would the rest (actual buyers and sellers) even agree to participate in this speculative ...

1

You are mostly right, I don't really get what you don't understand. The answer in the book is quite clear, but let me put it that way : Selling a put and buying a call on the same underlying $S$ with same maturity and same stike $K$ is always equivalent to a long position in a forward contract on $S$ with delivery price $K$. The easiest way to see that is ...

1

To start, it very much depends on your outlook. Do you believe that the future price movement is independent of previous price movement? If so you probably wouldn't look for trending or consolidating markets (it would be entirely random). On the other hand, maybe you have a fractal view of the market (search for fractional Brownian motion, regime switching ...

1

Read paper written by Malkiel, "The Efficient Market Hypothesis and Its Critics". It is wonderful paper on EMH. http://eml.berkeley.edu/~craine/EconH195/Fall_14/webpage/Malkiel_Efficient%20Mkts.pdf It will help you to gain conceptual clarity in EMH.

1

Market is efficient when all available public information gets priced-in relatively fast by market participants. This yields the fair price. Efficiency depends on the speed of the information dissemination. Equilibrium is a balance between supply and demand, which can be skewed by short term liquidity issues. So market can be efficient and not in equilibrium ...

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# Tag Info

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An oracle (at least in this context) is simply an operation that has some property that you don't know, and are trying to find out. The term "black box" is used equivalently, to convey the idea that it's just a box that you can't see inside, and hence you don't know what it's doing. All you know is that you can supply inputs and receive outputs. In the ...

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There are two questions here. The first asks how you might actually implement this in code, and the second asks what's the point if you know which oracle you're passing in. Implementation Probably the best way is to create a function IsBlackBoxConstant which takes the oracle as input, then runs the Deutsch Oracle program to determine whether it is constant....

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The simplest solution is to use an ancilla in the $|+\rangle$ state. Swap that ancilla for the oracle's output qubit, conditioned on the control qubit being false, before and after applying the oracle. Since toggling the $|+\rangle$ state has no effect, this deactivates the oracle when the control is set. Here's this technique applied to a simple comparison ...

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Overview To recap the one-bit Deutsch Oracle problem, there are four possible oracle functions: constant-0, constant-1, identity, and negation. The task is to determine whether the oracle function is constant (constant-0 & constant-1) or variable/balanced (identity & negation). You can do this using phases as follows: Rewrite the oracle function as ...

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You can implement a controlled version of any gate in Q# using Controlled functor; for the oracles, you'll use something like Controlled X(controlQubits, targetQubit), where controlQubits is an array of qubits that have to be in 1 state for the X gate to be applied and targetQubit is the qubit to which the gate is applied. Q# also has a neat library ...

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Let's say that the first bit of $s$ $s_0=1$ (the argument will be exactly the same for any bit, just for convenience). You can split the space of inputs $x \in \{0,1\}^n$ in two halves: one half where $x_0 = 0$ and the other half where $x_0 = 1$. For each bitstring $x$ from the first half you'll have a bitstring $\tilde{x}$ from the second half which will ...

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As a bare minimum, you would need access to a controlled version of your oracle. This cannot be created from the oracle itself (I'm sure there's already an SE question about this part, but cannot immediately lay my hands on it). A typical construction would allow you to create (Hadamard - controlled oracle - Hadamard) would create an output $$\cos\frac{f(x)... 4 Although popular explanations of Grover's algorithm talk about searching over a list, in actuality you use it to search over possible inputs 0..N-1 to a function. The cost of the algorithm is O(\sqrt{N} \cdot F) where N is the number of inputs you want to search over and F is the cost of evaluating the function. If you want that function to search over ... 4 Consider the last part of your question: Why can't we have a query of the form$$O_{x}|i \rangle = (-1)^{x_{i}} |i \rangle$$This transformation is certainly unitary. Andrew Childs notes [...] that we can't distinguish between x and \bar{x} (bitwise complement of x) if we exclude the |b \rangle register. I don't see why this should be the case.... 3 This is not so straightforward, I suspect. The issue is being able to distinguish between the constant case (e.g. every input gives output 0) and the case where only one input returns 1, and all others return 0. To distinguish these cases is essentially a Grover Search (the return of 1 being essentially a marked item that you want to search for the existence ... 3 One problem is that you are resetting the \left|z\right\rangle register after applying the Controlled X(z, y) operation. Right before you reset, your \left|z\right\rangle register is entangled with the other two registers, such that resetting in that way collapses any superposition on the \left|x\right\rangle \left|y\right\rangle registers. While that'... 3 It seems to me that the diagonalisation arguments that can be used are only slightly different from a standard one, e.g. such as can be found in these lecture notes about the Baker–Gill–Solovay Theorem (i.e., that there are oracles A for which \mathsf P^A = \mathsf{NP}^A and also oracles A for which \mathsf P^A \ne \mathsf{NP}^A... 3 You could do something like: assume the most significant bit of s is 1. write a function that says "if the most significant bit of x is 0, return x. if the most significant bit of x is 1, return x\oplus s. This is easily implemented because you start by doing a transversal set of cNOT gates to copy x from the input register to the output ... 3 An application of an oracle does not return a value; rather, it modifies the state of the system in a non-collapsing way. The oracles are a bit similar to controlled gates in this respect (in fact, a lot of oracles rely on controlled gates for their implementation). Consider, for example, CNOT gate: it does not measure the control qubit and apply an X gate ... 3 There is no way to build the oracle in a way which would not defeat the point of Deutsch's algorithm - that's why it is an oracle-based algorithm. The only way would be if you would come up with an incredibly hard to compute function (this is, an incredibly long circuit) which would take one input bit x and give one output bit f(x) (but on the way could ... 2 I don't have an example for Deutsch's algorithm handy, but here and here are two tutorials which walk you through implementing the Deutsch-Jozsa algorithm and the oracles it uses in Q#. The idea for these two algorithms is the same: you have to provide the oracle to the algorithm as an operation implemented elsewhere. This way the algorithm doesn't know ... 2 You have two examples on the IBM Q Experience page about the algorithm. They show an example of a function. This could inspire you for your simulations I hope. 2 Since this question seems to be in the context of Grover's search, I will explain using what happens in Grover's search, however, this is way more general. The oracle function f itself can be thought of a specification as to what should happen given various basis states (in fact, just classical bit strings to classical bit strings). For example, f can ... 2 Let f be your favorite \mathrm{SAT} problem. For example, one that I like is: Are there integers x_1, x_2, x_3, each -2^{50}\le x_1,x_2,x_3 \le 2^{50}, with x_1^3+x_2^3+x_3^3=42? Write f as a sequence of irreversible \mathsf{NAND} gates, etc., and convert them to a sequence of reversible \mathsf{CCNOT} gates, etc. to determine a unitary ... 2 You are given a quantum circuit for U compiled into the H/CNOT/T gateset. Derive a controlled version of U by adding a control qubit q, replacing every H with a controlled H, every CNOT with a CCNOT, and every T with a controlled T. In all cases the new control goes on the q. Recompile the modified gates down into the H/CNOT/T gate set. Prepend and ... 2 This picture can be confusing, since it makes some assumptions that are not clear when you quickly scan through the text. The picture of the oracle U_f describes its effect on the basis states that go into it, not on arbitrary superposition states. If both x and y are basis states, the oracle will behave exactly as the rectangle says, but the ... 2 The first example you're looking at is closer to the process of building an oracle for practical applications than the second one. The oracle circuit has to encode the Boolean formula for a specific instance of the problem you're trying to solve, and not the known solutions themselves. When you encode the oracle, you don't know the answer to the problem, but ... 2 Suppose we have two quantum circuits, the first computes (or at least approximates) the classical \sqrt{\cdot} function$$S|x\rangle|0\rangle=|x\rangle |\sqrt{x}\rangle,$$while the second circuit A computes (again could probably just approximate) the \arccos(\cdot) function$$A|x\rangle|0\rangle=|x\rangle |\arccos(x)\rangle. Lastly, suppose we have ...

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If your function is defined in a way that requires you to iterate over all possible inputs to implement the oracle, you clearly lose the advantage of using Grover's algorithm. This question goes into detail about using Grover's search for unordered database search, which is pretty much your example. Instead, the function should be defined in a way that ...

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In your card example, since you've created the encoding, you've kind of already made Grover's algorithm useless. That is, you already know that 00 is the spade of Ace, so you don't need to search. Grover's algorithm would only help find the spade of Ace if you didn't already know which bitstring encoded it. As an example: Suppose you shuffle the cards and ...

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Given controlled access to the phase oracle, this is possible with surprisingly small overhead by avoiding phase estimation altogether. The technique you are after relies on applying "quantum singular value transformations" to objects that are known as "block encodings", and it was invented by Gilyén et al. in 2018. The idea was originally introduced in this ...

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It's hard to know what are you referring to without context, but a "quantum oracle" is just a type of (generally unitary) gate. As such, it does not provide information about the system, and neither it induces collapse. You can find more details about oracles in this other question (and links therein). You cannot know the state of the system without ...

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I think that ahelwer's answer touches on some the ways that we think about the complexity of algorithms. However — given that we don't literally have "oracles" in the real world which we wish to query, you might wonder why we would worry about query complexity, or the idea of oracles at all. I will try to give some perspective on this, and in ...

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If we express the action of $O_x$ on the basis $\mid i, \pm \rangle$ instead of $\mid i , b \rangle$ \begin{eqnarray*} O_x \mid i , + \rangle = \mid i , + \rangle\\ O_x \mid i , - \rangle = (-1)^{x_i} \mid i , - \rangle\\ \end{eqnarray*} Because we say auxiliary qubits must be started and ended with $0$, \begin{eqnarray*} O_x'' &=& (1 \otimes S_x ...

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# How do you determine the value of k if the remainder is 3 given (x^3+kx^2+x+5) div(x+2)?

Feb 13, 2017

The answer is $k = 2$

#### Explanation:

Let $f \left(x\right) = {x}^{3} + k {x}^{2} + x + 5$

When $f \left(x\right)$ is divided by $\left(x + 2\right)$ , the remainder is $= 3$

This is the remainder theorem

so,

$f \left(- 2\right) = - 8 + 4 k - 2 + 5 = 3$

$4 k = 3 + 5 = 8$

$k = 2$