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# Far-field probing of leaky topological states in all-dielectric metasurfaces

• Nature Communicationsvolume 9, Article number: 909 (2018)
• doi:10.1038/s41467-018-03330-9
Accepted:
Published online:

## Abstract

Topological phase transitions in condensed matter systems give rise to exotic states of matter such as topological insulators, superconductors, and superfluids. Photonic topological systems open a whole new realm of research and technological opportunities, exhibiting a number of important distinctions from their condensed matter counterparts. Photonic modes can leak into free space, which makes it possible to probe topological photonic phases by spectroscopic means via Fano resonances. Based on this idea, we develop a technique to retrieve the topological properties of all-dielectric metasurfaces from the measured far-field scattering characteristics. Collected angle-resolved spectra provide the momentum-dependent frequencies and lifetimes of the photonic modes that enable the retrieval of the effective Hamiltonian and extraction of the topological invariant. Our results demonstrate how the topological states of open non-Hermitian systems can be explored via far-field measurements, thus paving a way to the design of metasurfaces with unique scattering characteristics controlled via topological effects.

## Introduction

Topological phase transitions in two-dimensional (2D) condensed matter systems have attracted an enormous interest1,2,3,4 crowned by the Nobel Prize in Physics in 2016. With the advent of photonic topological insulators5,6,7,8,9,10,11,12 the research domain expanded to include topological transitions for light13,14. However, most of the topological photonic systems considered until today have largely ignored the fact that photonic modes can couple to the continuum of free-space modes, and the exploration of the topological modes has been limited to the near-field properties. Only recently have topological phenomena in photonic systems associated with leaky states attracted attention in the context of non-Hermitian systems15,16, including Floquet systems17,18 and non-radiative modes in the continuum19.

While the leakage of photonic modes to the free-space continuum can be considered as a challenge, as it makes the system non-Hermitian, it may also offer an alternative route to explore the topology of photonic bands with far-field measurements, provided that the radiative decay of the topological modes is controllable and nondestructive. This is especially relevant in the context of metasurfaces—ultrathin photonic structures enabling a variety of novel optical devices, including flat lenses and wave-plates to control polarization and angular momentum of optical beams20,21,22,23,24,25,26. Endowing metasurfaces with topological properties may open a whole new realm of opportunities in the field of light scattering and wavefront control, as the synthetic gauge field for light have been shown to controllably affect all degrees of freedom associated with electromagnetic radiation24,27. Provided the metasurface is engineered to sustain topological photonic modes above the light line, they may couple to the radiative continuum, giving rise to unique scattering features, such as Fano resonances discriminating light by its polarization, angular momentum, or any other synthetic degree of freedom.

In recent years, it has become a traditional approach to look at the topological properties of photonic systems through the prism of the bulk-boundary correspondence principle28, focusing on investigation of their edge characteristics, including chiral and helical edge states. Although this strategy to explore topological properties6,9,29,30 is quite appealing due to the distinctive features such as backscattering-immune propagation, it can divert us from new and potentially beneficial properties of bulk modes stemming from their topological nature.

To explore these ideas, we design and fabricate a metasurface that supports radiative photonic modes in the near-infrared spectral range and exhibits a topological transition. As the parameters of the metasurface are tuned, the coupling of topological modes to the radiation continuum enables the observation of a topological transition, which is accompanied by the inversion of bright and dark modes, referred to as band crossing. Angle-resolved spectroscopy allows for the direct extraction of spectral positions, intensity, and bandwidth of the peaks corresponding to the excitation of bulk states of the metasurface from the transmission and reflection spectra. As demonstrated below, these parameters provide valuable information about the nature of the eigenmodes supported by the structure, enabling the univocal retrieval of topological invariants from far-field measurements.

## Results

### Topological metasurface

The metasurface consists of cylindrical Si pillars arranged into hexagon clusters with edge length R, placed at the sites of a triangular lattice with period a (Fig. 1a, b). The topological properties of its infinite 2D analog possessing C6v symmetry have been the subject of several recent studies31,32,33,34,35,36,37. For a/R = 3, this system represents a conventional honeycomb lattice (with the unit cell containing two cylinders), which exhibits Dirac cones centered at K and K′ points of the Brillouin zone. However, if the lattice symmetry is reduced by clustering six neighboring pillars so that $a∕R≠3$ (distorted lattice), the size of the unit cell increases leading to the reshaping of the first Brillouin zone as shown in Fig. 1c. As a consequence, the Dirac points appear in the vicinity of the Γ point due to the bands folding in the distorted lattice as shown in Fig. 1d. Additionally, the interaction between the valleys of the former honeycomb lattice caused by such symmetry reduction leads to the opening of photonic bandgaps (Fig. 1e, g). This interaction can be viewed as synthetic spin–orbit coupling between two pseudo-spin states corresponding to the former valley degree of freedom of the unperturbed honeycomb lattice. Previous studies suggested that the shrunken structure with a/R > 3 is topologically trivial, whereas the expanded system with a/R < 3 is topologically nontrivial31 (see also Supplementary Note 1), which, as shown below, remains true for the case of the open system. The important distinction, however, is that in the case of the open metasurface, the folded modes appear within the light cone, and, therefore, are radiatively coupled to the continuum of free space, which enables their far-field characterization38 by spectroscopic methods (Fig. 1f, h).

To perform spectroscopy measurements of metasurfaces with two topologically distinct geometries, we fabricated two sets of samples (Fig. 1a, b) with the same lattice constant a = 750 nm, radius of the pillars r = 75 nm, and height of the pillars h = 1000 nm. The sizes of the clusters R shown in Fig. 1a, b were chosen to be a/R = 3.15 and a/R = 2.85 for shrunken and expanded structures, respectively.

### Effective Hamiltonian

Although the topological properties of the perturbed honeycomb lattice have been previously explored under the tight binding approximation31, here we take a different approach, fully considering the electromagnetic nature of the system. Correspondingly, the effective Hamiltonian near the Γ point of the Brillouin zone is obtained directly from Maxwell’s equations by the plane wave expansion method8,12 (Supplementary Note 2). To describe the properties of photonic bands of bulk modes in the 2D photonic crystal in the vicinity of the Γ point, we construct an effective 4 × 4 Hamiltonian. Focusing on vertical dipolar modes with E z component of the electric field directed along the axis of Si rods, we find the results in compliance with the tight binding model (Supplementary Note 1 and refs.31,32). The derived effective 4 × 4 Hamiltonian assumes the following form31:

$Ĥ= Ĥ - K ^ K ^ † Ĥ + ,$
(1)

where the 2 × 2 matrices $Ĥ ±$ and $K ^$ are

$Ĥ ± = μ ( k ) v ∓ k x - i k y v ∓ k x + i k y - μ ( k ) ,$
(2)
$K ^ = α k x + i k y 2 0 0 - α k x - i k y 2 ,$
(3)

μ(k) = μ + βk2, μ and β are the mass term and band parabolicity, respectively. This form of the Hamiltonian corresponds to the basis choice

$ψ = p - , d - , p + , d + T ,$
(4)

where, $p ± = p x ±i p y$ are the circularly polarized dipolar modes and $d ± = d x 2 - y 2 ±i d x y$ are the circularly polarized quadrupolar modes, which both originate from modes supported by an isolated meta-molecule (the hexamer of 6 dielectric rods).

The four eigenstates of the Hamiltonian Eq. (1) exhibit a pairwise degeneracy at the Γ point k = 0 (Supplementary Note 1 and 2), with eigenvalues equal to μ (dipolar modes) and −μ (quadrupolar modes). Numerically computed field distribution of the modes (shown in Fig. 1i, j) demonstrate in-plane and out-of-plane variation, respectively, and confirm their dipolar and quadrupolar form and strong confinement to the pillars. Transition from shrunken to expanded design is accompanied by the band inversion for dipolar and quadrupolar eigenstates, and by the reversal of the sign of the mass term μ. In addition, the degeneracy between left- and right-handed circularly polarized modes is removed for nonzero values of k due to the term being proportional to α. However, for the topological invariant calculation, this term is inessential and can be dropped31. As a result of this approximation, the Hamiltonian splits into two independent 2 × 2 blocks, and the effective pseudo-spin can be introduced. A straightforward calculation39 of spin Chern number (Supplementary Note 1) yields

$C= 1 2 sgn μ - sgn β ,$
(5)

thus, to evaluate the topological invariant one has to extract the mass term μ and the parabolicity of the bands β from the experimentally measured spectra.

### Retrieval of topological state via far-field spectroscopy

The modes of interest are located above the light cone, and, for the case of a metasurface of a finite thickness, they couple to the radiation continuum of an open background medium38. As a result, the modes acquire a finite lifetime and the corresponding eigenfrequencies become complex-valued. Thus, the silicon pillars constituting the metasurface effectively act as laterally coupled cavities, and the leaky modes of the structure can be excited by the polarization currents induced at the interface of the metasurface by the incident fields. However, taking into consideration the symmetry of the modes, normally incident light can only couple to the dipolar modes, which thus represent super-radiant modes, whereas coupling to the quadrupolar modes is suppressed because of the symmetry mismatch with the incident field, and they represent sub-radiant modes. The quadrupolar modes can still be excited indirectly through coupling to the dipolar modes at oblique incidence, because of the hybridization of dipolar and quadrupolar modes away from the Γ point. This mechanism is referred to as an extrinsic coupling to dark modes in Fano-resonant systems and is caused by the symmetry reduction due to the finite value of the in-plane wave vector components.

Of special interest are the effects of the topological transition on the radiative coupling of the modes to the continuum, which should follow the band inversion described above, giving rise to switching of the bright (dipolar) and dark (quadrupolar) modes, and, thus, enabling control over the far-field scattering properties of the metasurface.

To support our claims, the complex photonic band structures for both the shrunken and expanded metasurfaces were calculated in Comsol Multiphysics with radiative decay fully considered. The results are shown in Fig. 1e, g. Indeed, the dipolar and quadrupolar modes at the Γ point appear to be bright and dark, respectively, as reflected by the radiative quality factors (essentially, the lifetimes of the modes). Even at oblique incidence, when modes are hybridized, the respective bands may have radiative quality factors that differ by orders of magnitude. The numerically calculated extinction spectra at oblique incidence are shown in Fig. 1f, h, and confirm excitation of the two eigenmodes, which manifest as two peaks. The predominantly dipolar and quadrupolar modes can be easily discriminated by their distinct bandwidth and the amplitude of the corresponding peaks. This allows one to establish the symmetry of the mode without the need to look into the near-field pattern and to extract the mass term μ needed for evaluation of the topological invariant.

To formalize our description in the context of the topological properties of leaky modes, we develop a unified approach utilizing the temporal coupled mode theory (CMT) along with the electromagnetic effective Hamiltonian description. We write the CMT equations for the amplitudes of the right-handed and left-handed circularly polarized leaky modes coupled to the external source in the following block-diagonal form:

$-iε ψ ± =-i Ĥ ± ψ ± +χ E in 0 - γ 0 + γ r 0 0 γ 0 ψ ± ,6$

where $ψ ± = p ± , d ± T$ is the mode composed of p (dipole) and d (quadrupole) eigenmodes of the system. The first term of Eq. (6) describes the evolution of the coupled modes in a closed system, $χ$ describes the coupling strength of the system to the external field $E in = E 0 ∕ 2$ (E0 is the field amplitude), γ0 represents non-radiative losses in the structure, and $γ r = χ 2 ∕2$ describes the radiative losses. The use of this form of dynamic equation can be also justified by applying electromagnetic perturbation theory for an open system as shown in Supplementary Note 4.

Once Eq. (6) is solved, one can obtain expressions for the transmission (and reflection) coefficients:

$t ± = t 0 1 - χ ψ ± ( p ) ∕ E in ,$
(7)

where ψ±(p) is the first (dipolar) component of the two-component wave function $ψ ±$. Since the results are the same for left-handed and right-handed circular polarizations, from now on we omit the ± subscript. Plugging the Hamiltonian into Eq. (7), we derive the expression for the extinction in the form

$R ̃ ≡1- t 2 t 0 2 =2 γ 0 χ 2 [ ε + μ ( k ) ] 2 + v 2 k 2 + γ 0 2 μ ( k ) 2 - ε 2 + v 2 k 2 + γ 0 γ 0 + χ 2 ∕ 2 2 + 2 γ 0 ε + χ 2 ∕ 2 ( μ ( k ) + ε ) 2 .$
(8)

Equation (8) suggests in particular that two peaks in the $R ̃$ spectrum can be observed due to excitation of dipolar and quadrupolar eigenmodes. Thus, the frequency of the most pronounced peak at incidence close to normal is around f0 + μ, whereas the frequency of the less intense peak is found around f0 − μ, where f0 is the mid-gap frequency.

Overall, there are six parameters in the effective model that control the entire dispersion and topological properties of the far-field response: μ, $γ r = χ 2 ∕2$, γ0, v, and β (see Eqs. (2) and (6)). The numerical values of parameters comprising the effective Hamiltonian can be obtained by fitting the experimental data. Next, by using the developed technique, we analyze the experimental data for both shrunken (a/R = 3.15) and expanded (a/R = 2.85) structures and extract their topological characteristics.

To confirm our theoretical predictions, a set of shrunk and expanded silicon samples on top of sapphire (Al2O3) substrates was fabricated and tested experimentally. The structures were illuminated using the halogen source Ocean Optics HL-2000-LL. The transmission was measured for both structures by the spectrometer Ocean Optics NIR Quest NQ 512–2.2 in the range of the wavelengths from 896 to 2142 nm corresponding to the frequency range (1.40 ÷ 3.35) × 1014 Hz. The measured transmittance was normalized to the one of a sapphire substrate. Figure 2 shows the color map of the quantity $R ̃ =1-T∕ T 0$ as a function of the wavelength and the incidence angle for both the shrunken and expanded structures.

In the wavelength range from 1600 to 1900 nm, two peaks are clearly observed. The detailed spectroscopy results for the whole studied spectral range are provided in Supplementary Note 3. In agreement with our numerical calculations, the experimental data clearly show that the most intense and broad peak for the shrunken structure is the low-frequency one, whereas for the expanded structure the high-frequency peak is more pronounced. Thus, the shrunken structure is characterized by a negative effective mass μ, while the expanded structure is described by a positive effective mass μ.

Next, we applied the above analytical model to fit the measured data with Eqs. (6) and (8), as detailed in the Supplementary Note 3. The results of the simplest fitting algorithm for the two particular angles of incidence, 0° and 16°, are illustrated in Fig. 3 for the cases of both shrunken and expanded structures. The extracted parameters of the effective Hamiltonian are listed in Table 1. Importantly, these numbers appear to be nearly independent of the fitting algorithm (see details in Supplementary Note 3). From the measured transmittance spectra, we recover again that the shrunken structure is topologically trivial, since the effective mass μ and parabolicity parameter β have the same sign, yielding zero spin Chern number. On the contrary, the expanded structure appears to be topologically nontrivial, due to the opposite signs of μ and β, yielding a spin Chern number of 1. It is important to mention that for non-Hermitian systems (including the one considered here) the definition of topological invariant has been a subject of active discussion15,16,17,18,39. In Supplementary Note 7, following ref. 39, we show that the topological invariant remains unaffected by the presence of loss for the case of the metasurface studied here.

Interestingly, the topological properties have been directly extracted previously from the bulk spectrum for 1D systems only40. A different approach based on the edge properties was taken by Mittal et al.41, who introduced a unit quantum flux at the edge and observed a shift in the position of the edge spectrum. Our results prove for the first time that it is possible to retrieve the topological properties of an open 2D system via far-field measurements in a robust and direct way.

### Far-field and near-field observation of edge states

In order to confirm that the structure under study indeed supports topological edge modes, we have performed additional experimental studies of the domain wall formed between the two topologically distinct structures (shrunken and expanded) and further visualized edge modes with the far-field and near-field experimental techniques. The far-field experimental studies have been performed on a sample consisting of an array of domain walls formed by repetitive stitching of 12 unit cells of shrunken and expanded domains along a zig-zag cut of the lattice. The schematic of the experimental setup is shown in Fig. 4a with an SEM image of the sample near the domain wall in Fig. 4b. The results of numerical calculations performed within the use of the CMT are presented in Fig. 4c and reveal the presence of the edge states within the bandgap of bulk modes. The examination of numerically computed field profiles of the edge modes overlaid with the SEM image confirms their localization at the domain wall as shown in false-color plot in Fig. 4b. The experimental far-field measurements of sample’s reflectance are presented in Fig. 4b and confirm the presence of edge states within the topological bandgap. Interestingly, despite the presence of the additional scattering channels due to diffraction, the modes have relatively narrow bandwidth and they can be easily distinguished from the bulk modes by their distinctive linear (1D Dirac-like) dispersion at small tangential wavenumbers. Thus, the leakage of the edge states appears to be strong enough so that they can be excited and measured in far-field region, yet, the non-Hermiticity it introduces is not strong enough to destroy the topological state of the system. Thus, the presence of the edge states in our system serves as another piece of evidence of the topological properties of the structure despite its open nature.

Additional proof-of-concept experimental near-field studies of the edge modes were conducted in the microwave spectral range. Note that, due to the dielectric nature of the structure and scalability of Maxwell’s equations, the obtained results can be directly transferred to the structures operating in the optical domain.

We considered a metasurface consisting of dielectric rods arranged into the lattice with the permittivity of the rods ε = 10, lattice period a = 37.5 mm, radius and height of dielectric rods are r = 1.5 mm and h = 10 mm, respectively. The cluster size R (Fig. 1a, b) was chosen such that a/R = 3.33 and a/R = 2.73 for shrunken and expanded structures, respectively. For these parameters, the bandgap is characterized by the mid-gap frequency of 6.79 GHz with a total bandwidth of 310 MHz.

The experimental sample was constructed by embedding commercially available dielectric powder (Eccostock HiK with the permittivity ε = 10 and loss tangent tan δ = 0.0007) (Ceramic powder datasheet https://www.eccosorb.com/products-eccostock-hik-powder.htm) into the machine-processed substrate made from Styrofoam material (permittivity of 1.15 and negligible losses) in which the holes were drilled (Fig. 5a). All the measurements were performed in an anechoic chamber using an automatic mechanical near-field scanning setup and an electric field probe connected to the receiving port of the vector network analyzer (Agilent E8362C). The probe was oriented normally with respect to the interface of the structure. The near-field was scanned at 2 mm distance from the back interface of the metasurface to avoid a direct contact between the probe and the sample. To excite the edge state, the subwavelength dipole source whose polarization could be adjusted from circular to linear was placed next to the domain wall in the center of the expanded unit cell. Figure 5b shows the results of full wave simulations of the field distribution of the edge state. The corresponding experimental map presented in Fig. 5c clearly shows that the mode excited inside the bandgap at frequency f = 6.84 GHz is indeed localized to the domain wall and represents the edge state. To confirm the topological properties of the edge states, we performed near-field excitation both by linearly and circularly polarized dipole fields. As can be seen from Fig. 5b, c, the linearly polarized dipole leads to bi-directional excitation of the edge state. In contrast, the circularly polarization excitation leads to one-way propagation of the edge state evidencing the locking of the polarization and the propagation direction, thus, confirming the topological nature of the corresponding electromagnetic mode42,43.

These results prove independently that the metasurface considered here exhibits a topological transition when its design is changed from shrunken to expanded structure, thus validating the introduced far-field measurement technique for the detection of a topological transition in open photonic systems such as metasurfaces.

## Discussion

In this paper, we have introduced the concept of topological metasurfaces and demonstrated that their scattering characteristics, radiative quality factors of the modes in particular, can be controlled by synthetic gauge fields. The developed formalism allows us to relate topological properties and the far-field scattering, thus enabling the extraction of the effective Hamiltonian and the topological invariant (pseudo-spin Chern number) of the metasurface from the measured transmission/reflection spectra. Coupling of photonic modes of the structure to the free-space modes allows for the direct probing and visualization of the topological phase transitions in the open photonic system. We believe that our results open a new avenue for designing metasurfaces with desirable scattering properties controllable via their topological characteristics. Provided the properties of metasurfaces are designed to stem from edge states, their response may exhibit topological robustness to continuous modifications and disorder, which can be of immense value for practical applications.

## Methods

### Modeling and calculations

To derive the effective Hamiltonian and topological properties of the photonic crystal, we use the tight binding approach (Supplementary Note 1), as well as the plane wave expansion method (Supplementary Note 2). To calculate the far-field scattering properties of the structure, we employ the CMT (Supplementary Note 3). Perturbative analysis of the radiative losses for the metasurface is performed by the guided mode expansion method (Supplementary Note 4). The calculation of scattering from the topological meta-grating based on the CMT is detailed in Supplementary Note 6. The role of non-Hermiticity and topological invariants in the presence of radiative losses are discussed in Supplementary Note 7.

### Sample fabrication

A scheme of the fabrication procedure is outlined in Supplementary Note 5 and Supplementary Fig. 6.

### Data availability

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

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

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## Acknowledgements

The work was supported by the National Science Foundation grants CMMI-1537294 and EFRI-1641069. Research carried out in part at the Center for Functional Nanomaterials, Brookhaven National Laboratory, which is supported by the U.S. Department of Energy, Office of Basic Energy Sciences, under Contract No. DE-SC0012704. The theoretical models were developed with the support of the Russian Science Foundation (Grant No. 16-19-10538).

## Author information

### Author notes

1. These authors contributed equally: Maxim A. Gorlach and Xiang Ni.

### Affiliations

1. #### The Department of Electrical Engineering, Grove School of Engineering, City College of the City University of New York, NY, 10031, USA

• Maxim A. Gorlach
• , Xiang Ni
• , Daria A. Smirnova
• , Dmitry Korobkin
• , Andrea Alù
•  & Alexander B. Khanikaev
2. #### ITMO University, Saint Petersburg, 197101, Russia

• Maxim A. Gorlach
• , Dmitry Zhirihin
• , Alexey P. Slobozhanyuk
• , Pavel A. Belov
•  & Alexander B. Khanikaev
3. #### The Graduate Center of the City University of New York, NY, 10016, USA

• Xiang Ni
•  & Alexander B. Khanikaev

• Andrea Alù

• Andrea Alù

### Competing interests

The authors declare no competing interests.

### Corresponding authors

Correspondence to Andrea Alù or Alexander B. Khanikaev.


text:
FEM, CFD, CAE, and all that.

## Tag: algorithm

### POD snippet in R

I’ve been working on proper orthogonal decomposition (POD) for a while. Though it’s nice to have packages such as modred out there, I cannot help myself writing a snippet using R, since lots of my previosu data processing was in that language. Without all the theory chitchat  like K-L transformation, from the the implementation-on-discrete-data perspective the (snapshot) POD method boils down to just singular value decomposition (SVD), and it’s out of box in R, which makes the task mostly straightforward. That’s why a POD function in R essentially calls svd in R (and it further calls LAPACK routines DGESDD and ZGESVD.):

pod <- function(arr, nmod=dim(arr)[1], nv=nmod){
pod <- svd(arr, nmod, nv)
return(pod)
}

Here “nmod” is the number of POD modes to be retrieved. Though it’s default to be the length of each snapshot, it’s rarely necessary this large. Mostly of the time a reasonable correlated set of snapshots requires less than 10 modes to obtain most of the system energy.

With pod of data “arr” obtained, we can do several thing immediately. We  can obtain the modes by retrieving the right singular vector:

pod_u<- function(pod, nnod){
u <- matrix(unlist(pod["u"]), nrow = nnod )
return(u)
}

or get the singular values:

pod_sv<- function(pod){
sv <- as.vector(unlist(pod["d"]))
return(sv)
}

or the coefficients of the pod modes for the approximation:

pod_coef<- function(pod, nshot, nmod){
v <- matrix(unlist(pod["v"]), nrow = nshot )
sv <- pod_sv(pod)[1:nmod]
coef <- diag(sv) %*% t(v)
return(coef)
}

and eventually the approximation itself

pod_approx<- function(pod, nshot, nmod, nnod){
u <- pod_u(pod, nnod)
coef <- pod_coef(pod, nshot, nmod)
approx <- u %*% coef
return(approx)
}

### Jacobi method demo

Code below is to solve the static version of the equation in previous FDM code demo, with B.C. changed as homogeneous. The Jacobi method is used to replace the LAPACK solver which gives inexact method. The exact solution is
$u(x)=-x^2/2+x/2$
Using initial guess as
$u_0(x)=\sin(2\pi x)+0.5\sin(10\pi x)+0.3\sin(20\pi x)+0.1\sin(40\pi x)$
one can see that most of the computing cost is spent on smoothing lowest mode.

#include <stdio.h>
#include <math.h>
#include <iostream>
#include <fstream>
//#include <vector>
//#include <sunperf.h>
using namespace std;
//typedef std::vector<float> fvector_s;

//Maximum of an array.
float max(float a[], int n) {
float v = a[0];
for (int i=1; i<n; i++) {
if (a[i] > v) {
v = a[i];
}
}
return v;
}

int main ()
{
int n=500;
float tmp, b[n], a_d[n], a_u[n-1], a_l[n-1],    \
tol=1.e-7, x0[n], x1[n], x2[n], error=1.;
float h=1./(n+1), pi=3.1416;
int count=0;

//initializing a
for(int i=0; i<n; ++i){
a_d[i]=2.0;
}
for(int i=0; i<n-1; i++){
a_l[i]=-1.;
a_u[i]=-1.;
}

//initializaing b[]
for(int i=0; i<n; ++i){
b[i]=h*h;
}

//initial guess
for(int i=0; i<n; ++i){
x0[i]=sin(2*pi*h*(i+1))+0.5*sin(10*pi*h*(i+1))+\
0.3*sin(20*pi*h*(i+1))+0.1*sin(40*pi*h*(i+1));
x1[i]=x0[i];
}

ofstream outfile;
outfile.open("jac.out");
do{
count+=1;
for(int i=0; i<n; ++i){
if(i==0)
x2[i]=(b[i]-a_u[i]*x1[i+1])/a_d[i];
else if(i==n-1)
x2[i]=(b[i]-a_l[i-1]*x1[i-1])/a_d[i];
else
x2[i]=(b[i]-a_u[i]*x1[i+1]-a_l[i-1]*x1[i-1])/a_d[i];
}
for(int i=0; i<n; i++){
x1[i]=x2[i]-x1[i];
}
error=max(x1, n)/max(x0, n);
for(int i=0; i<n; i++){
x1[i]=fabs(x2[i]-(-(i*h)*(i*h)/2+i*h/2));
}
outfile << count << "\t" << error << endl;
for(int i=0; i<n; ++i){
x1[i]=x2[i];
}
}while(error>tol);
cout << "Number of iterations:" << count << endl;
for(int i=0; i<n; i++){
outfile << count << "\t" << x2[i] << endl;
}

outfile.close();
return 0;
}

Result of Jacobi method at different iteration

### Alpha shape

Alpha shape is a generalized way determining the geometric objects by a set of points, of which convex hull is a special case. It can be applied to FEM for boundary recognition. Essentially this method identifies the boundary of a set of points by eliminating those triangles (generated by triangulation of the points) which can be put into empty circles whose diagram are defined by threshold alpha. It is intuitive for a more simple criteria: eliminating the triangles whose circumcircles contains no points in the defined set. Below is a simple demo by Mathematica which tries to identify the boundary of a subset of points within circle radius 5 from a set points randomly distributed. The red arc isolates the point set $P$  inside, the first step is to eliminate the triangles whose circumcirlces contains no points in $P$, the next step is to eliminate the triangles contain nodes other than those in $P$. More delicate results can be reached by tuning alpha circle instead of using simple circumcirlces.

### Bowyer/Watson algorithm: 3D

3D version implementation of Bowyer/Watson algorithm is almost the same as the 2D. Instead of eliminating lines that belong only to the triangles whose circumcirlce contains the newly added point, in 3D we eliminate the facets that belong only to the tetrahedra whose circumsphere contains the newly added point, then form new tetrahedra by appending new point to those facets’ nodes. Here is an example generated by Mathematica scripts I compiled. Below on the right is tetrahedronization of 200 random points, together with enclosing tetrahedron. On the left is he tetrahedrons generated by first two points.

### DistMesh

DistMesh is no doubt the most interesting way meshing a domain, simple and elegant. There is a MATHEMATICA version of the original Matlab code, but only in 2D. Base on that, I wrote a script for mesh test. Hopefully continue working on this one would bring some application to my research. Here is one simple case of generating a non-uniform mesh within a circle, whose distance funtion is the easist to define. At the end of termination, most of the ill-shaped triangles are corrected.

format: plain


text:
## Welcome to Serendeputy!

Serendeputy is your personal news assistant.

- learns what you like and don't like,
- lovingly compiles a list of news and blogs for you.

How it works.

What to do:
2. Click smileys and frownies
3. Find favorite topics and sources
4. See how much better your deputy is getting at finding you good stuff.

# Stats Stack Exchange

I have on question regarding standardized coefficients (beta) in linear models. I have already asked one question here. From the answers I assume that I should use R's scale() function on the dependent variable as well as on all independent variables...
From: Stats Stack Exchange | By: MagnusMetz | Tuesday, October 21, 2014
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I need to describe what the difference between two groups (patients and normal controls) consists of in terms of latent variables that I can describe within each group. For instance, given this PCA variable map: how do I compare this to with the same...
From: Stats Stack Exchange | By: Fredrik Karlsson | Wednesday, October 22, 2014
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I know that $Var(\theta)\geq 1/I(\theta)$ where $I(\theta)$ is Fisher information. Let take an example of natural exponential family with density $f(x)=\lambda\exp(-\lambda x)$. In this case we have: $-E[\frac{\partial^2 \log(f)}{\partial\lambda^2}]=\frac{1}{\lambda^2}$...
From: Stats Stack Exchange | By: Ahmed | Wednesday, October 22, 2014
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I'm trying to understand the theory of estimators. As I understand it now, if you have an r.v. $X$ and take $n$ i.i.d. samples then an estimator for $E[X^{2}]$ would be $\overline{X^{2}}$ since $E[\overline{X^{2}}] = E[X^{2}]$ (probably only true for...
From: Stats Stack Exchange | By: Addem | Wednesday, October 22, 2014
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I am working on research paper for diagnosis of cancer. List of Known prognostic factors Age of patient Size of tumor Grade of tumor Lymphnode involvement and list of Unknown factors which are to be assessed with prognosis by correlating with known prognostic...
From: Stats Stack Exchange | By: made_in_india | Wednesday, October 22, 2014
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I want to compare two profile likelihood curves and determine if they are significantly different from one another. For example are the following curves significantly different from one another: I realize I can find a 95% confidence interval for a value...
From: Stats Stack Exchange | By: Ben Haley | Tuesday, October 21, 2014
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Attempting to understanding a statistical concept which I'm positive is basic stats, but that I currently don't understand. Say that there's a one in ten million likelihood of an outcome happening during an event, that happens a given count of times,...
From: Stats Stack Exchange | By: blunders | Monday, October 20, 2014
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In one use of k-fold cross-validation for evaluating classifiers, one trains k models, each on n(k-1)/k examples, and tests each on n/k examples. The average accuracy on those k test sets of size n/k is used as an estimate of the accuracy of a classifier...
From: Stats Stack Exchange | By: DavidDLewis | Wednesday, October 22, 2014
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I want to find the dominant one among two theories which give predictions about the relationships between one dependent variable( Y ) and eight independent variables ( x1 , x2 ... x8 ). The predictions of the two theories are mutually exclusive , i.e....
From: Stats Stack Exchange | By: Sandip Sinha | Wednesday, October 22, 2014
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Suppose I have normally distributed data. For each element of the data I want to check how many SDs it is away from the mean. There might be an outlier in the data (likely only one, but might be also two or three) or not, but this outlier is basically...
From: Stats Stack Exchange | By: Oliver | Wednesday, October 22, 2014
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Good morning everyone, I have a question in regards to Statistically Significant Sample Sizes to a population. I am working with data in excel and wanting to put the formula for this into the workbook as well but have run into a roadblock of sorts. I...
From: Stats Stack Exchange | By: Filmore34 | Wednesday, October 22, 2014
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Let's say I am trying to figure out whether two classes can be differentiated. My methods may not be perfect, but I would like to know whether my features "mean" anything that may possibly be added to reinforce another system (for instance). I know that...
From: Stats Stack Exchange | By: ido | Wednesday, October 22, 2014
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Could you please answer some of my questiona regarding SVM-RFE (svm with recursive feature elimination). I am using SVM-RFE with linear kernel for the binary classification and feature selection problem. All feature are rescale with mean 0 and standard...
From: Stats Stack Exchange | By: TmN | Wednesday, October 22, 2014
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I know it might be trivial but does the density of daily values impact the forecast accuracy? For example, if a call center receives less than 50 calls for weekdays and less than 10 calls for weekend, is the forecast accuracy diminished compared with...
From: Stats Stack Exchange | By: user12 | Wednesday, October 22, 2014
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Suppose I have 1000 draws each of two random variables X and Y. If I wanted to sample the sum of these variables, I would simply calculate 1000 samples, i.e. $$S_{i}=X_{i}+Y_{i}, i=1,2,…,1000$$ And that would give me draws from the pdf of the sum...
From: Stats Stack Exchange | By: mzuba | Wednesday, October 22, 2014
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I have a dataset with approximately 4000 rows and 150 columns. I want to predict the values of a single column (= target). The data is on cities (demography, social, economic, ... indicators). A lot of these are highly correlated, so I want to do a PCA...
From: Stats Stack Exchange | By: chrmar | Wednesday, October 22, 2014
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I'm not well versed in statistics so I'm not sure if my question is worded exactly correctly but basically here's the problem I'm trying to solve: imagine you have two equal sized arrays of size n. Each array is filled with random numbers from 0 to 1....
From: Stats Stack Exchange | By: David Webb | Wednesday, October 22, 2014
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Suppose n people go to a fancy restaurant. Each person is wearing a hat and checks his/her hat at the door as he/she arrives. The hat-check attendant gets tipsy throughout the evening, forgetting which hat belongs to whom, and returns a random hat to...
From: Stats Stack Exchange | By: Zhe Huang | Wednesday, October 22, 2014
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I am trying to see differences in the feeding-rate of one bird species between big forest patches and small ones. I have several forest patches of both sizes, and three years of study. Some individuals have been recorded (to assess the feeding-rate)...
From: Stats Stack Exchange | By: Javiero84 | Wednesday, October 22, 2014
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I want to find the dominant among two theories. I have used a unique "t - test" as shown in the file below : https://docs.google.com/viewer?a=v&pid=sites&srcid=ZGVmYXVsdGRvbWFpbnxzYW5kaXBzaW5oYW5vd3xneDo1ZTU5ZWVkNWRmOTgzZDMx Kindly comment about...
From: Stats Stack Exchange | By: Sandip Sinha | Wednesday, October 22, 2014
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I feel this is a simple problem, yet I cannot seem to solve it. Any help is greatly appreciated.
From: Stats Stack Exchange | By: confused | Wednesday, October 22, 2014
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Let say we have a dataset, $\mathbf{X}$ of $m$ instances, and $n$ features, and a target scalar variable $\mathbf{y}$ ($m$ instances). Now I want to do a regression so, I try to fit a hyperplane $y = \mathbf{x} .\mathbf{w}$ + c. Note : $\mathbf{w}$...
From: Stats Stack Exchange | By: user76170 | Wednesday, October 22, 2014
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I am trying to do Logistic Regression in R. My data set contains more than 50 variables. Some of them are factor (qualitative variable) and others are independent variable(quantitive ). I would like to get the significance of the variables from their...
From: Stats Stack Exchange | By: Mohammad Saifullah | Wednesday, October 22, 2014
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Imagine a hypothetical scenario in which a ball is thrown along a straight line. During flight, the position is continually sampled; however, at some distance, the sampling fails and only noise is detected. This distance is unknown and variable. One...
From: Stats Stack Exchange | By: user59071 | Wednesday, October 22, 2014
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I am trying understand how to correctly build a mixed-effects logistic regression model in R. I believe my model is pretty simple and straight forward but I'm lacking in experience and uncertain I'm doing it correctly. Not being a statistician and struggling...
From: Stats Stack Exchange | By: Michael | Wednesday, October 22, 2014
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I need to estimate a panel model. I have run the "normal" fixed effects model using plm in R and also wfe. I also wanted to try pggls considering its tolerance of heteroskedasticity and autocorrelation. However, the results I am getting with pggls are...
From: Stats Stack Exchange | By: jayhawk | Wednesday, October 22, 2014
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In a mixed effects model the recommendation is to use a fixed effect to estimate a parameter if all possible levels are included (e.g., both males and females). It is further recommended to use a random effect to account for a variable if the levels...
From: Stats Stack Exchange | By: gung | Tuesday, October 21, 2014
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I am aware of some nice examples of pairs of correlated random variables which are marginally normal but not jointly normal. See this answer by Dilip Sarwate, and this one by Cardinal. I am also aware of an example of two normal random variables whose...
From: Stats Stack Exchange | By: user65257 | Tuesday, October 21, 2014
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Say I take 500 bootstraps of a population and calculate 95% confidence intervals (CIs) for each sample. I would expect that 95% of the bootstrap sample CIs to contain the true population mean. However, I'm then asked what the probability is of 100% of...
From: Stats Stack Exchange | By: Makaira Murakami | Tuesday, October 21, 2014
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Given input vector $x$, let the maximum of $x$ occur at index $i$ in the input vector. I am trying to quantify the peakedness of this maximum and do that I have thought of determining following quantities. I am interested in determining numbers $l$ and...
From: Stats Stack Exchange | By: morpheus | Tuesday, October 21, 2014
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Above are three plots of the Linear model I am trying to analyze. The first one is a basic plot of the linear data: LinearModel = read.csv(file= "C:/Users/Nikhil/Documents/LinearModelCase2.csv", header=TRUE, sep=",") plot(LinearModel$X,LinearModel$LinearModel)...
From: Stats Stack Exchange | By: Nikhil Agrawal | Tuesday, October 21, 2014
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I have two series of trading profit results. I use the geometric mean to calculate the average in percent (CAGR). I would like to divide it by the standard deviation by combining the two series, but I´m having trouble calculating the combined standard...
From: Stats Stack Exchange | By: Macce | Tuesday, October 21, 2014
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The 21,000 estimate for Oct. was certainly not via quad or power regression. I wonder how they got that number? http://www.telegraph.co.uk/news/worldnews/ebola/11121045/Graphic-how-Ebola-cases-have-grown-since-March.html...
From: Stats Stack Exchange | By: JackOfAll | Tuesday, October 21, 2014
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I've seen a couple of seemingly unrelated notes about working with large volumes of data and it struck me that I couldn't find much content on problems specific to statistical analysis of Big Data. Is there a compiled list somewhere (or book, article)...
From: Stats Stack Exchange | By: ivanmp | Tuesday, October 21, 2014
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I have a database with several continuous variables measured in two times. I searched for a change in time in my dependent variables in this way: difJS<-lmer(JS~Time+(Time|id)+(Time|occupation),dat,REML=T) If I detect a significant fixed effect of...
From: Stats Stack Exchange | By: Andrea Gragnano | Monday, October 20, 2014
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As far as I know, both Gaussian mixtures as well as Gaussian processes can be used for regression. My question is: what is better and why? The answers might contain theoretic insights, practical experience or reference to further resources....
From: Stats Stack Exchange | By: Karel Macek | Tuesday, October 21, 2014
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I'm a psychology PhD student doing analysis on a relatively large set of data, obtained via online surveys. The purpose of the study is largely to determine normative data for a population of adults, on a number of psychology scales. However, I'm not...
From: Stats Stack Exchange | By: Jim | Tuesday, October 21, 2014
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(R studio) Hi, I'm running LDA on a dataset with 250,000 observations, 2 classes and 30 variables. My goal is to create a classification model using the LDA function. After loading my variables I receive a warning that my X's are collinear. (should this...
From: Stats Stack Exchange | By: Marvin Crisostomo | Tuesday, October 21, 2014
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I am interested in a model like: $y_{i} = \sum_{k\in K}{\beta_{k} z_{k}}$, with $z_{k} \tilde{} N(\mu_{k}, \sigma_{k})$. where $\beta \equiv(\beta_{k})_{k\in K}$ is not known, but all else is. I assume that there is a prior $\pi (\beta)$ of an arbitrary...
From: Stats Stack Exchange | By: Pedro Forquesato | Tuesday, October 21, 2014
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I have fitted a non linear assymptotic equation to a set of data and my interest is in getting the standard deviations of the fitted parameters. Is this possible in nls?
From: Stats Stack Exchange | By: Azzy Azali Azal | Tuesday, October 21, 2014
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I am trying to build a, regressive, predictive model for a target time-series that is heavily skewed. You could think of the target as being like earthquake magnitudes or heavy rainfall. Most of the time we sit in the relatively boring head of the distribution,...
From: Stats Stack Exchange | By: lb n-plus-1 | Tuesday, October 21, 2014
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I am interested in finding the median absolute distance to quantiles. So, for $Q_\alpha$ the $0 \le \alpha \le 1$ quantile, I would like to find $Q_\gamma^*$ such that $Q_\gamma^*$ satisfies \underset{\gamma}{median}|Q_\alpha-Q_\gamma|=...
From: Stats Stack Exchange | By: Deathkill14 | Tuesday, October 21, 2014
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I have several questions concerning analysis of data, especially when there are replications and/or pseudoreplications. First, I read an example in « pseudoreplication is a pseudoproblem » where we wish to determine which of two urns contains the greater...
From: Stats Stack Exchange | By: user3866113 | Tuesday, October 21, 2014
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I have 10 items in my store and I am running 3 promotions. To maximize my sell and profit, I want to decide price for the items on daily or weekly basis. No of items=10 Promotions running-3 Other factors can influence- competitor price of that area (...
From: Stats Stack Exchange | By: panda | Tuesday, October 21, 2014
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I want to use some count data to train a classifier. The count data range from 0 to 400 something. There are a bunch of smaller counts (0's and 1's). I wonder what would be a good way to categorize it into 4 groups.Thanks in advance!...
From: Stats Stack Exchange | By: Alex | Tuesday, October 21, 2014
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A survey found that the average communication of male is 35 and the average communication of female is 65. The data obtained from 100 students from 2 samples and that the standard deviation were _ and _ respectively at a= 0.5. Can it be concluded that...
From: Stats Stack Exchange | By: Leah Young | Tuesday, October 21, 2014
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i want to analyze the correlation between overall attitudes with purchase intention. both is continuous variables. Overall attitude using 7-liker scale while purchase intention 5 liker scale. So, can i do correlation between this even they have different...
From: Stats Stack Exchange | By: user59001 | Tuesday, October 21, 2014
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This question is based on Honglak Lee's paper "Convolutional Deep Belief Networks for Scalable Unsupervised Learning of Hierarchical Representations". In chapter "4.3 Handwritten digit classification", it is written: We trained 40 first layer bases from...
From: Stats Stack Exchange | By: Baptiste Wicht | Tuesday, October 21, 2014
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I am trying to conduct an A/B(/C) test to compare the performance of 3 different website pages but I'm facing issues regarding zero inflated data. I have data for each page regarding 1.) the number of people who clicked on the page 2.) performed the...
From: Stats Stack Exchange | By: user3682157 | Tuesday, October 21, 2014
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I attempted to build a deep network (e.g. deep autoencoder) for some object classification, my result showed that the deep networks is worst than shallow network. However, from what I have read from lecture, deep network perform well. This raise me a...
From: Stats Stack Exchange | By: RockTheStar | Monday, October 20, 2014
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text:
# Deceleration parameter

The deceleration parameter ${\displaystyle q}$ in cosmology is a dimensionless measure of the cosmic acceleration of the expansion of space in a Friedmann–Lemaître–Robertson–Walker universe. It is defined by:

${\displaystyle q\ {\stackrel {\mathrm {def} }{=}}\ -{\frac {{\ddot {a}}a}{{\dot {a}}^{2}}}}$

where ${\displaystyle a}$ is the scale factor of the universe and the dots indicate derivatives by proper time. The expansion of the universe is said to be "accelerating" if ${\displaystyle {\ddot {a}}>0}$ (recent measurements suggest it is), and in this case the deceleration parameter will be negative.[1] The minus sign and name "deceleration parameter" are historical; at the time of definition ${\displaystyle {\ddot {a}}}$ was expected to be negative, so a minus sign was inserted in the definition to make ${\displaystyle q}$ positive in that case. Since the evidence for the accelerating universe in the 1998–2003 era, it is now believed that ${\displaystyle {\ddot {a}}}$ is positive therefore the present-day value ${\displaystyle q_{0}}$ is negative (though ${\displaystyle q}$ was positive in the past before dark energy became dominant). In general ${\displaystyle q}$ varies with cosmic time, except in a few special cosmological models; the present-day value is denoted ${\displaystyle q_{0}}$.

The Friedmann acceleration equation can be written as

${\displaystyle 3{\frac {\ddot {a}}{a}}=-4\pi G\sum _{i}(\rho _{i}+3p_{i})=-4\pi G\sum _{i}\rho _{i}(1+3w_{i}),}$

where the sum ${\displaystyle i}$ extends over the different components, matter, radiation and dark energy, ${\displaystyle \rho _{i}}$ is the energy density of each component, ${\displaystyle p_{i}}$ is its pressure, and ${\displaystyle w_{i}}$ is the equation of state for each component. the value of ${\displaystyle w_{i}}$ is 0 for non-relativistic matter (baryons and dark matter), 1/3 for radiation, and -1 for a cosmological constant; for more general dark energy it may differ from -1, in which case it is denoted ${\displaystyle w_{DE}}$ or simply ${\displaystyle w}$.

Defining the critical density as

${\displaystyle \rho _{c}={\frac {3H^{2}}{8\pi G}}}$

and the density parameters ${\displaystyle \Omega _{i}=\rho _{i}/\rho _{c}}$, dividing the above gives

${\displaystyle q={\frac {1}{2}}\sum \Omega _{i}(1+3w_{i})=\Omega _{rad}(z)+{\frac {1}{2}}\Omega _{m}(z)+{\frac {1+3w_{DE}}{2}}\Omega _{DE}(z)\ .}$

where the density parameters are at the relevant cosmic epoch. At the present day ${\displaystyle \Omega _{rad}\sim 10^{-4}}$ is negligible, and if ${\displaystyle w_{DE}=-1}$ (cosmological constant) this simplifies to

${\displaystyle q_{0}={\frac {1}{2}}\Omega _{m}-\Omega _{\Lambda }.}$

where the density parameters are present-day values; this evaluates to ${\displaystyle q_{0}\approx -0.55}$ for the parameters estimated from the Planck spacecraft data. (Note that the CMB, as a high-redshift measurement, does not directly measure ${\displaystyle q_{0}}$; but its value can be inferred by fitting cosmological models to the CMB data, then calculating ${\displaystyle q_{0}}$ from the other measured parameters as above).

The time derivative of the Hubble parameter can be written in terms of the deceleration parameter:

${\displaystyle {\frac {\dot {H}}{H^{2}}}=-(1+q).}$

Except in the speculative case of phantom energy (which violates all the energy conditions), all postulated forms of mass-energy yield a deceleration parameter ${\displaystyle q\geqslant -1.}$ Thus, any non-phantom universe should have a decreasing Hubble parameter, except in the case of the distant future of a Lambda-CDM model, where ${\displaystyle q}$ will tend to -1 from above and the Hubble parameter will asymptote to a constant value of ${\displaystyle H_{0}{\sqrt {\Omega _{\Lambda }}}}$.

The above results imply that the universe would be decelerating for any cosmic fluid with equation of state ${\displaystyle w}$ greater than ${\displaystyle -{\tfrac {1}{3}}}$ (any fluid satisfying the strong energy condition does so, as does any form of matter present in the Standard Model, but excluding inflation). However observations of distant type Ia supernovae indicate that ${\displaystyle q}$ is negative; the expansion of the universe is accelerating. This is an indication that the gravitational attraction of matter, on the cosmological scale, is more than counteracted by the negative pressure of dark energy, in the form of either quintessence or a positive cosmological constant.

Before the first indications of an accelerating universe, in 1998, it was thought that the universe was dominated by matter with negligible pressure, ${\displaystyle w\approx 0.}$ This implied that the deceleration parameter would be equal to ${\displaystyle \Omega _{m}/2}$, e.g. ${\displaystyle q_{0}=1/2}$ for a universe with ${\displaystyle \Omega _{m}=1}$ or ${\displaystyle q_{0}\sim 0.1}$ for a low-density zero-Lambda model. The experimental effort to discriminate these cases with supernovae actually revealed negative ${\displaystyle q_{0}\sim -0.6\pm 0.2}$, evidence for cosmic acceleration, which has subsequently grown stronger.

## ReferencesEdit

1. ^ Jones, Mark H.; Lambourne, Robert J. (2004). An Introduction to Galaxies and Cosmology. Cambridge University Press. p. 244. ISBN 978-0-521-83738-5.


text:
# A framework for the dynamic implementation of finite automata for performance enhancement

@inproceedings{Ngassam2004AFF,
title={A framework for the dynamic implementation of finite automata for performance enhancement},
author={Ernest Ketcha Ngassam and Bruce W. Watson and Derrick G. Kourie},
booktitle={Int. J. Found. Comput. Sci.},
year={2004}
}
• Published in Int. J. Found. Comput. Sci. 1 December 2005
• Computer Science
The aim of this work is to provide a model for the dynamic implementation of finite automata for enhanced performance. Investigations have shown that hardcoded finite automata outperforms the traditional table-driven implementation up to some threshold. Moreover, the kind of string being recognized plays a major role in the overall processing speed of the string recognizer. Various experiments are depicted to show when the advantages of using hardcoding as basis for implementing finite automata…
5 Citations

## Figures from this paper

Hardcoding and dynamic implementation of finite automata
• Computer Science
• 2004
It is shown that the history of the states previously visited in the dynamic framework in order to predict the suitable algorithm for acceptance testing can be relied on and help further the investigations on the problem of the dynamic implementation of finite automata.
Dynamic Allocation of Finite Automata States for Fast String Recognition
• Computer Science
Int. J. Found. Comput. Sci.
• 2006
It is shown that the algorithm, referred to as the Dynamic State Allocation algorithm outperforms the traditional table-driven algorithm for strings that tend to repeatedly access the same set of states, provided that the string is long enough to amortize the allocation cost.
Reordering finite automata states for fast string recognition
• Computer Science
Stringology
• 2005
The spatial and temporal locality of reference on which cache memory relies to minimize cache swaps, is exploited to design a new algorithm for finite automaton string recognition that outperforms the traditional table-driven algorithm for strings that tend to repeatedly access the same set of states.
Proceedings of the Eindhoven FASTAR Days 2004
The algorithm is an extension of an incremental construction algorithm for acyclic deterministic automata (ADFAs) and similar extensions of two other construction algorithms for ADFAs are presented.
A framework for benchmarking FA-based string recognizers
• Computer Science
SAICSIT '10
• 2010
A unified framework for frequently evaluating existing FA-based string recognizers such that FA-implementers could capture appropriate problem domains that guarantee an optimal performance of available recognizers is proposed.

## References

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Hardcoding finite automata
The main result of this work is that the hardcoded implementations of finite automata outperform the table-driven implementation up to some threshold, which emphasizes that many applications using finite Automata as basic model may be optimized by replacing the table -driven implementation with a hardcoded implementation, resulting to better performances.
Hardcoding finite state automata processing
• Computer Science
• 2003
This paper presents various experiments in hardcoding the transition table of a finite state machine directly into string-recognizing code and measures are provided to show the time efficiency gains by various hardcoded versions over the traditional table-driven approach.
Preliminary Experiments in Hardcoding Finite Automata
• Computer Science
CIAA
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Various experiments in hardcoding a single row of a transition table of a finite state machine directly into symbol- recognizing code are presented. Measurements are provided to show the time
The working set model for program behavior
A new model, the “working set model,” is developed, defined to be the collection of its most recently used pages, which provides knowledge vital to the dynamic management of paged memories.
Stretching and jamming of automata
• Computer Science
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This paper presents two transformations on automata, called stretching and jamming, which will, under certain conditions, reduce the size of the transition table, and under other conditions reduce the string recognition time.
Fast Pattern Matching in Strings
• Computer Science
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An algorithm is presented which finds all occurrences of one given string within another, in running time proportional to the sum of the lengths of the strings, showing that the set of concatenations of even palindromes, i.e., the language $\{\alpha \alpha ^R\}^*$, can be recognized in linear time.
Flexible pattern matching in strings - practical on-line search algorithms for texts and biological sequences
• Computer Science
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This book presents a practical approach to string matching problems, focusing on the algorithms and implementations that perform best in practice, and includes all of the most significant new developments in complex pattern searching.
Phd by thesis
• R. Lathe
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Deposits of clastic carbonate-dominated (calciclastic) sedimentary slope systems in the rock record have been identified mostly as linearly-consistent carbonate apron deposits, even though most
299-300
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Taxonomies and Toolkits of Regular Languages Algorithms
• Taxonomies and Toolkits of Regular Languages Algorithms
• 1995


text:
# Measurement and Evolution

## Generalised Measurement Operators

The extension of measurement beyond orthogonal projectors alone arises from the extension of quantum mechanics to open systems, where we are performing the measurement on a subsystem S alone. We can understand this as allowing our system to interact with a 'pointer' or measurement device; consider a particle of mass $m$, helping us to measure an observable $\mathbf{M}$. The interaction is turned on during a window defined by the function $\lambda(t): \lambda(t) \in \{0,1\}.$ Thus, the system and pointer have a Hamiltonian

We assume that $\commutator{\mathbf{M}}{\mathbf{H_{0}}}=0$ for simplicity. We have evolution during the measurement $\mathbf{U}(T)\approx \exp(-i\lambda T\mathbf{M}\otimes\mathbf{P})$, which we can rewrite as using the spectral decomposition of $\mathbf{M}.$

The operator $\mathbf{P}$ translates the position of the particle, and the 'amount' of evolution depends on the eigenvalues $M_{a}$. More abstractly, given a set of orthogonal projectors $\{\mathbf{E}_{a}, a=0,1,2\cdots N-1\}$ which satisfy $\mathbf{E}_{a}\mathbf{E}_{b}=\delta_{a,b}\mathbf{E}_{a}, \sum_{a}\mathbf{E}_{a}=\mathbf{I}.$

We can measure these projectors by coupling to a system with n basis states $\{\ket{a}\}$, and then doing a transformation $\mathbf{U}=\sum_{a,b}\mathbf{E}_{a}\otimes\ketbra{b+a}{b}$.

This is a unitary $\mathbf{U}:\ket{\Psi}=\ket{psi}\otimes\ket{0}\rightarrow\ket{\Psi'}=\sum_{a}\mathbf{E}_{a}\ket{psi}\otimes\ket{a}.$

Thus, the state of the system before the measurement outcome is known is given by and thus we can rewrite this in the density matrix formalism as $\rho\rightarrow\sum_{a}\mathbf{E}_{A}\rho\mathbf{E}_{a}.$

### Beyond Orthogonality

We can extend this description of measurement beyond the case where the measurements we are doing are all orthogonal. This is called a 'generalised measurement' made up of a set of operators $\{\mathbf{M}_{a}\}.$ For normalisation, these operators need to satisfy a completeness relation $\sum_{a}\mathbf{M}_{a}^{\dagger}\mathbf{M}_{a}=1$, such that the evolution is given by

We have that $Prob(a)=\|\mathbf{M}_{a}\ket{\psi}\|^{2}$, and $Prob(b|a)=\frac{\|\mathbf{M}_{b}\mathbf{M}_{a}\ket{\psi}\|^{2}}{\|\mathbf{M}_{a}\ket{\psi}\|^{2}}.$

For each $\mathbf{M}_{a}$ there is an associated Hermitian, positive operator $\mathbf{E}_{a}\equiv\mathbf{M}_{a}^{\dagger}\mathbf{M}_{a}$ such that $Prob(a)=tr\left(\rho\mathbf{E}_{a}\right).$ These are also called Positive Operator Valued Measurements (POVMS).

## Quantum Channels

Quantum channels are a method of modelling the evolution of one subsystem, based on unitary evolution of the composite system. In fact, the mathematics of quantum channels is identical to the evolution of the state under a POVM before collapse given above:

$\mathcal{E}:\rho\rightarrow\rho'$ is a linear map acting on the density operator of the subsystem. These channels are also called superoperators, or Complete Positive Trace-Preserving (CPTP) Maps. They obey 4 simple properties

1. Linearity: $\mathcal{E}(\alpha\rho_{1}+\beta\rho_{2})=\alpha\mathcal{E}(\rho_{1})+\beta\mathcal{E}(\rho_{2})$
2. Hermiticity Preserving: $\rho=\rho^{\dagger}\implies\mathcal{E}(\rho)=\mathcal{E}(\rho^{\dagger})=\mathcal{E}(\rho)^{\dagger}$
3. Positivity Preserving: $\rho\geq 0 \implies \mathcal{E}(\rho)\geq 0$
4. Trace Preserving: $tr\left(\mathcal{E}(\rho)\right)=tr(\rho)$

We can prove that any quantum channel admits a decomposition in to a set of operators $\{\mathbf{M}_{a}\}$ called 'Kraus' operators, introudcing a unitary acting on the system $\ket{\psi}_{A}\otimes\ket{0}_{B}$ and tracing out B. These operator sum rotations are not unique, and we can see that the action of a channel is invariant under a chnge of basis $\ket{a}=\sum_{u}\ket{\mu}V_{\mu a}$, giving new operators $\mathbf{N}_{\mu}=\sum_{a}V_{\mu a}\mathbf{M}_{a}.$

Channels can be compose $\mathcal{E}' = \mathcal{E}_{2}\circ\mathcal{E}_{1}: \mathcal{E}'(\rho) = \sum_{\mu a}\mathbf{N}_{\mu}\mathbf{M}_{a}\rho\left(\mathbf{N}_{\mu}\mathbf{M}_{a}\right)^{\dagger}$ where $\mathbf{M}_{a},\mathbf{N}_{\mu}$ are the Kraus operators of channel 1 and 2 respectively.

### Reversibility

Under what conditions is a Quantum channel reversible? This means the channel has an inverse $\mathcal{E}_{2}$ such that

This imposes the condition that $\mathbf{N}_{\mu}\mathbf{M}_{a}=\lambda_{\mu a}\mathbf{I}\;\forall\mu, a.$ Using the completeness relation for Kraus operators, we can show

Using the polar decomposition of $\mathbf{M}_{a}$, we can thus see that the Kraus operators are proprotional to unitary operators, and thus $\mathcal{E}_{1}$ is a simple unitary map.

### The Dual of a Channel

The Heisenberg representation of quantum mechanics is one in which the time dependence of the evolution is carried by the opertors, instead of the states as in the 'Schrödinger' picture. In this case, we have the Heisenberg equaiton of evolution

The two pictures are defined such that they agree on the expectation values at any time, and that the two picutres are indistinguishable for $t=0.$

We can define an alternative depiction of a channel where general observables evolve under a channel $\mathcal{E}^{*}:\mathbf{A}'=\sum_{a}\mathbf{M}^{\dagger}_{a}\mathbf{A}\mathbf{M}_{a}.$ Thus, we have

In this case, instead of requiring the dual channel be trace-preserving, ew require it is 'unital' Not all channels are unital, but all dual channels are. A channel is unital if the completeness relation is satisfied for both 'permutations' of $\mathbf{M}_{a}, \mathbf{M}_{a}^{\dagger}.$ These channels are analagous to doubly stochastic maps.

### Quantum Operations

Channels sharing a mathematical form with maps allow us to consider the more general class of 'quantum operations'. For example, consider applying a POVM that can potentially give information about two observables $a,\mu$, but discarding the value of $\mu.$ This gives us an effective channel which has outcome $a$ with probability $tr\mathcal{E}_{a}(\rho).$ Just summing over $\mu$, the Kraus operators of $\mathcal{E}_{a}$ will not satisfy the completeness relation, and so we need to renormalise the state by the probability of the outcome. We can use this channel form to concatenate successive measurements too.

### Linearity

Why are quantum channels generlly linear maps? One note is that the description of a density operator as an ensemble of states is incompatible with a nonlinaer-evolution; indeed, given the description of the density matrix as a convex sum of points, a quantum channel must be linear.

Consider an initial state $\rho_{i}$ prepared with probability $p_{i}$, and then subsequently measured to obtain outcome $a$ ith probability $p(a|i).$ We have the post measurement state $\mathcal{E}_{a}(\rho_{i})/ p(a|i)$, and thus the ensemble post measurement is given by: where $p(i|a)$ is the probability that $i$ was prepared given outcome a. We can also write this as using a convex combination of initial states $\{\rho_{i}\}.$

Using Bayes' rule, $p_{i}p(a|i)=p_{a}p(i|a)$, and thus $\mathcal{E}_{a}$ is a linear map:

### Complete Positivity

Complete positivity is the stronger requirement that a quantum channel is positive, even if acting on only part of a composite system.

For example, consider a channel $\mathcal{E}: \rho\in\mathcal{H}_{A}\rightarrow\rho'\in\mathcal{H}_{A'}.$ What happens if we append the input space to $\mathcal{H}_{A}\otimes\mathcal{H}_{B}$ and apply the operation $\mathcal{E}\otimes\mathbf{I}$ i.e. do nothing to this additional space? As long as this extended operation is positive, then the channel is called completely positive.

This is easy to show for a channel in the Kraus representation: the extended map $\mathcal{E}\otimes\mathbf{I}$ simply has Kraus operators $\mathbf{M}_{a}\otimes\mathbf{I}.$ The complete positivity requirement is a fairly loose constraint; physically, it can be understood as saying it is possible to manipulate systems in isolation.

Not all positive maps are complete positive. An interesting example is the Transpose map, which is defined as a map $T:\ketbra{i}{j}\rightarrow\ketbra{j}{i}.$ This map is positive, as

But, if we perform the 'partial transpose' $T\otimes\mathbf{I}$, this map is not complete positive. In fact, it is negative for entangled states! This is called the Peres-Horodecki criterion for entanglement.

## Alternative Representations of Channels

A quantum channel acting on a subsystem $A$ derives from a unitary map acting on an extension of A, and is itself a CPTP map in the space of density operators. We can thus interpret any evolution of A as a unitary evolution of its purification AR.

This can be understood using the Choi-Jamiolkowski isomporhism. Consider the action of an extended channel $\mathbf{I}\otimes\mathcal{E}$ acting on a maximally entangled state $\ket{\Phi}_{RA}=\sum_{i=0}^{d-1}\ket{i}_{R}\ket{i}_{A}\rightarrow \sum_{a}\left(\ketbra{\Psi_{a}}{\Psi_{a}}\right)_{RA'}.$

We note that $\ket{\phi}_{A}=\sum_{i}\phi_{i}\ket{i}_A= \sum_{i}\phi_{i} \left( _{R}\braket{i}{\Phi}_{RA}\right) = _{R}\braket{\phi^{*}}{\Phi}_{RA}.$ Using the linearity of $\mathcal{E}$, we can thus write

What this tells is that the mapping of a state $\ket{\phi}\in\mathcal{H}_{A}\rightarrow\ket{\phi'}\in\mathcal{H}_{A'}$ can be obtained using the state $\ket{\Psi}_{RA'}$ as $\mathbf{M}_{A}= _{R}\braket{\phi^{*}}{\Psi_{a}}_{RA'}.$ These operators $\mathbf{M}_{a}$ form the Kraus operators of the channel, and thus there is a direct connection between the state $\ket{\Psi}_{RA'}$ and $\mathcal{E}.$

Thus, we can draw an equivalence between the freedom to choose a set of Kraus oeprators, and the freedom to choose a decomposition of $\ket{\Psi}_{RA'}$ into an ensemble of pure states. Recall that two ensembles are related by a unitary change of basis $\ket{\gamma_{\mu}}=\sum_{a}V_{\mu a}\ket{\Psi}_{a}$, and so are two sets of Kraus operators $\mathbf{N}_{\mu}=\sum_{a}V_{\mu a}\mathbf{M}_{a}.$

This also makes it clear that, for a mapping from $A:|A|=d \rightarrow A':|A'|=d'$, we need $d^{2}(d'^{2}-1)$ parameters.

### Stinespring dilation

Consider a channel $\mathcal{E}:A\rightarrow A' : |A|>|A'|.$ In this case we can define an 'environment' E, such that the actionof the channel can be described by an inner-product preserving map or isometry V

We can see that thus is a 'unitary' isometry $: V^{\dagger}V=\mathbf{I}_{A}.$ This isometry is called the 'Stinespring Dilation' of a channel $\mathcal{E}=\{\mathbf{M}_{a}\}.$ Another way of understanding this is to conceptualise the Stinespring dilation as the purification of the Choi state.

It can be shown that


text:
# Reinforcement Learning

tags
Machine Learning

Reinforcement Learning is the field of learning and decision-making under uncertainty. An agent acts with a learnable behaviour policy in an environment with initially unknown dynamics and reward. The agent observes the environment’s state (sometimes partially: POMDPs) and chooses an action.

behaviour
$$\Pi(a | s)$$
action
$$a_t \in A$$
reward
$$r_{t+1} \in R$$
state
$$s_{t+1} \in S$$
dynamics
$$T(s_{t+1} | s_t, a_t)$$
reward
$$R(r_{t+1} | s_t, a_t)$$

The dynamics and rewards here reflect the markovian assumption. States can be represented in many ways: in a tabular fashion (state 1 $$\rightarrow$$ state 2) or as features.

## Key Challenges

1. Exploration vs Exploitation
2. Function approximation
3. Credit Assignment

## Making Complex Decisions

Earlier, we were concerned with environments with one-shot, episodic decision problems. Sequential decision problems incorporate utilities, uncertainty and sensing. These include searching and planning problems as special cases.

### Utilities over time

A finite horizon for decision making means that there is a fixed time $$N$$ after which nothing matters. In these scenarios, the optimal action in a given state could change over time, i.e. the optimal policy is non-stationary.

It turns out that under stationarity, there are only 2 coherent ways to assign utilities to sequences:

$$U_h([s_0, s_1, \dots, s_n]) = R(s_0) + R(s_1) + \dots + R(s_n)$$
discounted rewards
$$U_h([s_0, s_1, \dots, s_n]) = R(s_0) + \gamma \cdot R(s_1) + \dots + \gamma^2 \cdot R(s_n)$$

This discount factor $$\gamma$$ is a number between 0 and 1. Assuming stationarity has several problems. First, if the environment does not contain a terminal state, then utilities of undiscounted rewards go to infinite, and comparing two infinitely state sequences would be impossible. With discounted rewards, the utility of an infinite sequence can be made finite.

However, if the environment contains a terminal state, and the agent is guaranteed to reach a terminal state eventually, then this policy is called a proper policy, and the above issue goes away. Infinite sequences can be compared in terms of the average reward obtained per time step.

### Optimal policies and the utilities of states

First, we can derive the expected utility of executing a policy $$\pi$$ in $$s$$:

$$U^\pi (s) = \mathbb{E} \left[ \sum_{t=0}^\infty \gamma^t R(S_t) \right]$$

where the expectation is with respect to the probability distribution over state sequences. determined by $$s$$ and $$\pi$$. Then $π^*(s) = argmax_π U^π (s)*. A consequence of using discounted utilities with infinite horizons is that the optimal policy is independent of the starting state. This allows us to compute the true utility of the state as $$U^{\pi^*} (s)$$. The utility function allows the agent to select actions by using the principle of maximum expected utility from the earlier chapter: $$\pi^*(s) = argmax_{a \in A(s) } \sum_{s^{i}} P(s’ |s, a)U(s’)$$. ### Summary Problem Bellman Equation Algorithm Prediction Bellman Expectation Equation Iterative Policy Evaluation Control Bellman Expectation Equation + Greedy Policy Improvement Policy Iteration Control Bellman Optimality Equation Value Iteration ## Passive Reinforcement Learning We start with a passive learning agent using a state-based representation in a fully observable environment. In passive learning, the agent’s policy $$\pi$$ is fixed: in state $$s$$, it always executes the action $$\pi(s)$$. Its goal is simply to learn how good the policy is – the utility function $$U^\pi (s)$$. The passive learning task is similar to policy evaluation, but the agent does not know the transition model $$P(s'|s, a)$$ and the reward function $$R(s)$$. The agent executes a number of trials using the policy $$\pi$$, and experiences a sequence of state transitions. At each state its percepts receives the current state and the reward of the state. We write the utility as: $$U^\pi (s) = E\left[\sum_{t=0}^\infty \gamma^t R(S_t) \right]$$ ### Direct Utility Estimation (MC Learning) The main idea of direct utility estimation is that the utility of a state is the expected total reward from that state onward, and each trial provides a sample of this quantity for each state visited. Direct utility estimation reduces the reinforcement learning problem to a supervised inductive learning problem, where each example has the state as input, and the observed reward-to-go as output. However, it misses an important source of information: that the utility of states are not independent. This means it misses many opportunities for learning. For example, if a state has high expected utility, then neighbouring states should also have high expected utility. The utility of each state equals its own reward plus the expected utility of its successor states: i.e. it obeys the Bellman Equation for a fixed policy. We can view directed utility estimation as searching for $$U$$ in a hypothesis space that is much larger than it needs to be, since it includes many functions that violate the Bellman equations. ### Adaptive Dynamic Programming An ADP agent takes advantage of the constraints among the utilities of states by learning the transition model that connects them and solving the corresponding MDP using a dynamic programming method. For a passive learning agent, the task is as simply as plugging in the learnt transition model and the rewards into the Bellman equations to calculate the utility of each state. The task of learning the model is easy, because the environment is fully observable. This means we have a supervised learning task where the input is a state-action pair, and the output is the resulting state. We keep track of how often each action outcome occurs and estimate the transition probability $$P(s’ | s, a)$$ from the frequency with which $$s’$$ is reached when executing $$a$$ in $$s$$.  function PASSIVE-ADP_AGENT(percept) returns an action inputs: percept, indicating state s' and reward signal r' persistent: \pi, a fixed policy mdp: MDP with model P, rewards R, and discount \gamma U: a table of utilities, initially empty N_{sa}: a table of frequencies for each state-action pair N_{s'|s,a}: a table of outcome frequencies s, a: the previous state and action if s' is new then$U[s'] <- r'; R[s'] <- r'
if s is not null then
increment N_{sa}[s, a] and N_{s'|s,a}[s', s, a]
for each t such that N_{s'|s, a}[t,s,a] is nonzero do
P(t|s, a) <- N_{s'|s, a}[t,s,a] / N_{sa}[s, a]
U <- POLICY-EVALUATION(\pi, U, mdp)
if s'.TERMINAL? then s,a <- null else s,a <- s', \pi[s']
return a

Code Snippet 1: A passive RL agent based on ADP.

This approach is computationally intractable for large state spaces. In addition, it uses the maximum-likelihood estimation for learning the transition model.

A more nuanced approach would be Bayesian reinforcement learning, which assumes a prior probability $$P(h)$$ for each hypothesis $$h$$ about what the true model is. The posterior probability $$P(h|e)$$ is obtained via Bayes’ rule. Then $$\pi^* = argmax_\pi \sum_h P(h|e) u_h^\pi$$.

Another approach, derived from robust control theory, allows for a set of possible models $$H$$ and defines an optimal robust policy as one that gives the best outcome in the worst case over $$H$$: $$\pi^* = argmax_\pi min_h u_h^\pi$$.

### Temporal-difference Learning

TD learning involves using the observed transitions to adjust the utilities such that the constraint equations are met.

When a transition occurs from state $$s$$ to state $$s’$$, we apply the update rule:

$$U^\pi(s’) \leftarrow U^\pi(s) + \alpha (R(s) + \gamma U^\pi(s’) -U^\pi(s))$$

Where $$\alpha$$ is the learning rate. The difference in utilities gives rise to the name temporal-difference.

  function PASSIVE-TD-AGENT(percept) returns an action
inputs: percept, with current state s' and reward r'
persistent: \pi, a fixed policy
U, a table of utilities, initially empty
N_s, a table of frequencies
s, a, r, the previous state, action and reward

if s' is new then U[s'] <- r'
if s is not null then
increment N_s[s]
U[s] <- U[s] + \alpha N_s[s] (r + \gamma U[s'] - U[s])
if s'.TERMINAL? then s, a r <- null else s,a,r <- s', \pi[s'], r'
return a

TD learning learns slower than ADP and shows much higher variability, but is simpler and requires less computation. TD learning does not need a transition model to perform updates.

ADP and TD are closely related. Both try to make local adjustments to the utility estimates in order to make each state “agree” with its successors. However, TD adjusts a state to agree with its observed successor, while ADP adjusts the state to agree with all of the successors that might occur, weighted by their probabilities.

ADP can be made more efficient by approximating the algorithms for value or policy iteration. For example, the prioritized sweeping heuristic prefers adjustments to states that have undergone a large adjustment in their own utility schemes. This enables them to handle state spaces that are far too large for a full ADP. An approximation algorithm can use a minimum adjustment size that decreases as the environment model becomes more accurate, eliminating very long value iterations that occur early in learning due to large changes in the model.

## Active Reinforcement Learning

A passive learning agent has a fixed policy that determines its behaviour. An active agent must learn what actions to take.

First, the agent will need to learn a complete model with outcome probabilities for all actions, rather than the model for the fixed policy. The learning mechanism for the passive ADP agent will work for this

Next, the agent has a choice of actions. The utilities it learns are defined by the optimal policy, governed by the Bellman Equations. Having obtained a utility function for the given model, the agent can extract an optimal action by one-step look-ahead to maximise the expected utility.

### Potential Pitfalls

A greedy agent, that picks the best action given the learned model, very seldom converges to the optimal policy for the environment and sometimes converges to horrible policies.

This is because the learned model is not the same as the true environment. What is optimal in the learned model might not be optimal in the true environment.

An agent therefore has to make a tradeoff between exploitation to maximise its reward, and exploration to maximise its long-term well-being. The question on whether there is an optimal exploration policy is a subfield of statistical decision theory called the bandit problem.

An agent has to be greedy in the limit of infinite exploration, or GLIE. This is the scenario where the learned model is the true model. There are several GLIE schemes, one of the simplest is to have the agent choose a random action a fraction $$\frac{1}{t}$$ of the time and to follow the greedy policy otherwise. This can be extremely slow to converge.

A more sensible approach is to assign some eight to actions that the agent has not tried very often,while tending to avoid actions that are believed to be of low utility. This can be achieved by altering the constraint equation to assign higher utility estimates to unexplored state-action pairs.

$$U^+(s) \leftarrow R(s) + \gamma max_{a} f\left( \sum_{s’} P(s’ | s, a) U^+(s), N(s, a) \right)$$

$$f(u, n)$$ is called the exploration function. It determines how greed is traded off against curiosity. The function should be increasing in $$u$$ and decreasing in $$n$$.

### Learning an action-utility function

An active TD agent is no longer equipped with a fixed policy, so if it learns a utility function $$U$$, it will need to learn a model in order to be able choose an action based on $$U$$ via one-step look-ahead. The update rule for TD remains unchanged. IT can be shown that the TD algorithm will converge to the same values as ADP as the number of training sequences tends to infinity.

## RL Applications

Personalizer
ICML 2017 Tutorial on Real World Interactive Learning
Robotics
Deep Reinforcement Learning, Decision Making, and Control
Tutoring Systems
Reinforcement Learning with People - NIPS 2017 - YouTube

<biblio.bib>


text:
# Induced EMF and farady's law

1. Aug 16, 2007

### pardesi

well iwas going through the standard example that is there is a uniform magnetic field in aregion going into the palne and ther is a rectangular 'loop' with one side which can slide on metallic rods with constant speed $$v$$ and the the relation

$$\epsilon=BLv$$
well the example is (almost) here
well i could intutively understand that from lorentz' force concept that when the side starts moving so do the electrons so they experience a force and a current is produced.

well i wanted to go about it from farady's alw.well what i couldn't get is farady's law states when the "flux across a circuit...".but here our loop itself is changing so how do i utilize the law

2. Aug 16, 2007

### marlon

3. Aug 16, 2007

### pardesi

but here the 'loop' itself is not constant

4. Aug 16, 2007

### marlon

What do you mean by the loop is not constant ?

Generally, the flux changes due to :
1) change in the position of the loop with respect to the B-field (rotation or translation)

2) the loop changes its form (rectangle becomes circle eg).

The basic ingredient is the fact that the flux must change with respect to time ! If this does not happen, no potential is created.

marlon

5. Aug 16, 2007

### pardesi

i mean here the rectangle grows in size it does have a change in area but this change is due to addition of 'more conducting element' as oppsoed to whenit changes toa circle in which the lopp is 'constant' but it's shape changes

6. Aug 16, 2007

### marlon

First of all, if it changes in surface, this means that the flux WILL change and a voltage will be created.

But how does the rectangle increases its surface ? I don't get the "due to more conducting elements" reasoning.

marlon

7. Aug 16, 2007

### pardesi

well waht i meant was consider the rectangular loop PQRS no suppose i start stretching it or change it into a circle in presence of a mganetic field then the flux changes and an 'emf' is induced across the loop PQRS.
but here the loop PQRS not only schanges its' shape but also gets some additional new elements.

8. Aug 17, 2007

### marlon

How does the loop acquire additional new elemnts and what are these elements ? I really don't get it. Besides, what exactly do you want to know other than the consequence of the applicaton of Faraday's law which we covered here.

marlon

9. Aug 17, 2007

### pardesi

well i got thsi ...i read that in griffith where it is explained nicely...well it is sheer coincidence he writes that the motional emf has the same formula as that for the other two cases...
great

10. Aug 17, 2007

### marlon

By motional emf you mean the moving loop right ? What same formula ? You mean Faraday's Law ?

Actually, could you explain the solution to your problem to me because i don't get it.

marlon

11. Aug 17, 2007

### pardesi

yes motional emf means moving coil but stationary fie;d(here).
and the formula refers to faraday's law
if u could get a copy of griffith then he very beutifully denies to accept that the emf induced due to motion of the cil as the one given by faraday's law.infact goes on to say that in some sense it is not the 'true' farady's law that holds here.
and most probably for the reason which i was worrying about
because if u go through faraday's law it starts witha line that "if through a loop..." so the loop is fixed and by loop i mean a collection some points.
but while we consider the motional emf as in my case the rectangle is taken to be the loop whose one side is moving.so the 'points' which make up the loop are not the same.
crudely speaking the loop itself is changing
so there is no point in assuming 'farday's law' for granted...though he discovered for this case also.it is just sheer coincidence taht thsi happened

12. Aug 17, 2007

### marlon

That's incorrect. What matters is the variation of the flux with respect to time, inside the loop. The loop may vary in structure as well.

marlon

13. Aug 17, 2007

### pardesi

flux through what

14. Aug 17, 2007

### pervect

Staff Emeritus
If you're hopelessly confused by the area changing (honestly, I don't see why), just perform a line integral of the electric field around some closed curve.

http://www.ucl.ac.uk/Mathematics/geomath/level2/mult/mi3.html

You will find that the intergal over a curve of the electric field component pointing along the curve is non-zero.

If you like, you can use a vector identity to convert this line integral into an area integral. This is known as stokes theorem. This is why people are talking about integrating the flux, this is justified by Stokes theorem.

Last edited: Aug 17, 2007
15. Aug 17, 2007

### pardesi

i am not worried about the changing area...and stokes theorem is fine
also when u ask me to calculate the line integral it has to be about a fixed loop
what i am worried about is when we say that flux through a loop changes ...then obviously the flux has to be through a fixed loop,obviously i f i vary my loop with time my flux has to change(in most cases) ,so what i am saying is though the faraday's law holds true in case of motional emf...but that's just because it coincidentally happened to be so...
what i am opposing is 'direct' use of farady's law in getting the induced emf in case of motional emf..though it gives u the right answer but that won't be using faraday's laws in the right sense,as most books do (thankfully griffith didn't) and for the reason i stated above

16. Aug 17, 2007

### pardesi

to clarify things further as in this case the we assume the loop to be the rectangle PQRS then calculate the flux through it at any time t to be $$\Phi=Blx$$
then we write $$E=\frac{d \Phi}{dt}=Blv$$ where $$v$$ is the speed of the movable side.
but this makes no sense because the flux through the loop is not changing as the field is constant here.what actually is changing is the loop which we are considering so the calculated flux changes.though the flux through PQRS is not changing.
also here the emf induced is best explained by lorentz force.
it is just by sheere coincidence(and i repeat) that what we do is working out right.

17. Aug 19, 2007

### marlon

But can't you see the contradictio in terminis in the above two sentences ? First you say the flux is constant, then you say it changes. Again, the flux IS changing because the number of magnetic fieldlines in the loop changes with time. The reason of this change is the varying loop ! The flux can change due to 1) the B field or 2) the loop that encloses the field lines !

marlon

18. Aug 19, 2007

### pardesi

yes what i am saying is the flux through any loop is not changing with time.what is changing is the loop which we have taken into consideration.
say for example there is magnetic field uniform in a room.at time t i take my loop to be a square one inside the room.the next moment 'i' will take my loop to be a square one outside the room.so where will the induced emf be .

again i say the thing is the proof by faraday' law is not all correct(at least to me).lorentz force proof seems right.

19. Aug 19, 2007

### marlon

Again, if the loop is changing the flux is also changing. I would advise you to carefully investigate the definition of FLUX. After having done that, you will understand that what you are saying is incorrect.
Yes because the flux has changed. Again, make sure you look at the definition of flux.

marlon

20. Aug 19, 2007

### pardesi

flux $$\phi = \int_{S} \vec B . \vec dS$$
by loop i mean any fixed loop .consider any loop u want to the flux through it doesn't change with time.what is changing is the loop u are considering .so just because u r changing the loop of your consideration why should an emf be developed.

when u say the flux has changed can u tell me flux through what has changed?


text:
# Quark Matter 2012

12-18 August 2012
US/Eastern timezone

## How does longitudinal drag change the radiative energy loss rate?

16 Aug 2012, 16:00
2h

Poster Jets

### Speaker

Guang-You Qin (Duke University/Wayne State University)

### Description

Jets propagating through dense matter are modified due to the scattering between the patrons of the jet with the constituents of the medium. Such scattering leads to an exchange of momenta between the medium and the jet, and has components that are both parallel and perpendicular to the direction of jet propagation. These scatterings introduce both drag and diffusion, and also change the rate of parton splitting leading to radiative energy loss. In all current calculations of jet modification, only the effect of transverse scattering (quantified by $\hat{q}$) on stimulated emission has been considered. The effect of longitudinal exchange, leading to a drag and diffusion of the produced patrons, is included separately. We compute, for the first time, the gluon emission rate from a hard quark in the presence of both transverse and longitudinal exchange. Depending on the size of the longitudinal drag and diffusion coefficients, longitudinal scattering is shown to influence the radiative energy loss of a hard quark in a dense medium in a non-intuitive way. Ramifications of this effect for the phenomenology of jet quenching in heavy-ion collisions will also be presented.

### Primary author

Guang-You Qin (Duke University/Wayne State University)

### Co-author

Abhijit Majumder (Wayne State University)

### Presentation Materials

There are no materials yet.