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# GGX BRDF is greater than 1?

I am implementing a GGX BRDF, this is the formula I used is from:https://schuttejoe.github.io/post/ggximportancesamplingpart1/

And this is my implementation:

float BRDF(float3 viewDir, float roughness, float3 normal, float3 Li) {
float3 halfDir = normalize(Li + viewDir);
float nl = saturate(dot(normal, Li));
float nh = saturate(dot(normal, halfDir));

half lv = saturate(dot(Li, viewDir));
half lh = saturate(dot(Li, halfDir));
half nv = abs(dot(normal, viewDir));

float a2 = roughness * roughness;

float D = GGX(nh, roughness);
return G2 * D /(4 * abs(nv) * abs(nl));
}

{

float denomA = NdotV * sqrt(a2 + (1.0f - a2) * NdotL * NdotL);
float denomB = NdotL * sqrt(a2 + (1.0f - a2) * NdotV * NdotV);

return 2.0f * NdotL * NdotV / (denomA + denomB);
}

float GGX(float NdotH, float a) {
float a2 = sqr(a);
return a2 / (UNITY_PI * sqr(sqr(NdotH) * (a2 - 1) + 1));
}

I assume the fresnel term is constant: 1. However, I found that my BRDF sometimes give me number greater than 1.

• Shouldn't the return value be a float3? Also, why do you assume fresnel to be constant 1? That would only be true for an incident angle of $90°$. Have a look at Natty Hoffman's Siggraph Physically Based Shading Course, he displays some Fresnel values plotted for different angles (p. 15) blog.selfshadow.com/publications/s2013-shading-course/hoffman/…
– Tare
Mar 7 '19 at 7:44
• You are confusing brdf with probabilities. gamedev.stackexchange.com/questions/62438/… gamedev.net/forums/topic/257668-brdf-range----greater-than-01- Mar 7 '19 at 10:59
• A brdf can be larger than $1$ and be energy conserving. Consider the ideal specular reflection/refraction brdf which can be considered to go to infinity at a single point. It's the integral of the brdf multiplied by the cosine for all directions that needs to be $\leq 1$. Mar 7 '19 at 12:30
• Thank you very much. I understand now. If I do importance sampling on brdf, then my brdf/pdf can rescale my value back to [0,1] right? But if I do importance sampling on the light source, my brdf/pdf can still has some value greater than 1 because the pdf is the pdf of sampling my light source, not the pdf of sampling brdf. Am I correct? Mar 7 '19 at 15:07


text:
### Uzdevums:

3♦
Read the extract from an article and try to choose the correct words to fill in the gaps in the text!

First nationalism

The native peoples of British Columbia still maintain that nearly all the province's 950,000 square kilometres (367,000 square miles) are rightfully theirs - and they are battling for generous
for their historic loss.
That struggle has been

on by the success of the group that started it. In a treaty ratified earlier this year by the federal Parliament, the Nisga'a, a 6,000-member tribe in the north-west, finally won a settlement of a claim it first voiced in the mid-19th century. The treaty
Nisga'a ownership of 1,992 square kilometres of land, gives the tribe the right to self-government and to manage its own lands and natural resources - plus a payout of C$190m ($128m).

Lai risinātu uzdevumus, lūdzu reģistrējies!

format: plain


text:
Translator Disclaimer
VOL. 47 | 2004 Multiple comparisons with the best ROC curve
Jason C. Hsu, Peihua Qiu, Lin Yee Hin, Donald O. Mutti, Karla Zadnik

Editor(s) Y. Benjamini, F. Bretz, S. Sarkar

## Abstract

The accuracy of a medical diagnostic tool depends on its specificity, the probability that it classifies a normal person as normal, and its sensitivity, the probability that it classifies a diseased person as diseased. The receiver operating characteristic (ROC) curve of such a tool is its sensitivity plotted against (1$-$specificity) as the threshold defining "normal" versus "diseased" ranges over all possible values. A common, global measure of the accuracy of a diagnostic tool is the area under the curve (AUC), the curve being the ROC curve. Thus, one way to compare the accuracies of medical diagnostic tools is to compare their AUCs. By comparing each diagnostic tool with the truly most accurate diagnostic tool, one can eliminate diagnostic tools that are not the most accurate, and discover diagnostic tools which are either the most accurate or practically the most accurate. This article shows how the method of multiple comparison with the best (MCB) for normal error general linear models can be adapted to compare diagnostic tools in terms of AUCs of their ROC curves. MCB of AUCs of ROC curves is illustrated by comparing diagnostic variables for predicting the need for emergency Cesarean section, and for predicting the onset of juvenile myopia.

## Information

Published: 1 January 2004
First available in Project Euclid: 28 November 2007

zbMATH: 1268.62077
MathSciNet: MR2118592

Digital Object Identifier: 10.1214/lnms/1196285626

Subjects:
Primary: 92B15
Secondary: 62P10


text:
## Posts Tagged ‘Pell equations’

### Ramanujan’s pi approximations and Pell equations

Ramanujan gave many fascinating formulas and approximations to pi. Using one of his examples, we can give its family. First, define the fundamental units,

$U_{2} = 1+\sqrt{2}$

$U_{29} = \frac{5+\sqrt{29}}{2}$

$U_{58} = 99+13\sqrt{58}$

$U_{174} = 1451+110\sqrt{174}$

These are involved in fundamental solutions to Pell equations.  For example, for $x^2-58y^2 = -1$, it is {x, y} = {99, 13}, (see the values above), while for $x^2-174y^2 = 1$ it is {x, y} = {1451, 110}. Using these solutions to Pell equations, then,

$\pi \approx \frac{1}{\sqrt{58}} \ln \Big[ 2^6 (U_{29})^{12} \Big]$

$\pi \approx \frac{1}{2\sqrt{58}} \ln \left[ 2^9 \left((U_2)^3 U_{29} \sqrt{U_{58}} \,\right)^6 \right]$

$\pi \approx \frac{1}{3\sqrt{58}} \ln \left[ 2^6 (U_{29})^{12} (U_{174})^2 \left( \sqrt{\frac{9+3\sqrt{6}}{4} } + \sqrt{\frac{5+3\sqrt{6}}{4}}\right)^{24}\right]$

$\pi \approx \frac{1}{4\sqrt{58}} \ln \Big[ 2^9 \left((U_2)^3 U_{29} \sqrt{2U_{58}} \,\right)^3 \left(\sqrt{v+1} +\sqrt{v}\right)^{12}\Big]$

where,

$v = 2^{-1/2}(U_2)^6(U_{29})^3$

Nice, isn’t it?  The second to the last approximation is by Ramanujan which is accurate to 31 digits, while the last is by this author and is accurate to 42 digits.  (Can anyone find a nice expression for the next step? )  The expression inside the log function is the exact value of,

$\left(\frac{\eta(\tau)}{\eta(2\tau)}\right)^{24}$

where $\eta(\tau)$ is the Dedekind eta function, and $\tau = \frac{\sqrt{-58}}{2}$, $\tau = \frac{2\sqrt{-58}}{2}$$\tau = \frac{3\sqrt{-58}}{2}$$\tau = \frac{4\sqrt{-58}}{2}$, respectively.

The fundamental discriminant d = -4∙58 has class number h(d) = 2.  Another one with the same class number is d = -4∙37.  Hence, given,

$U_{37} = 6+\sqrt{37}$

$U_{111} = 295+28\sqrt{111}$

then,

$\pi \approx \frac{1}{\sqrt{37}} \ln \Big[ 2^6 (U_{37})^{6} \Big]$

$\pi \approx \frac{1}{3\sqrt{37}} \ln \left[ 2^6 (U_{37})^{6} (U_{111})^2 \left( \sqrt{\frac{37+20\sqrt{3}}{4} } + \sqrt{\frac{33+20\sqrt{3}}{4}}\right)^{12}\right]$

where the expression inside the log function is now the absolute value of the eta quotient at $\tau = \frac{1+\sqrt{-37}}{2}$ and $\tau = \frac{1+3\sqrt{-37}}{2}$.


text: Abstract:We discuss two ways to construct standard probability measures, called push-down measures, from internal probability measures. We show that the Wasserstein distance between an internal probability measure and its push-down measure is infinitesimal. As an application to standard probability theory, we show that every finitely-additive Borel probability measure $P$ on a separable metric space is a limit of a sequence of countably-additive Borel probability measures $\{P_n\}_{n\in \mathbb{N}}$ in the sense that $\int f \,{\rm d} P=\lim_{n\to \infty} \int f\, {\rm d} P_n$ for all bounded uniformly continuous real-valued function $f$ if and only if the space is totally bounded.


text:
# Profile René F. Swarttouw

René Swarttouw (b. 1964 in The Hague, The Netherlands) is Assistant Professor in the Department of Mathematics of the VU University, Amsterdam, The Netherlands.

Swarttouw is mainly a teacher of mathematics and has published a few papers on special functions and orthogonal polynomials. He is coauthor of the book Hypergeometric Orthogonal Polynomials and Their $q$-Analogues (Koekoek, Lesky, and Swarttouw (2010)).

Swarttouw is coauthor of the following DLMF chapter


text:
TITLE

# A note on equi-integrability in dimension reduction problems

AUTHOR(S)
Braides, Andrea; Zeppieri, Caterina
PUB. DATE
June 2007
SOURCE
Calculus of Variations & Partial Differential Equations;Jun2007, Vol. 29 Issue 2, p231
SOURCE TYPE
DOC. TYPE
Article
ABSTRACT
In the framework of the asymptotic analysis of thin structures, we prove that, up to an extraction, it is possible to decompose a sequence of â€˜scaled gradientsâ€™ $${\left(\nabla_\alpha u_\varepsilon\big|\frac{1}{\varepsilon}\nabla_\beta u_\varepsilon\right)}$$ (where $$\nabla_\beta$$ is the gradient in the k-dimensional â€˜thin variableâ€™ x Î²) bounded in $${L^p(\Omega;\mathbb{R}b^{m\times n})}$$ (1 < p < + âˆž) as a sum of a sequence $${\left(\nabla_\alpha v_\varepsilon\big|\frac{1}{\varepsilon}\nabla_\beta v_\varepsilon\right)}$$ whose p-th power is equi-integrable on Î© and a â€˜restâ€™ that converges to zero in measure. In particular, for k = 1 we recover a well-known result for thin films by Bocea and Fonseca (ESAIM: COCV 7:443â€“470; 2002).
ACCESSION #
24476003

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### Communities

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Rigorous Science

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text:
## Introduction

Berry curvature and orbital magnetic moment are two pseudovectors that are observed in the presence of spatial inversion asymmetry. These enable us to control valley contrasting phenomena in two-dimensional materials1,2,3; therefore, they are essential in the field of valleytronics. The presence of the Berry curvature induces electrons to have anomalous velocity perpendicular to an applied electric field, thus exhibiting the valley Hall effect. The valley Hall effect has been demonstrated at the gapped Dirac band such as gapped bilayer graphene and monolayer graphene/h-BN moiré superlattice4,5,6. In contrast to this, an orbital magnetic moment will induce an energy shift under the application of the perpendicular magnetic field; this is known as valley Zeeman splitting and has been demonstrated in gapped bilayer graphene7,8,9 as well as transition metal dichalcogenides10. Both Berry curvature and orbital magnetic moment, in principle, strongly depends on the local structure of the band. Here we choose graphene/hexagonal-boron nitride (h-BN) as a model system to investigate the orbital magnetic moment in relation to the band structure.

## Results

### Band structure and orbital magnetic moment in graphene/h-BN

A pristine monolayer graphene is an inversion symmetric crystal (Fig. 1a). At the corner of the hexagonal Brillouin zone, different bands are connected at the Dirac point (DP) located at the K- and K′-point, and the energies between different valleys (K and K′) are degenerated. When graphene is transferred on a h-BN insulator through van der Waals force, two significant changes occur in the band structure. First, the interaction between graphene and h-BN breaks the spatial inversion symmetry of graphene. Second, the lattice mismatch between two crystals generates a moiré pattern (Fig. 1b). Owing to the moiré superlattice potential, the band structure of graphene is completely reconstructed11,12,13, as shown in Fig. 1c, d, and calculated at the K-point using an effective continuum theory12. Herein, to emphasize the effect of inversion asymmetry, two different calculations are performed: one that holds spatial inversion symmetry (inversion symmetric model shown in Fig. 1c) and one that uses an inversion asymmetric (Fig. 1d) model (see Supplementary Note 1). The inversion asymmetric model was demonstrated to comprehensively match the experimental results on band structure as well as Landau quantization11,12,13,14,15,16. The plot of energy with respect to the density of states (DOS) derived from Fig. 1d is shown in Fig. 1e. Because of the moiré potential, series of sub-DPs, such as secondary-DPs (SDPs) or tertiary-DPs (TDPs), are generated at both low- and high-energy sides of the main DP11,12,13,17,18,19. These result in local minima in the DOS (Fig. 1e). Moreover, saddle points between the DPs, known as van Hove singularities (vHSs), are represented as local maxima. The most striking effect of inversion asymmetry indicated in Fig. 1c, d is the opening of the band gap at several points11,12,13, which is indicated by the red- and blue-dashed circles. First, a band gap is opened at the main DP (i.e., the K-point in the first band) and both electron- and hole-side SDPs (i.e., the X-point; blue-dashed circles in Fig. 1d). This gapped DP has been reported to exhibit a finite Berry curvature4,5,6.

In this study, we observed that there was another point (i.e., a second vHS in the hole-side located at the Y-point (red-dashed circle in Fig. 1d)) that exhibited a pronounced effect of inversion asymmetry; this point has not been discussed to date. In the symmetric model (Fig. 1c), this point is a point of contact between the hole-side of the second and third bands. Inversion asymmetry induces a significant gap opening; as a result, this point becomes the vHS point in the inversion asymmetric model. This is a unique property of this particular vHS because the influence of inversion symmetry on other vHSs from the first to the third bands is limited. The opening of a gap with inversion asymmetry naturally results in the generation of an orbital magnetic moment m(k) at this vHS. This is an analog of a spin angular moment of the electron. However, m(k) is orbital in nature, and it is responsible for the splitting between the K and K′ valley instead of the up- and down-spin. A general form of the m(k) is expressed as the following equation2,3,9:

$${\mathbf{m}}\left( {\mathbf{k}} \right) = - {\mathrm{i}}\left( {{\mathrm{e}}/2\hbar } \right) \times \left\langle {\nabla _{\mathrm{k}}u\left| { \times \left[ {H\left( {\mathbf{k}} \right) - \varepsilon \left( {\mathbf{k}} \right)} \right]} \right|\nabla _{\mathrm{k}}u} \right\rangle ,$$
(1)

where $$\left. {\left| {u\left( {\mathbf{k}} \right)} \right.} \right\rangle$$ is the periodic part of the Bloch function, H(k) is the Bloch Hamiltonian, and ε(k) is the dispersion of band. We calculated m(k) for the band structure in Fig. 1d and obtained m(k) with respect to the wavevector of the first, second, and third bands in the hole-side of the main DP, as depicted in Fig. 1f–h, respectively. m(k) is large for the gapped DPs, including both the main DP and SDPs3. Moreover, the calculation shows that there is a noticeably large m(k) presented at the second vHS(h; i.e., the Y-point in the second band). The second vHS(h) yields an m(k) value of 66 μB, corresponding to a valley g-factor of ~130. This value is noticeably large as an orbital magnetic moment of vHS, considering that m(k) is negligibly small at other vHSs such as Y-point in the first band (see Supplementary Note 2 for the detail comparison).

### All-electrical magnetothermoelectric measurement

To verify the existence of orbital moment at the vHS, we performed magnetothermoelectric measurement, as schematically illustrated in Fig. 2a; the optical micrograph of the device is shown in Fig. 2b. Two different flakes of graphene encapsulated with h-BN were placed on the SiO2/doped-Si substrate. We created trench in the h-BN to reduce capacitive coupling between the two graphene flakes (see Methods section and Supplementary Note 3). The graphene depicted on the right has a graphite gate (GrG), which can apply a local back-gate voltage of VR to control the carrier density nR. The carrier density of the graphene on the left nL is controlled using the back-gate voltage VL. The lattice of the graphene on the left is aligned with the underlying h-BN substrate, to ensure a large period of moiré potential. In contrast, the period of the graphene on the right is small because of the significant rotational misalignment between graphene and h-BN. As illustrated in Fig. 2c, the nonaligned graphene is used as a local heater by applying current to induce Joule heating. The heat radiated from the nonaligned graphene induces heating of the aligned graphene, thereby creating a temperature gradient perpendicular to the channel of the aligned graphene. In the presence of this temperature gradient and the magnetic field perpendicular to the plane B, thermoelectric voltage Vind is generated along the channel direction of the aligned graphene, due to the Nernst effect20,21,22,23,24. This voltage can be detected using the metal electrode connected at both ends of the graphene. The thermopower is significantly enhanced at the vHS as it is demonstrated by photo-Nernst measurement25. Therefore, our all-electrical magnetothermoelectric measurement provides sensitive probes for vHSs.

The two-terminal resistance R of the aligned graphene is measured under a sweep of nL at 2.0 K; the results are plotted in Fig. 2d. In addition to the resistance peak caused by the main DP at nL = 0, other peaks from the SDP at the electron- and the hole-side at nL = +2.20 and −2.25 × 1012 cm−2, respectively, are visible. Although its origin is not clear at this moment, there is a slope background in R that increases towards the left in the figure; we infer this is related to the inhomogeneity of metal/graphene contact. Based on these nL values, the period of moiré potential λ is determined using the relationship $$n_{{\mathrm{SDP}}} = 8/\left( {\sqrt 3 \lambda ^2} \right)$$ where nSDP represents the carrier density at the resistance peak position of the SDP14,15,16. Using the obtained λ value of ~14 nm, we determined the lattice alignment angle between graphene and h-BN as θ ~ 0°17. During thermoelectric measurement, a constant power P = 1 mW is applied to the nonaligned graphene using a source meter, while maintaining its carrier density at charge neutrality nR = 0 at B = 0 and quantum Hall filling factor ν = −6 such that nR = 6eB/h in the magnetic field, where e depicts electron charge and h the Planck constant. This maintain the two-terminal resistance of heater graphene to be kept ~10 kΩ throughout the measurement from no magnetic field to high magnetic field (see Supplementary Note 4). Under this condition, the increase in the average temperature of the aligned graphene is estimated to be ~10 K. Vind as a function of nL is measured at different magnetic fields B, and the results are presented in Fig. 2e by the solid black lines. For B = 0 T, the Vind signal is negligibly small. The small signal around nL = +2.20 and −2.25 × 1012 cm−2 at B = 0 T can be attributed to a Seebeck effect of SDP, originating from an asymmetry of the contacts. On increasing B, initially, the Vind signals that appeared at the main DP and the SDP had similar amplitudes. Thereafter, the Vind signal at the DPs exhibits a peak in positive B and a dip in negative B. For comparison, the Vind signal obtained from nonaligned graphene/h-BN device is provided in Supplementary Note 5. In addition, a larger signal appears at a position away from the SDPs, as indicated by the arrows in Fig. 2e. Evidently, the signs of the values of Vind at these locations differ from those of the DPs, such that the signal from the three DPs exhibit dips in negative B values, whereas the signals indicated by the arrows exhibit peaks. These results are compared to the calculated DOS derived from Fig. 1e plotted versus carrier density n, as depicted in Fig. 2f. Here, the carrier density was calculated by integrating DOS such that $$n\left( {E_F} \right) = {\int}_0^{E_F} {{\mathrm{DOS}}\left( E \right)dE}$$ The unit of this axis is normalized using the total number of electron states divided by the area of a filled Bloch band n0. Therefore, n/n0 = ±4 and 8 correspond to the SDP and TDP, respectively. A comparison of Fig. 2d–f indicates that additional large signals of Vind are observed at three vHSs, namely the first vHS(e), first vHS(h), and second vHS(h). The first vHSs on both the electron- and hole-side had previously been investigated25; here, we investigated the second vHS(h) which was observed26,27 but not discussed before. In addition to these, it is noteworthy that the Vind signal largely increases around nL = −4.65 × 1012 cm−2 in Fig. 2e due to the presence of TDP, the location of which can be seen from Fig. 2f; however, a detail investigation of this TDP is difficult on this device since the demonstration of carrier density higher than that of Fig. 2e is limited by the breakdown of SiO2 dielectric.

The Vind versus nL are measured in a magnetic field from B = + 0 to 2.23 T and results are shown in Fig. 2g. The magnetothermopower from the second vHS(h) and first vHS(h) exhibits a considerably different behavior. Their positions are depicted by a solid magenta circle and a blue circle, respectively, in Fig. 2g. The dip-shape signal generated from the second vHS(h) exhibits significant splitting with increasing B (see also refs. 26,27), whereas the Vind signals from other vHSs, i.e., the first vHS(h) and the first vHS(e) (see Fig. 3a, b), did not exhibit any evident splitting with increasing B.

### Valley splitting of the thermoelectric voltage at vHS

To investigate the origin of the magnetic field-induced splitting in Vind at the second vHS(h), we present a comparison between the experimental values and the calculated DOS in aligned graphene/h-BN obtained from the continuum model12. In Fig. 3a, we depict the plot of Vind as a function of nL and B plotted for +B and −B. The signals arising from three vHSs—the first vHS(e), first vHS(h), and second vHS(h)—are extracted from the data, and plotted with green, blue, and magenta solid circles, respectively in Fig. 3b (details of the extraction procedure as well as the Vind data in high magnetic field region are presented in Supplementary Note 6). We present a calculated DOS as a function of n/n0 and B in Fig. 3c–e. The calculated DOS as a function of energy E is also provided in Supplementary Note 7. Here, we plotted the total DOS, including the DOS for both K and K′ valleys (Fig. 3c), DOS for K valley (Fig. 3d), and DOS for K′ valley (Fig. 3e), separately. A good coincidence was observed between Fig. 3a, c; the B-induced splitting of vHSs as well as the Landau fan diagram of the DPs were completely reproduced by the calculation. By plotting the K- and K′-valley contributions to the DOS separately in Fig. 3d, e, we observed that the DOS peak at the second vHS(h) for the K(K′) valley changed to the larger (smaller) n/n0 value with the magnetic field. Furthermore, we plotted individual traces of Fig. 3d, e in Fig. 3f, for a comparison with the experimental data of Vind shown in Fig. 3g (The data plotted in Fig. 3g is same data as Fig. 2g). These data also exhibited good agreement with each other. Therefore, we conclude that the magnetic field induced splitting of the second vHS(h) indicated by our results is caused by the large magnetic field-induced valley splitting.

## Discussion

The significant valley splitting corresponding to a valley g-factor of ~130 extracted from Fig. 1g indicates the emergence of a large m(k) at the second vHS(h). To obtain a finite m(k) from Eq. 1, the matrix element for the inter sub-band coupling should be non-zero28 (see Supplementary Note 8 for a detailed discussion). This can only be achieved at a point of contact between the bands (i.e., DP) in the graphene/h-BN under the inversion symmetric model. As seen from the comparison between Fig. 1c, d, the second vHS(h) in Fig. 1d is a point of contact between hole-side second and third band at Y-point under inversion symmetry; thus, it is a sub-DP. This sub-DP is gapped under the introduction of inversion asymmetry and becomes a vHS point. Therefore, the inter sub-band coupling at this vHS point is still non-zero, even after the opening of the gap. This hidden DP nature of the Y-point at the hole-side second and third band demonstrates the emergence of an orbital magnetic moment at the second vHS(h) in the graphene/h-BN moiré superlattice. Our all-electrical magnetothermoelectric measurements unveil this unique band property of graphene/h-BN moiré superlattice. Since there is a growing interest in twisted bilayer or twisted multilayer graphene29,30,31,32,33,34,35,36, this method may also provide a sensitive probe to detect orbital magnetic moment with respect to different band structures of such new material systems.

## Methods

### Sample fabrication

A device was fabricated using sequential dry release transfer of an individual flake. Two different flakes of monolayer graphene were successively transferred on top of the h-BN/graphite structure on the SiO2/doped-Si substrate. Finally, both flakes of graphene were encapsulated with another h-BN. The flake transfer was performed using a poly(propylene) carbonate (PPC)-based dry release transfer method37,38, and the thickness of h-BN was chosen to be ~30–40 nm. Using e-beam (EB) lithography and reactive ion etching employing a mixture of CF4 and Ar gases, a trench was fabricated in the h-BN located between the two graphene flakes. Thus, the connection through h-BN between the two graphene flakes was eliminated. Finally, a contact pattern was created using an EB lithography with PMMA resist, and a 90-nm-thick Au/5-nm-thick Cr stack was deposited using EB evaporation. It should be noted that during device fabrication, the device was annealed at 350 °C under an Ar/3% H2 atmosphere after each flake transfer. This annealing caused the transferred graphene flakes to rotate within the plane toward the preferred orientation with respect to the adjacent h-BN39,40. In the device presented in Fig. 2a, b, the graphene on the left is almost perfectly aligned with h-BN, to generate moiré superlattice potential; however, the graphene on the right was not aligned.

### Experimental setup

A liquid He-cooled variable temperature cryostat equipped with a superconducting magnet was used for transport measurement. The differential resistance of graphene was measured by applying an AC current Iac = 10 nA, with a frequency of 18 Hz; the AC voltage was measured using a lock-in amplifier. A source measure unit (Keithley 2400) was used to apply constant power to the heater graphene.


text:
COMPSTAT 2018: Start Registration
View Submission - COMPSTAT2018
A0430
Title: A maximum likelihood estimation of Brun's constant under the twin primes distribution Authors:  Ryuichi Sawae - Okayama University of Science (Japan) [presenting]
Daisuke Ishii - Okayama University of Science (Japan)
Abstract: The results of research on the distribution of prime numbers are numerous, including a very important theorem in number theory. On the other hand, twin primes are pairs of primes of the form \$(p, p+2)\$, but it has not been proven that there are infinite number of twins primes until now. Let the set \$K_2\$ be \$\{(3, 5), (5, 7), (11, 13), (17, 19),\ldots\}\$ of twin-prime pairs. Then, although the sum of the reciprocals of all the primes is divergent, the sum of the reciprocals of \$K_2\$, i.e. \$B_2 = (1/3 + 1/5) + (1/5 + 1/7) + (1/11 + 1/13) + (1/17 + 1/19) + \ldots\$ , is convergent. The limit of this sum is known as Brun's sum or Brun's constant. For the maximum likelihood estimation of Brun's constant, we list up the twin primes by a fast Eratosthenes sieve, we use a fast computer algorithm for the reciprocals calculation and the error analysis for the Hardy-Littlewood approximation.


text:
# On the regularity of the generalised golden ratio function

Abstract : Given a finite set of real numbers $A$, the generalised golden ratio is the unique real number $\mathcal{G}(A) > 1$ for which we only have trivial unique expansions in smaller bases, and have non-trivial unique expansions in larger bases. We show that $\mathcal{G}(A)$ varies continuously with the alphabet $A$ (of fixed size). What is more, we demonstrate that as we vary a single parameter $m$ within~$A$, the generalised golden ratio function may behave like $m^{1/h}$ for any positive integer $h$. These results follow from a detailed study of $\mathcal{G}(A)$ for ternary alphabets, building upon the work of Komornik, Lai, and Pedicini (2011). We give a new proof of their main result, that is we explicitly calculate the function $\mathcal{G}(\{0,1,m\})$. (For a ternary alphabet, it may be assumed without loss of generality that $A = \{0,1,m\}$ with $m\in(1,2)]$.) We also study the set of $m \in (1,2]$ for which $\mathcal{G}(\{0,1,m\})=1+\sqrt{m},$ we prove that this set is uncountable and has Hausdorff dimension~$0$. We show that the function mapping $m$ to $\mathcal{G}(\{0,1,m\})$ is of bounded variation yet has unbounded derivative. Finally, we show that it is possible to have unique expansions as well as points with precisely two expansions at the generalised golden ratio.
Type de document :
Article dans une revue
Bulletin of the London Mathematical Society, London Mathematical Society, 2017, 49 (1), pp.58 - 70. 〈10.1112/blms.12002〉
Domaine :

https://hal.archives-ouvertes.fr/hal-01620763
Contributeur : Wolfgang Steiner <>
Soumis le : vendredi 20 octobre 2017 - 23:11:07
Dernière modification le : jeudi 11 janvier 2018 - 06:27:38

### Citation

Simon Baker, Wolfgang Steiner. On the regularity of the generalised golden ratio function. Bulletin of the London Mathematical Society, London Mathematical Society, 2017, 49 (1), pp.58 - 70. 〈10.1112/blms.12002〉. 〈hal-01620763〉

### Métriques

Consultations de la notice


text:
# deconv

Deconvolution and polynomial division

## Syntax

``````[q,r] = deconv(u,v)``````

## Description

example

``````[q,r] = deconv(u,v)``` deconvolves a vector `v` out of a vector `u` using long division, and returns the quotient `q` and remainder `r` such that `u = conv(v,q) + r`. If `u` and `v` are vectors of polynomial coefficients, then deconvolving them is equivalent to dividing the polynomial represented by `u` by the polynomial represented by `v`.```

## Examples

collapse all

Create two vectors `u` and `v` containing the coefficients of the polynomials $2{x}^{3}+7{x}^{2}+4x+9$ and ${x}^{2}+1$, respectively. Divide the first polynomial by the second by deconvolving `v` out of `u`, which results in quotient coefficients corresponding to the polynomial $2x+7$ and remainder coefficients corresponding to $2x+2$.

```u = [2 7 4 9]; v = [1 0 1]; [q,r] = deconv(u,v)```
```q = 1×2 2 7 ```
```r = 1×4 0 0 2 2 ```

## Input Arguments

collapse all

Input vectors, specified as either row or column vectors. `u` and `v` can be different lengths or data types.

• If one or both of `u` and `v` are of type single, then the output is also of type single. Otherwise, `deconv` returns type `double`.

• The lengths of the inputs should generally satisfy `length(v) <= length(u)`. However, if `length(v) > length(u)`, then `deconv` returns the outputs as `q = 0` and `r = u`.

Data Types: `double` | `single`
Complex Number Support: Yes

## Output Arguments

collapse all

Quotient, returned as a row or column vector such that ```u = conv(v,q)+r```.

Data Types: `double` | `single`

Remainder, returned as a row or column vector such that ```u = conv(v,q)+r```.

Data Types: `double` | `single`

## Version History

Introduced before R2006a


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🤖

# Fall 2022

## LIN/CSE 667: Advanced Topics in Computational Linguistics

Instructor Name: Dr. Cassandra Jacobs Class Day and Time: MW 9:00-10:20AM Number of Credits: 1-3 units Email Address: cxjacobs@buffalo.edu Office Location: 614 Baldy Hall Office Hours: On Zoom by appointment

### Course description

This course aims to provide students with an overview of the key areas which make up the field called Computational Linguistics, an understanding of the major challenges of the field as well as the major application areas for language processing techniques, and the skills to implement fundamental language processing algorithms. This course is dual listed between CSE 667LEC and LIN 667LEC.

### Required Text and Materials

All reading materials will be made available on UBLearns as well as the course webpage and will consist primarily of journal articles or conference proceedings.

## Theme: Linguistic probes and linguistic representations in large language models

The landscape for natural language processing (NLP) has changed dramatically in the past decade with the explosion of neural language models for downstream NLP tasks. A persistent challenge, however, is that prior methods, e.g., structured perceptron approaches, random forests, etc., were significantly more transparent from the perspective of the modeler. Given the nature of the model representations, it was trivial to inspect why a classifier made the decision it did and cheap and easy to perform causal inference by performing feature ablation. While deep learning largely rejects interpretability, it is of great importance to scientists and practitioners of NLP to be able to understand what is encoded in neural language models concretely, e.g., linguistic or statistical regularities. This course is a deep dive into the literature on probing for linguistic factors (e.g., syntactic structure) in the decisions of neural language models, in addition to the statistical regularities that they encode (e.g., linguistic bias).

## Goals

Course Learning OutcomeInstructional MethodsAssessment Methods
Obtain familiarity with probing methods to test for the presence of linguistic structure in neural language model representationsClassroom presentation; leading class discussion; course reading posts; classroom participationCompletion rubrics
Understand the effects of including linguistic features or metadata as part of neural language model feature engineering; better understand types of probes and fine-tuning methodsCourse readings and classroom participationForum participation on Blackboard/UBLearns
Obtain fluency in communication of scientific results and discussion of scientific topicsClassroom presentation; leading class discussion; final projectPresentation rubric; forum participation on Blackboard/UBLearns
Obtain competency in the design and completion of a research projectFinal projectCompletion rubrics

## Weekly class/lecture structure

The course is primarily conducted using a mix of instructor-guided discussion and student-led discussion. Each week on Wednesdays, we will discuss a broad linguistic domain relevant to doing NLP with deep learning, why interpretability is critical for that domain, and the technological tools that have been developed to probe different structures in that linguistic domain. Particular attention will be paid to the nature of the datasets leveraged or generated during lecture. On Mondays, two students will each present one paper from the current theme. Then, by the following Wednesday, every non-presenter will submit a review for papers from the preceding week.

1WednesdayTopic 1: Instructor-guided discussion
2MondayTopic 1: Student-led discussion
2WednesdayTopic 2: Instructor-guided discussion
3MondayTopic 2: Student-led discussion
3WednesdayTopic 3: Instructor-guided discussion (Topic 3)
Topic 1: Paper reviews
4MondayTopic 3: Student-led discussion
4WednesdayTopic 4: Instructor-guided discussion
Paper reviews (Topic 2)

40%Weekly reviews posted to UBLearns
20%Student-led presentation & discussion
20%Final project paper
10%Final project presentation
10%Participation in class and monthly self-assessments

## Weekly paper reviews

As part of this course, you will be asked to provide an ACL Rolling Review-style paper review for one of two selected papers from each week. The goal of these assignments is to encourage you to analyze research work from a scientific perspective. Do not be afraid if the task is challenging — it is expected that the quality of your reviews will improve over the course of the semester as you gain familiarity with the literature and obtain feedback on your reviews. More details are available here: https://aclrollingreview.org/reviewform

• Week 2 (September 5-): Interpreting Neural NLP models
• September 5: Labor Day! No class!
• September 7: Instructor-guided discussion on Interpreting Neural NLP - Discussion due September 6 at 10pm
• Sarthak Jain and Byron C. Wallace. 2019. Attention is not Explanation. In Proceedings of the 2019 Conference of the North American Chapter of the Association for Computational Linguistics: Human Language Technologies, Volume 1 (Long and Short Papers), pages 3543–3556, Minneapolis, Minnesota. Association for Computational Linguistics. https://aclanthology.org/N19-1357/
• Sarah Wiegreffe and Yuval Pinter. 2019. Attention is not not Explanation. In Proceedings of the 2019 Conference on Empirical Methods in Natural Language Processing and the 9th International Joint Conference on Natural Language Processing (EMNLP-IJCNLP), pages 11–20, Hong Kong, China. Association for Computational Linguistics. https://aclanthology.org/D19-1002/
• Tenney, I., Das, D., & Pavlick, E. (2019, July). BERT Rediscovers the Classical NLP Pipeline. In Proceedings of the 57th Annual Meeting of the Association for Computational Linguistics (pp. 4593-4601). https://aclanthology.org/P19-1452/
• Durrani, N., Sajjad, H., Dalvi, F., & Belinkov, Y. (2020, November). Analyzing Individual Neurons in Pre-trained Language Models. In Proceedings of the 2020 Conference on Empirical Methods in Natural Language Processing (EMNLP) (pp. 4865-4880). https://aclanthology.org/2020.emnlp-main.395/
• Week 3 (September 12-): Syntactic Structure 1
• September 12: Instructor-guided discussion on grammatical categories & in-class discussion of week 2 readings
• September 14: Instructor-guided discussion on larger syntactic structures - Discussion due September 13 at 10pm - In-class “peer review” walkthrough for Week 5 assignment
• Kim, N., & Smolensky, P. (2021, February). Testing for Grammatical Category Abstraction in Neural Language Models. In Proceedings of the Society for Computation in Linguistics 2021 (pp. 467-470). https://aclanthology.org/2021.scil-1.59/ - Instructor review / presentation demo
• Kim, N., Rawlins, K., Van Durme, B., & Smolensky, P. (2019, July). Predicting the argumenthood of English prepositional phrases. In Proceedings of the AAAI Conference on Artificial Intelligence (Vol. 33, No. 01, pp. 6578-6585). https://dl.acm.org/doi/abs/10.1609/aaai.v33i01.33016578
• Marius Mosbach, Stefania Degaetano-Ortlieb, Marie-Pauline Krielke, Badr M. Abdullah, and Dietrich Klakow. 2020. A Closer Look at Linguistic Knowledge in Masked Language Models: The Case of Relative Clauses in American English. In Proceedings of the 28th International Conference on Computational Linguistics, pages 771–787, Barcelona, Spain (Online). International Committee on Computational Linguistics. https://aclanthology.org/2020.coling-main.67
• Gulordava, K., Bojanowski, P., Grave, É., Linzen, T., & Baroni, M. (2018, June). Colorless Green Recurrent Networks Dream Hierarchically. In Proceedings of the 2018 Conference of the North American Chapter of the Association for Computational Linguistics: Human Language Technologies, Volume 1 (Long Papers) (pp. 1195-1205). https://aclanthology.org/N18-1108/
• Week 4 (September 19-): Syntactic Structure 2
• September 19: Instructor-guided discussion on syntactic processes
• [No paper review due this week]
• September 21: Instructor-guided discussion on syntactic processes - Discussion due September 20 at 10pm
• Aina, L., & Linzen, T. (2021, November). The Language Model Understood the Prompt was Ambiguous: Probing Syntactic Uncertainty Through Generation. In Proceedings of the Fourth BlackboxNLP Workshop on Analyzing and Interpreting Neural Networks for NLP (pp. 42-57). https://aclanthology.org/2021.blackboxnlp-1.4/ - Arthur Domino
• Finlayson, M., Mueller, A., Gehrmann, S., Shieber, S. M., Linzen, T., & Belinkov, Y. (2021, August). Causal Analysis of Syntactic Agreement Mechanisms in Neural Language Models. In Proceedings of the 59th Annual Meeting of the Association for Computational Linguistics and the 11th International Joint Conference on Natural Language Processing (Volume 1: Long Papers) (pp. 1828-1843). https://aclanthology.org/2021.acl-long.144/
• Zhang, Y. (2020, November). Latent Tree Learning with Ordered Neurons: What Parses Does It Produce?. In Proceedings of the Third BlackboxNLP Workshop on Analyzing and Interpreting Neural Networks for NLP (pp. 119-125). https://aclanthology.org/2020.blackboxnlp-1.11/
• Adina Williams, Andrew Drozdov*, Samuel R. Bowman; Do latent tree learning models identify meaningful structure in sentences?. Transactions of the Association for Computational Linguistics 2018; 6 253–267. https://doi.org/10.1162/tacl_a_00019
• Week 5 (September 26-): Multilingual representations
• September 26: Student-led discussion on Syntactic Structure 2 - Recorded, watch on your own
• Arthur Domino
• September 26: Syntactic Structure 1 paper review due
• September 28: Instructor-guided discussion on multilingual models - Recorded, watch on your own
• Ethan A. Chi, John Hewitt, and Christopher D. Manning. 2020. Finding Universal Grammatical Relations in Multilingual BERT. In Proceedings of the 58th Annual Meeting of the Association for Computational Linguistics, pages 5564–5577, Online. Association for Computational Linguistics. https://aclanthology.org/2020.acl-main.493/ - Sabiha Shaik
• Taraka Rama, Lisa Beinborn, and Steffen Eger. 2020. Probing Multilingual BERT for Genetic and Typological Signals. In Proceedings of the 28th International Conference on Computational Linguistics, pages 1214–1228, Barcelona, Spain (Online). International Committee on Computational Linguistics. https://aclanthology.org/2020.coling-main.105/
• Tanti, M., van der Plas, L., Borg, C., & Gatt, A. (2021, November). On the Language-specificity of Multilingual BERT and the Impact of Fine-tuning. In Proceedings of the Fourth BlackboxNLP Workshop on Analyzing and Interpreting Neural Networks for NLP (pp. 214-227). https://aclanthology.org/2021.blackboxnlp-1.15/
• Blevins, T., & Zettlemoyer, L. (2022). Language Contamination Explains the Cross-lingual Capabilities of English Pretrained Models. arXiv preprint arXiv:2204.08110. https://arxiv.org/abs/2204.08110 - Sarah Sues
• Week 6 (October 3-): Semantic Knowledge 1
• October 3: Student-led discussion on Multilingual representations
• Sarah Sues
• Sabiha Shaik
• October 3: Syntactic Structure 2 paper review due
• October 5: Instructor-guided discussion on lexical semantics - Discussion due October 4 at 10pm
• Ivan Vulić, Edoardo Maria Ponti, Robert Litschko, Goran Glavaš, and Anna Korhonen. 2020. Probing Pretrained Language Models for Lexical Semantics. In Proceedings of the 2020 Conference on Empirical Methods in Natural Language Processing (EMNLP), pages 7222–7240, Online. Association for Computational Linguistics. https://aclanthology.org/2020.emnlp-main.586
• Aina Garí Soler and Marianna Apidianaki. 2020. BERT Knows Punta Cana is not just beautiful, it’s gorgeous: Ranking Scalar Adjectives with Contextualised Representations. In Proceedings of the 2020 Conference on Empirical Methods in Natural Language Processing (EMNLP), pages 7371–7385, Online. Association for Computational Linguistics. https://aclanthology.org/2020.emnlp-main.598/ - Benjamin Conrow-Graham
• Week 7 (October 10-): Semantic Knowledge 2
• October 10: Student-led discussion on Semantic Knowledge 1
• Jacob Springborn
• Benjamin Conrow-Graham
• October 10: Multilingual representations paper review due
• October 12: Instructor-guided discussion on reasoning - Discussion due October 11 at 10pm
• Nora Kassner, Benno Krojer, and Hinrich Schütze. 2020. Are Pretrained Language Models Symbolic Reasoners over Knowledge?. In Proceedings of the 24th Conference on Computational Natural Language Learning, pages 552–564, Online. Association for Computational Linguistics. https://aclanthology.org/2020.conll-1.45/
• Bhagavatula, C., Le Bras, R., Malaviya, C., Sakaguchi, K., Holtzman, A., Rashkin, H., ... & Choi, Y. (2019, September). Abductive Commonsense Reasoning. In International Conference on Learning Representations. https://openreview.net/forum?id=Byg1v1HKDB
• Nora Kassner and Hinrich Schütze. 2020. Negated and Misprimed Probes for Pretrained Language Models: Birds Can Talk, But Cannot Fly. In Proceedings of the 58th Annual Meeting of the Association for Computational Linguistics, pages 7811–7818, Online. Association for Computational Linguistics. https://aclanthology.org/2020.acl-main.698 - Venkata Sai Rohit Ayyagari
• Week 8 (October 17-): Fine-tuning
• October 17: Student-led discussion on Semantic Knowledge 2
• Venkata Sai Rohit Ayyagari
• October 17: Semantic Knowledge 1 paper review due
• October 19: Instructor-guided discussion on fine-tuning language models - Discussion due October 18 at 10pm
• Merchant, A., Rahimtoroghi, E., Pavlick, E., & Tenney, I. (2020, November). What Happens To BERT Embeddings During Fine-tuning?. In Proceedings of the Third BlackboxNLP Workshop on Analyzing and Interpreting Neural Networks for NLP (pp. 33-44). https://aclanthology.org/2020.blackboxnlp-1.4/ - Sean Afridi
• Marius Mosbach, Anna Khokhlova, Michael A. Hedderich, and Dietrich Klakow. 2020. On the Interplay Between Fine-tuning and Sentence-level Probing for Linguistic Knowledge in Pre-trained Transformers. In Findings of the Association for Computational Linguistics: EMNLP 2020, pages 2502–2516, Online. Association for Computational Linguistics. https://aclanthology.org/2020.findings-emnlp.227/
• Durrani, N., Sajjad, H., & Dalvi, F. (2021, August). How transfer learning impacts linguistic knowledge in deep NLP models?. In Findings of the Association for Computational Linguistics: ACL-IJCNLP 2021 (pp. 4947-4957). https://aclanthology.org/2021.findings-acl.438/ - Shubham Pandey
• Yu, L., & Ettinger, A. (2021, August). On the Interplay Between Fine-tuning and Composition in Transformers. In Findings of the Association for Computational Linguistics: ACL-IJCNLP 2021 (pp. 2279-2293). https://aclanthology.org/2021.findings-acl.201/
• Week 9 (October 24-): Bias 1
• October 24: Student-led discussion on Fine-tuning
• Sean Afridi
• Shubham Pandey
• October 24: Semantic Knowledge 2 paper review due
• October 26: Instructor-guided discussion on identifying bias - Discussion due October 25 at 10pm
• Ethayarajh, K., Duvenaud, D., & Hirst, G. (2019, July). Understanding Undesirable Word Embedding Associations. In Proceedings of the 57th Annual Meeting of the Association for Computational Linguistics (pp. 1696-1705). https://aclanthology.org/P19-1166/
• Victor Steinborn, Philipp Dufter, Haris Jabbar, and Hinrich Schuetze. 2022. An Information-Theoretic Approach and Dataset for Probing Gender Stereotypes in Multilingual Masked Language Models. In Findings of the Association for Computational Linguistics: NAACL 2022, pages 921–932, Seattle, United States. Association for Computational Linguistics. https://aclanthology.org/2022.findings-naacl.69/
• Zhao, J., Wang, T., Yatskar, M., Ordonez, V., & Chang, K. W. (2018, June). Gender Bias in Coreference Resolution: Evaluation and Debiasing Methods. In Proceedings of the 2018 Conference of the North American Chapter of the Association for Computational Linguistics: Human Language Technologies, Volume 2 (Short Papers) (pp. 15-20). https://aclanthology.org/N18-2003/
• Week 10 (October 31-): Bias 2
• October 31: Class-led discussion on Bias 1 - Happy Halloween! 🎃
• October 31: Fine-tuning paper review due
• November 2: Instructor-guided discussion on debiasing - Discussion due November 1 at 10pm
• Cheng, P., Hao, W., Yuan, S., Si, S., & Carin, L. (2020, September). FairFil: Contrastive Neural Debiasing Method for Pretrained Text Encoders. In International Conference on Learning Representations. https://openreview.net/forum?id=N6JECD-PI5w
• Liang, S., Dufter, P., & Schütze, H. (2020, December). Monolingual and multilingual reduction of gender bias in contextualized representations. In Proceedings of the 28th International Conference on Computational Linguistics (pp. 5082-5093). https://aclanthology.org/2020.coling-main.446/
• Prost, F., Thain, N., & Bolukbasi, T. (2019). Debiasing embeddings for reduced gender bias in text classification. arXiv preprint arXiv:1908.02810. https://aclanthology.org/W19-3810/
• Week 11 (November 7-): Longer spans
• November 7: Class-led discussion on Bias 2
• November 7: Bias 1 paper review due
• November 9: Instructor-guided discussion on longer spans - Discussion due November 8 at 10pm
• Zhu, Z., Pan, C., Abdalla, M., & Rudzicz, F. (2020, November). Examining the rhetorical capacities of neural language models. In Proceedings of the Third BlackboxNLP Workshop on Analyzing and Interpreting Neural Networks for NLP (pp. 16-32). https://aclanthology.org/2020.blackboxnlp-1.3/
• Kim, T., Choi, J., Edmiston, D., & Lee, S. G. (2019, September). Are Pre-trained Language Models Aware of Phrases? Simple but Strong Baselines for Grammar Induction. In International Conference on Learning Representations. https://openreview.net/forum?id=H1xPR3NtPB&amp=1
• Week 12 (November 14-): Contextual representations
• November 14: Class-led discussion on Longer spans
• November 14: Bias 2 paper review due
• November 16: Instructor-guided discussion on contextual word representations - Discussion due November 15 at 10pm
• Mengjie Zhao, Philipp Dufter, Yadollah Yaghoobzadeh, and Hinrich Schütze. 2020. Quantifying the Contextualization of Word Representations with Semantic Class Probing. In Findings of the Association for Computational Linguistics: EMNLP 2020, pages 1219–1234, Online. Association for Computational Linguistics. https://aclanthology.org/2020.findings-emnlp.109
• Ethayarajh, K. (2019, November). How Contextual are Contextualized Word Representations? Comparing the Geometry of BERT, ELMo, and GPT-2 Embeddings. In Proceedings of the 2019 Conference on Empirical Methods in Natural Language Processing and the 9th International Joint Conference on Natural Language Processing (EMNLP-IJCNLP) (pp. 55-65). https://aclanthology.org/D19-1006/ - Vishal Rajasekar
• Week 13 (November 21 -): Contextual representations cont.
• November 21: Student-led discussion of Contextual representations
• Vishal Rajasekar
• November 21: Contextual Representations 1 paper review due
• November 23: Fall break! No class!
• Week 14 (November 28-): Geometry
• November 28: Instructor-guided discussion on metrics for interpretability
• Marvin Kaster, Wei Zhao, and Steffen Eger. 2021. Global Explainability of BERT-Based Evaluation Metrics by Disentangling along Linguistic Factors. In Proceedings of the 2021 Conference on Empirical Methods in Natural Language Processing, pages 8912–8925, Online and Punta Cana, Dominican Republic. Association for Computational Linguistics. https://aclanthology.org/2021.emnlp-main.701/
• Ferrando, J., Gállego, G. I., & Costa-jussà, M. R. (2022). Measuring the Mixing of Contextual Information in the Transformer. arXiv preprint arXiv:2203.04212. https://arxiv.org/abs/2203.04212
• November 28: Contextual Representations 2 paper review due
• November 30: Instructor-guided discussion on geometric properties of word vectors - Discussion due November 29 at 10pm
• Chen, B., Fu, Y., Xu, G., Xie, P., Tan, C., Chen, M., & Jing, L. (2020, September). Probing BERT in Hyperbolic Spaces. In International Conference on Learning Representations. https://openreview.net/forum?id=17VnwXYZyhH
• William Timkey and Marten van Schijndel. 2021. All Bark and No Bite: Rogue Dimensions in Transformer Language Models Obscure Representational Quality. In Proceedings of the 2021 Conference on Empirical Methods in Natural Language Processing, pages 4527–4546, Online and Punta Cana, Dominican Republic. Association for Computational Linguistics. https://aclanthology.org/2021.emnlp-main.372/
• Zhou, K., Ethayarajh, K., Card, D., & Jurafsky, D. (2022, May). Problems with Cosine as a Measure of Embedding Similarity for High Frequency Words. In Proceedings of the 60th Annual Meeting of the Association for Computational Linguistics (Volume 2: Short Papers) (pp. 401-423). https://aclanthology.org/2022.acl-short.45/
• Chang, T. A., Tu, Z., & Bergen, B. K. (2022). The Geometry of Multilingual Language Model Representations. arXiv preprint arXiv:2205.10964. https://arxiv.org/abs/2205.10964
• Week 15 (December 5-): Wrap-up
• December 5: Final project presentations
• December 7: Final project presentations

## Paper presentation and discussion guidelines

• 60%: Presentation quality
• Presentation is of sufficient length (~30 minutes of content or approximately 15-20 slides)
• Clearly describes the goals of the paper
• Describes the linguistic phenomena under question
• Summarizes each of the experiments conducted in the paper and their results
• Clearly states significance of the work with respect to the course questions
• 40%: Discussion quality
• Highlights any uncertainty about linguistic phenomena in paper
• Raises points for discussion from course discussion page

## Final project guidelines

A final project will help determine your mastery of the course material and refine your ability to initiate, design, analyze, and summarize unique research questions. The general topic of your project should be broadly related to the interpretability and probing literatures discussed in class. The work you do for your final project should be your own and not plagiarized or purely replicating work completed elsewhere (e.g., data science blogs) and must come with an associated thoughtful literature review detailing prior work and scientific motivation. You may attempt to replicate the results or reconstruct the models from published ACL proceedings paper on the basis of that paper alone.

Students who wish to complete a capstone project via 667 should note that the thoroughness of the literature review and a fair, in-depth analysis of the results is critical for the project completion and it is imperative that you meet with the instructor or update them regularly, beginning early in the course. If you intend to complete a capstone project, all relevant paperwork must be completed on time following the CSE or LIN department guidelines and you must tell the instructor the first week of class if it is your intention to complete a capstone project.

• Please use the ACL 2023 $LaTeX$ template from the ACL Github repository: https://github.com/acl-org/acl-style-files
• Papers will be allowed unlimited pages for references

## Accessibility Services and Student Resources:

If you have a disability and may require some type of instructional and/or examination accommodation, please inform me early in the semester so that we can coordinate the accommodations you may need.  If you have not already done so, please contact the Office of Accessibility Services (formerly the Office of Disability Services) University at Buffalo, 60 Capen Hall, Buffalo, NY 14260-1632; email: stu-accessibility@buffalo.edu Phone: 716-645-2608 (voice); 716-645-2616 (TTY); Fax: 716-645-3116; and on the web at http://www.buffalo.edu/studentlife/who-we-are/departments/accessibility.html. All information and documentation is confidential.

The University at Buffalo and the Graduate School of Education are committed to ensuring equal opportunity for persons with special needs to participate in and benefit from all of its programs, services and activities.

Academic integrity is critical to the learning process. It is your responsibility as a student to complete your work in an honest fashion, upholding the expectations your individual instructors have for you in this regard. The ultimate goal is to ensure that you learn the content in your courses in accordance with UB’s academic integrity principles, regardless of whether instruction is in-person or remote. Thank you for upholding your own personal integrity and ensuring UB’s tradition of academic excellence.

It is expected that you will behave in an honorable and respectful way as you learn and share ideas. Therefore, recycled papers, work submitted to other courses, and major assistance in preparation of assignments without identifying and acknowledging such assistance are not acceptable. All work for this class must be original for this class. Please be familiar with the University and the School policies regarding plagiarism. Read the Academic Integrity Policy and Procedure for more information. Visit The Graduate School Policies & Procedures page (http://grad.buffalo.edu/succeed/current-students/policy-library.html) for the latest information.

### Counseling Services:

As a student you may experience a range of issues that can cause barriers to learning or reduce your ability to participate in daily activities. These might include strained relationships, anxiety, high levels of stress, alcohol/drug problems, feeling down, health concerns, or unwanted sexual experiences. Counseling, Health Services and Health Promotion are here to help with these or other issues you may experience. You can learn more about these program and services by contacting:

### Counseling Services

120 Richmond Quad (North Campus), 716-645-2720

202 Michael Hall (South Campus), 716-829-5900

https://www.buffalo.edu/studentlife/who-we-are/departments/counseling.html

### Health Services

Michael Hall (South Campus), 716-829-3316

https://www.buffalo.edu/studentlife/who-we-are/departments/health.html

### Office of Health Promotion

114 Student Union (North Campus), 716-645-2837

https://www.buffalo.edu/studentlife/who-we-are/departments/health-promotion.html

### Sexual Harassment/Violence:

UB is committed to providing a safe learning environment free of all forms of discrimination and sexual harassment, including sexual assault, domestic and dating violence and stalking. If you have experienced gender-based violence (intimate partner violence, attempted or completed sexual assault, harassment, coercion, stalking, etc.), UB has resources to help. This includes academic accommodations, health and counseling services, housing accommodations, helping with legal protective orders, and assistance with reporting the incident to police or other UB officials if you so choose. Please contact UB’s Title IX Coordinator at 716-645-2266 for more information. For confidential assistance, you may also contact a Crisis Service Campus Advocate at 716-796-4399.

Please be aware UB faculty are mandated to report violence or harassment on the basis of sex or gender. This means that if you tell me about a situation, I will need to report it to the Office of Equity, Diversity and Inclusion. You will still have options about how the situation will be handled, including whether or not you wish to pursue a formal complaint. Please know that if you not wish to have UB proceed with an investigation, your request will be honored unless UB’s failure to act does not adequately mitigate the risk of harm to you or other members of the university community. You also have the option of speaking with trained counselors who can maintain confidentiality. UB’s Options for Confidentiality Disclosing Sexual Violence provides a full explanation of the resources available, as well as contact information. You may call UB’s Office of Equity, Diversity and Inclusion at 716-645-2266 for more information, and you have the option of calling that office anonymously if you would prefer not to disclose your identity.

### Technology Recommendations

To effectively participate in this course, regardless of mode of instruction, the university recommends you have access to a Windows or Mac computer with webcam and broadband. Your best opportunity for success in the blended UB course delivery environment (in-person, hybrid and remote) will require these minimum capabilities.

### Public health compliance in a classroom setting

UB student Behavioral Requirements in all Campus Public Spaces include:

1. Students are required to obtain and wear a high-quality, tight-fitting, high-filtration mask when aboard a UB bus or shuttle or in a clinical health care setting in accordance with current health and safety guidelines. Masks indoors and in other public campus settings are optional.
1. Students who are regularly on campus and not fully vaccinated are required to participate in surveillance testing.
1. Students are required to abide by New York State, federal and Center for Disease Control and Prevention (CDC) travel restrictions and precautionary quarantines.
1. Students are required to stay home if they are sick.
1. Students are required to follow campus and public health directives for isolation or quarantine.
1. Should a student need to miss class due to illness, isolation or quarantine, they are required to notify their faculty to make arrangements to make up missed work.
1. Living on campus is a privilege that comes with additional requirements. Residential students are required to follow specific Campus Living rules as outlined in the Campus Living Housing Agreement, the Guide to Campus Living and any posted signage.
1. Students dining at on-campus facilities are expected to follow posted information on any additional requirements specific to the dining environment.
1. Students are responsible for following any additional directives in settings such as labs, clinical environments etc.


text:
New Zealand
Level 6 - NCEA Level 1

# Use density to determine volume or mass

Lesson

The density of a substance is its mass for each unit of volume.

The Greek letter $\rho$ρ is often used for density. Given a quantity of a substance, we write

$\rho=\frac{m}{V}$ρ=mV

where $m$m is the mass of the sample and $V$V is its volume.

If the mass is measured in kilograms and volume in cubic metres, the density measure is in the units $kg.m^{-3}$kg.m3, (kilograms per cubic metre).

If the density is known and either the mass or the volume, then the remaining unknown quantity can be calculated. That is, if $\rho=\frac{m}{V}$ρ=mV is given, then $m=\rho V$m=ρV and $V=\frac{m}{\rho}$V=mρ.

The density of most substances varies with temperature. This is because the volume of a given mass changes with temperature. Water, for example, is at its most dense when its temperature is $4^\circ$4°C.

When a mass of a solid substance is immersed in a fluid and weighed while immersed, it is found to have lost weight equal to the mass of the fluid displaced. This is the principle of buoyancy supposed to have been discovered by Archimedes.

#### Example 1

At $4^\circ$4°C water has a density of $1.0$1.0 g/cm$^3$3.

What is the mass of $800$800 cm$^3$3 of water at this temperature?

Using $\rho=\frac{m}{V}$ρ=mV, we have $1=\frac{m}{800}$1=m800. Therefore the mass is $800$800 g.

#### Example 2

An instrument consisting of a sealed glass tube with weights at the lower end is used to measure the density of the wort in a brewing process, which in turn depends on the concentration of sugar in the brew.

Suppose the instrument weighs $250$250 g and its total volume is $275$275 cm$^3$3. If the instrument were to be immersed in water at its maximum density of $1.0$1.0 g.cm$^{-3}$3, what fraction of the tube would be below the surface of the water?

The tube sinks until $250$250 g of water has been displaced. So, a volume $V=\frac{250}{1}$V=2501 cm$^3$3 is displaced and this is $\frac{250}{275}=\frac{6}{7}$250275=67 of the tube length.

A calibration mark could be made on the tube to indicate the depth to which it sinks when immersed in a liquid with density $1.0$1.0

When used to test the density of a brewing wort, the tube would not sink down to the mark if the density of the wort were greater than $1$1. In effect, this procedure compares the density of a wort with the density of water and in this situation, the result is called the specific gravity, which is the ratio of the density being tested to that of water.

### Outcomes

#### GM6-2

Apply the relationships between units in the metric system, including the units for measuring different attributes and derived measures

#### 91030

Apply measurement in solving problems


text:
Browse Questions

# If 1000g of a water sample contains 10g of $CaCO_3$ then the hardness of that water sample is

$\begin{array}{1 1}(a)\;100ppm\\(b)\;500ppm\\(c)\;10000ppm\\(d)\;20000ppm\end{array}$

If 1000g of a water sample contains 10g of $CaCO_3$ then the hardness of that water sample is 10000ppm.
Hence (c) is the correct answer.


text:
March 26, 2019
March 26, 2019
Press Releases
March 26, 2019
Games Press

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# Reflections Based on Local Cubemaps in Unity

by Roberto Lopez Mendez on 01/15/16 01:14:00 pm

The following blog post, unless otherwise noted, was written by a member of Gamasutra’s community.
The thoughts and opinions expressed are those of the writer and not Gamasutra or its parent company.

Introduction

The topic of this blog was presented to students in a workshop at Brains Eden Gaming Festival 2014 at Anglia Ruskin University in Cambridge [1]. We wanted to provide students with an effective and low cost technique to implement reflections when developing games for mobile devices.

Early Reflection Implementations

From the very beginning, graphics developers have tried to find cheap alternatives to implement reflections. One of the first solutions was spherical mapping. Spherical mapping simulates reflections or lighting upon objects without going through expensive ray-tracing or lighting calculations. This approach has several disadvantages, but the main problem is related to the distortions when mapping a picture onto a sphere.  In 1999, it became possible to use cubemaps with hardware acceleration.

 Figure 1: Spherical mapping.

Cubemaps solved the problems of image distortions, viewpoint dependency and computational inefficiency related with spherical mapping. Cube mapping uses the six faces of a cube as the map shape. The environment is projected onto each side of a cube and stored as six square textures, or unfolded into six regions of a single texture. The cubemap is generated by rendering the scene from a given position with six different camera orientations with a 90 degree view frustum representing each a cube face. Source images are sampled directly. No distortion is introduced by resampling into an intermediate environment map.

 Figure 2: Cubemaps.

To implement reflections based on cubemaps we just need to evaluate the reflected vector R and use it to fetch the texel from the cubemap CubeMap using the available texture lookup function texCUBE:

 float4 col = texCUBE(CubeMap, R); Expression 1.

 Figure 3: Reflections based on infinite cubemaps.

With this approach we can only reproduce reflections correctly from a distant environment where the cubemap position is not relevant. This simple and effective technique is mainly used in outdoor lighting, for example, to add reflections of the sky. If we try to use this technique in a local environment we get inaccurate reflections.

 Figure 4: Reflection on the floor calculated wrongly with an infinite cubemap.

## Local Reflections

The main reason why this reflection fails is that in Expression 1 there is not any binding to the local geometry. For example, according to Expression 1, if we were walking on a reflective floor looking at it from the same angle, we would always see the same reflection on it. As the direction of the view vector does not change, the reflected vector is always the same and Expression 1 gives the same result. Nevertheless, this is not what happens in the real world where reflections depend on viewing angle and viewing position.

The solution to this problem was first proposed by Kevin Bjorke[2] in 2004. For the first time a binding to the local geometry was introduced in the procedure to calculate the reflection:

 Figure 5: Reflections using local cubemaps.

While this approach gives good results in objects’ surfaces with near to spherical shape, in the case of plane reflective surfaces the reflection shows noticeable deformations. Another drawback of this method is related to the relative complexity of the algorithm to calculate the intersection point with the bounding sphere which solves a second degree equation.

A few years later, in 2010, a better solution was proposed [3] in a thread of a developer forum at gamedev.net. The new approach replaced the previous bounding sphere by a box, solving the shortcomings of Bjorke’s method: deformations and complexity of the algorithm to find the intersection point.

 Figure 6: Introducing a bounding box.

A more recent work [4] uses this new approach to simulate more complex ambient specular lighting using several cubemaps and proposes an algorithm to evaluate the contribution of each cubemap and efficiently blend on the GPU.

At this point we must clearly distinguish between local and infinite cubemaps:

Figure 7 shows the same scene from Figure 4 but this time with correct reflections using local cubemaps.

 Figure 7: Reflection on the floor correctly calculated with a local cubemap.

The shader implementation in Unity of reflections using local cubemaps is provided below. In the vertex shader, we calculate the three magnitudes we need to pass to the fragment shader as interpolated values: the vertex position, the view direction and the normal, all of them in world coordinates:

vertexOutput vert(vertexInput input)
{
vertexOutput output;
output.tex = input.texcoord;
// Transform vertex coordinates from local to world.
float4 vertexWorld = mul(_Object2World, input.vertex);
// Transform normal to world coordinates.
float4 normalWorld = mul(float4(input.normal, 0.0), _World2Object);
// Final vertex output position.
output.pos = mul(UNITY_MATRIX_MVP,  input.vertex);
// ----------- Local correction ------------
output.vertexInWorld = vertexWorld.xyz;
output.viewDirInWorld = vertexWorld.xyz - _WorldSpaceCameraPos;
output.normalInWorld = normalWorld.xyz;
return output;
}

In the fragment shader the reflected vector is found along with the intersection point in the volume box. The new local corrected reflection vector is built and it is used to fetch the reflection texture from the local cubemap. Finally the texture and reflection are combined to produce the output colour:

float3 LocalCorrect(float3 origVec, float3 bboxMin, float3 bboxMax,
float3 vertexPos, float3 cubemapPos)
{
// Find the ray intersection with box plane
float3 invOrigVec = float3(1.0,1.0,1.0)/origVec;
float3 intersecAtMaxPlane = (bboxMax - vertexPos) * invOrigVec;
float3 intersecAtMinPlane = (bboxMin - vertexPos) * invOrigVec;
// Get the largest intersection values
// (we are not intersted in negative values)
float3 largestIntersec = max(intersecAtMaxPlane, intersecAtMinPlane);
// Get the closest of all solutions
float Distance = min(min(largestIntersec.x, largestIntersec.y),
largestIntersec.z);
// Get the intersection position
float3 IntersectPositionWS = vertexPos + origVec * Distance;
// Get corrected vector
float3 localCorrectedVec = IntersectPositionWS - cubemapPos;
return localCorrectedVec;
}

float4 frag(vertexOutput input) : COLOR
{
float4 reflColor = float4(1, 1, 0, 0);
// Find reflected vector in WS.
float3 viewDirWS = normalize(input.viewDirInWorld);
float3 normalWS = normalize(input.normalInWorld);
float3 reflDirWS = reflect(viewDirWS, normalWS);
// Get local corrected reflection vector.
float3 localCorrReflDirWS = LocalCorrect(reflDirWS, _BBoxMin,
_BBoxMax, input.vertexInWorld,
_EnviCubeMapPos)
// Lookup the environment reflection texture with the right vector.
reflColor = texCUBE(_Cube, localCorrReflDirWS);
// Lookup the texture color.
float4 texColor = tex2D(_MainTex, float2(input.tex));
return _AmbientColor + texColor * _ReflAmount * reflColor;
}

In the above code for the fragment shader the magnitudes _BBoxMax and _BBoxMin are the maximum and minimum points of the bounding volume. The variable  _EnviCubeMapPos is the position where the cubemap was created. These values are passed to the shader from the below script:

[ExecuteInEditMode]
public class InfoToReflMaterial : MonoBehaviour
{
// The proxy volume used for local reflection calculations.
public GameObject boundingBox;

void Start()
{
Vector3 bboxLenght = boundingBox.transform.localScale;
Vector3 centerBBox = boundingBox.transform.position;
// Min and max BBox points in world coordinates.
Vector3 BMin = centerBBox - bboxLenght/2;
Vector3 BMax = centerBBox + bboxLenght/2;
// Pass the values to the material.
gameObject.renderer.sharedMaterial.SetVector("_BBoxMin", BMin);
gameObject.renderer.sharedMaterial.SetVector("_BBoxMax", BMax);
gameObject.renderer.sharedMaterial.SetVector("_EnviCubeMapPos",
centerBBox);
}
}

The values for _AmbientColor and _ReflAmount as well as the main texture and cubemap texture are passed to the shader as uniforms from the properties block:

Shader"Custom/ctReflLocalCubemap"
{
Properties
{
_MainTex ("Base (RGB)", 2D) = "white" { }
_Cube("Reflection Map", Cube) = "" {}
_AmbientColor("Ambient Color", Color) = (1, 1, 1, 1)
_ReflAmount("Reflection Amount", Float) = 0.5
}
{
Pass
{
CGPROGRAM
#pragma glsl
#pragma vertex vert
#pragma fragment frag
#include "UnityCG.cginc"
// User-specified uniforms
uniform sampler2D _MainTex;
uniform samplerCUBE _Cube;
uniform float4 _AmbientColor;
uniform float _ReflAmount;
uniform float _ToggleLocalCorrection;
// ----Passed from script InfoRoReflmaterial.cs --------
uniform float3 _BBoxMin;
uniform float3 _BBoxMax;
uniform float3 _EnviCubeMapPos;

struct vertexInput
{
float4 vertex : POSITION;
float3 normal : NORMAL;
float4 texcoord : TEXCOORD0;
};

struct vertexOutput
{
float4 pos : SV_POSITION;
float4 tex : TEXCOORD0;
float3 vertexInWorld : TEXCOORD1;
float3 viewDirInWorld : TEXCOORD2;
float3 normalInWorld : TEXCOORD3;
};

ENDCG
}
}
}

The algorithm to calculate the intersection point in the bounding volume is based on the use of the parametric representation of the reflected ray from the local position (fragment). More detailed explanation of the ray-box intersection algorithm can be found in [4] in the References.

## Filtering Cubemaps

One of the advantages of implementing reflections using local cubemaps is the fact that the cubemap is static, i.e. it is generated during development rather than at run-time. This gives us the opportunity to apply any filtering to the cubemap images to achieve a given effect.

As an example, the image below shows reflections using a cubemap where a Gaussian filter was applied to achieve a “frosty” effect. The CubeMapGen [5] tool (from AMD) was used to apply filtering to the cubemap. Just to give an idea about how expensive this process can be it took more than one minute the filtering of a 256 pixels cubemap on a PC.

 Figure 8: Gaussian filter applied to reflections in Figure 3.

A specific tool was developed for Unity to generate cubemaps and save cubemap images separately to import later into CubeMapGen. Detailed information about this tool and about the whole process of exporting cubemap images from Unity to CubeMapGen, applying filtering and reimporting back to Unity can be found in the References section [4].

## Conclusions

Reflections based on static local cubemaps are an effective tool to implement high quality and realistic reflections and a cheap alternative to reflections generated at run-time.   This is especially important in mobile devices where performance and memory bandwidth consumption are critical to the success of many games.

Additionally, reflections based on static local cubemaps allow developers to apply filters to the cubemap to achieve complex effects that would otherwise be prohibitively expensive at run-time, even on high-end PCs.

The inherent limitation of static cubemaps when dealing with dynamic objects can be solved easily by combining static reflections with reflections generated at run-time. This topic will be examined in a future blog.

## REFERENCES

[1] Reflections based on local cubemaps. Presentation at Brains Eden, 2014 Gaming Festival at Anglia Ruskin University in Cambridge.  http://malideveloper.arm.com/downloads/ImplementingReflectionsinUnityUsingLocalCubemaps.pdf

[2] GPU Gems, Chapter 19. Image-Based Lighting. Kevin Bjork, 2004. http://http.developer.nvidia.com/GPUGems/gpugems_ch19.html.

[3] Cubemap Environment Mapping. 2010. http://www.gamedev.net/topic/568829-box-projected-cubemap-environment-mapping/

[4] Image-based based Lighting approaches and parallax-corrected cubemap. Sebastien Lagarde. SIGGRAPH 2012.

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