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A novel approach to likelihood-free inference: just use the likelihood!

A new arXival describes “a novel inference framework based on Approximate Bayesian Computation” for a modelling exercise in the field of strong gravitational lensing.  Since they acknowledge insightful discussions from the guys who seem to deliberate refuse to cite my work on ABC for astronomy, it’s no surprise that the statistical analysis again goes astray.  Basically, the proposal here is to use the negative log marginal likelihood—the marginalisation being over a set of nuisance parameters conditional on the observed data and a given set of hyper-parameters—as a distance in ABC.  Usually ABC is motivated because the likelihood is not available, but that doesn’t seem to be understood here.  The only thing that seems to stop the posterior collapsing to the mode is the early stopping of the ABC-SMC algorithm at an arbitrary number of steps.  There’s also some silliness with respect to the choice of prior for the source image (the component of the model that is marginalised out exactly) which is emphasised in the paper as disfavouring realistic source brightness distributions.

The only thing I found interesting here was the reference to Vegetti & Koopmans (2009)  [Note: this is clearly the paper the authors intended to cite, rather than the other Vegetti & Koopmans 2009: MNRAS 400 1583; this kind of mistake suggests the level of care going into these arXivals is less even than the few minutes I spend cranking out a blog post].  The Vegetti & Koopmans (2009) method involves construction of a source image prior via a Voronoi tessellation scheme with regularisation terms.  An interesting project would be to examine how the SPDE approach could allow for a more nuanced prior choice, introduction of non-stationarity etc (see Lindgren et al. 2011).

Actually there is another interesting thing that could be investigated with this model.  The authors choose to look at $-2 \log P(d|\theta)$ for their distance: the factor of 2 is of course irrelevant in the ABC context and with respect to collapse on the posterior mode; but for likelihood-based inference it would represent an extra-Bayesian calibration factor, which could actually be chosen to reduce exposure to excess concentration in the mis-specified setting via e.g. the loss-likelihood bootstrap.

More dumping on the NeurIPS Bayesian Deep Learning Workshop …

Today I noticed another paper on astro ph that irked me, and again it turns out to be accepted at this year’s NeurIPS Bayesian Deep Learning Workshop.  This particular arXival proposes to explore a Bayesian approach to construction of super-resolution images, in particular, to explore uncertainty quantification since “in many scientific domains this is not adequate and estimations of errors and uncertainties are crucial”.  What irked me?  One thing was the statement: “to the extent of our
knowledge, there is no existing work measuring uncertainty in super-resolution tasks”.  That might be true if you consider only a particular class of machine learning algorithms that have addressed the challenge of creating high resolution images from low resolution inputs, but this general problem (PSF deconvolution, drizzling, etc.) has been a core topic in astronomical imaging since the first CCDs, and in this context there are many studies of accuracy and uncertainty.  Likewise, the general problem of how to build confidence in reconstructions of images via statistical models without a ground-truth to validate against is also well explored in astronomy.  The first ever black hole image (‘the Katie Bouman news story’) addressed this challenge through a structured comparison on images separately created by four independent teams using different methods.

Another thing that irks me is that I find the breakdown between types of uncertainty—“Epistemic uncertainty relates to our ignorance of the true data generating process, and aleatoric uncertainty captures the inherent noise in the data.”—to be inadequate.  Here this is really just proposing a separation between the prior and the likelihood, which runs against the useful maxim for applied Bayesian modelling that the prior can only be understood in terms of the likelihood.  That said, I also wouldn’t call this a Bayesian method since the approximation of the posterior implied by dropout is like zero-th order.  Don’t get me wrong: dropout is a great technique for certain applications but I think the arguments to suggest it has a Bayesian flavour are rather unconvincing though attractive to citations.

On unnecessarily general paper titles

Last week a postdoc in my lab received a rejection letter from a high profile stats journal, with the reason for rejection given being that the problem was already solved in existing software such as INLA.  Which was odd because we use INLA all the time at work and the whole reason we decided to embark on the project described in the rejected manuscript was because INLA did not offer a solution for this particular problem.  My suspicion is that the editor or associate editor did a quick google search on the topic and found a paper with an unnecessarily general title.  That is, a paper whose title suggests that a general problem is solved therein, rather than the very restricted problem that is actually examined.  (In this case the problem is combination of area and point data, which is trivially solved in INLA under the Normal likelihood with linear link function, but is not solved in INLA for non-Normal likelihoods with non-linear link functions.)

For this reason I would say that I’m more than a little skeptical about the clickbait motivation for the title given to this recent arXival: “Uncertainty Quantification with Generative Models”.  Which is sufficiently broad as to encompass the entirety of Bayesian inference and most of machine learning!!   And in which you would probably expect to find something more substantial than a proposal to approximate posteriors of VAE style models via a mixture of Gaussians, obtained by local mode finding (optimisation from random starting points) followed by computation of the Hessian at those modes.  But apparently this novel idea is accepted to the Bayesian Deep Learning workshop at NeurIPS this year, so what do I know?!

If I’m going to start beef with the machine learning community then I may as well say something else on the topic.  Recently it came to light that an Australian engineering professor was fired from Swinburne University for having published a huge amount of duplicate work: i.e., submitting essentially the same paper to multiple journals in order to spin each actual project out into multiple near-identical publications.  The alleged motivation for doing so was the pressure to juke ones own research output stats (total pubs and total cites).  Which is funny because I don’t know of many machine learning professors who don’t have the same issue with their publications: multiple versions of the same paper given at NeurIPS, AISTATS, ICLR, etc.,  and then maybe submitted to a stats journal as well!

And in other indignities, the third author on this arXival is on a salary that is over 2.5 times my Oxford salary.

Priors make models like styles make fights …

A ubiquitous saying in boxing analysis is “styles make fights”, which means that to predict what a match up will look like you need to think about how the characteristic styles of the two opponents might work (or not) with respect to each other.  Two strong counter-punchers might find themselves circling awkwardly for twelve rounds, neither willing to come forward and press the action.  While a match up between two aggressive pressure fighters might turn on the question of whether their styles only work while moving forwards.  As an analyst of Bayesian science my equivalent maxim is “priors make models”.  Well-chosen priors can sensibly regularise the predictions of a highly flexible model, achieve powerful shrinkage across a hierarchical structure, or push a model towards better Frequentist coverage behaviour.  For that reason I don’t understand why cosmologists are so keen on ‘uninformative’ priors.  It’s like throwing away the best part of Bayesian modelling.

Anyway, two papers from the arXiv last week caught my eye.  The first proposes a statistic for ‘quantifying tension between correlated datasets with wide uninformative priors’.  So, aside from the focus on a type of prior (wide, uninformative) that I don’t care for, I’m also puzzled by the obsession of cosmologists with searching for ‘tension’ in the posteriors of models with shared parameters fitted to different datasets (or different aspects of the same dataset), as an indicator of either systematic errors or new physics.  As this paper makes clear, there is a huge variety of techniques proposed for this topic, but all of them come from the cosmology literature.  How is it that no other field of applied statistics has got itself twisted up in this same problem?  An example use of this statistic is given in which a model with shared parameters is fitted to four redshift slices from a survey and the decision whether or not to combine the posteriors is to be made according to how much their separately fitted posteriors overlap.

The other paper I read this week concerns the coupling of a variational autoencoder model as generative distribution for galaxy images with a physics based gravitational lensing model.  The proposal for this type of model and the authors’ advocacy for modern auto-diff packages like PyTorch makes a lot of sense.  However, it seems that a lot of work is still to be done to improve the prior on galaxy images and the posterior inference technique, because for the two examples shown suggests that there is a serious under-coverage problem in the moderate and high signal-to-noise regime.  Also, the cost of recovering a small number of HMC samples is very high here (many hours); I don’t think that HMC is a viable posterior approximation for this type of model. Any why bother when the coverage is so bad?  Most likely a better option will be some kind of variational approximation that will be quicker to fit and will improve coverage partly by accident and partly by design through its approximate nature; i.e., deliberately slowing the learning rate.  Sounds crazy to some, perhaps, but remember that the variational autoencoder here is trained via stochastic gradient descent with a predictive accuracy-based stopping rule, which is just another way of slowing the learning rate or artificially regularising a model.

How to push back against bullies in academia

I’m sure most of us know of at least one notorious workplace bully (or other type of shithead) in our field; the kind of professor about whom there’s an open secret of their bad behaviour, but given that universities don’t give a toss about this issue there is no chance of them being formally sanctioned.  An effective solution I’ve recently come across is the following: simply refuse to have anything to do with their work.  If you’re asked to review one of their papers, decline and let the journal know that you don’t feel you can ethically review their work.  If you’re asked to review on of their grant applications, do the same.  If you’re invited to speak in the same session as them at a conference, decline and let the organisers know your reasoning.

On this topic, one thing that amazes me is the fact that most universities don’t do exit interviews with departing staff members.  If you want to test a hypothesis (e.g. there is or isn’t a problem with workplace bullying in a given group or department) you need to gather data.  Simply relying on a passive monitoring system (i.e., self-reporting of incidents by victims) is nothing short of a deliberate strategy to avoid seeing the problem.

Copula models for astronomical distributions

Yesterday I read through this new arXival by an old friend from my ETH Zurich days, in which is presented a package (called LEO-py) for likelihood-based inference in the case of Gaussian copula models and linear regressions with missing data, censoring, or truncation.  I never quite understand the demand for astronomer-specific expositions and software on topics like this, since as soon as one understands what a hierarchical Bayesian model is and how to code one up in a standard statistical programming language like Stan or JAGS, then the world is your oyster.  (Indeed, back when we were at ETH, an errors-in-variables logistic regression model for predicting the barred galaxy fraction as a function of noisy-estimated stellar mass was one of my first forays into this field; of course it never saw the light of day because my supervisor at the time—she who shall not be named—was completely opposed to any statistical methods beyond ordinary linear regression!)  The key contribution here, to my mind, is rather the emphasis on copula models which are certainly under-utilised in the literature.  If this package helps popularise copulae (copulation?) that will be a very good contribution.

Note: Whenever I think of truncated astronomical data analysis problems I’m reminded of the example (described in JS Liu’s Monte Carlo Strategies in Scientific Computing) of a permutation test for doubly truncated (redshift, log-luminosity) data developed by Efron & Petrosian (1999).


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## Algebra 1

Since there are 18 electrons which have a charge of -1, the total negative charge is $18(-1)$. Since there are 16 protons which have a charge of +1, the total positive charge is $16(1)$. Add the positive and negative charges to find the ion's charge. $18(-1)+16(1)\longrightarrow$ multiply $=-18+16\longrightarrow$ add $=-2$

format: plain


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# which type of casting will give higher cooling rate, casting in thin water-cooled copper mold or casting in thick copper die?

It is will known that increasing cooling of a metal casting will lead to finer micro-structure which mean higher mechanical properties. If I have 2 molds one is thick copper die ,& second is just thin-walled copper cup or mold cooled by water. which one of them excepted to cool the molten metal i.e. aluminium faster ?

the calculated heat diffusivity of copper is : thermal conductivity x specific heat x density = 385 x 0.385 x 8960 = 1328096 m²/s

while for water it is just : 0.6 x 4.2 x 1000 = 2520 m²/s only !!

so, the copper can transfere heat 527 times faster than water.

by calculations it seems that copper die have higher cooling power as it has higher thermal diffusivity , but in real life it seems water have more cooling power. I am so confused.

• I suspect you mean $k/\rho \tilde{C}_p$ not $k \rho \tilde{C}_p$. Check your calculations and UNITS. Using water includes convection. For flowing water, convection is higher than conduction. – Jeffrey J Weimer Dec 11 '18 at 14:49
• Your basic assumption is incorrect ; Fast freezing and cooling of aluminum will produce lower strength although with finer grain size. It would be solution annealing the casting, more or less. If you want to heat-treat something , it would be best to do it as a separate operation. – blacksmith37 Dec 11 '18 at 16:16
• Water can be replaced with new water, copper not so much. Which is less fancier to say convection is the primary tranfer method of water. – joojaa Dec 12 '18 at 7:18
• A couple notes: aluminum is highly soluble in copper, so expect the mold properties to change with successive castings. It is also possible for intermetallics to form, which could scale off. Contained or sealed water has extremely high vapor pressures (multiple GPa) at liquid Al temperatures, so be wary of inadvertently creating a bomb with your system. Sufficient convection, and not directly exposing water to liquid metal, should prevent this issue. Stay safe! – wwarriner Dec 19 '18 at 21:31

## 2 Answers

A company I worked at used tool steel molds and channels in the mold with flowing water to increase cooling. Heed the advice of others, never let ANY water come into contact with molten aluminum. I have seen it hit a 20' ceiling (would have gone farther) and as far as 15' in circumference.

It depends on what you're casting. For a smaller castings with thin sections, the amount of heat that needs to be absorbed is low and the surface area and casting to mold volume ratio is high. So a chunky solid copper die may absorb the heat quickly enough.

But the heat transfer slows as the die heats up (temperature difference decreases). If the casting is a larger block, a lot more heat has to diffuse through a smaller surface area and as the surface of the die heats up, the cooling rate reduces. In this case having water that can take the heat and move it away from the die surface (by pump or convection), allows the heat to be distributed throughout the cooling system more evenly, keeping the temperature difference high.

You can also use chillers or cool the mould to increase the temperature difference and speed up the cooling, which will give you more nucleation points when solidifying and a smaller grains. But an increased cooling rate often actually decreases the strength due to increased porosity and other casting defects and may not even fill the die properly (I have seen people pre-heat moulds to slow down the cooling rate and get a better casting). Increasing the strength is really best done by alloying with other elements.


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# Normal subgroup of prime index

Generalizing the case $p=2$ we would like to know if the statement below is true.

Let $p$ the smallest prime dividing the order of $G$. If $H$ is a subgroup of $G$ with index $p$ then $H$ is normal.

This is a standard exercise, and the answer is that the statement is true, but the proof is rather different from the elementary way in which the $$p=2p=2$$ case can be proven.
Let $$HH$$ be a subgroup of index $$pp$$ where $$pp$$ is the smallest prime that divides $$|G||G|$$. Then $$GG$$ acts on the set of left cosets of $$HH$$, $${gH∣g∈G}\{gH\mid g\in G\}$$ by left multiplication, $$x⋅(gH)=xgHx\cdot(gH) = xgH$$.
This action induces a homomorphism from $$G→SpG\to S_p$$, whose kernel is contained in $$HH$$. Let $$KK$$ be the kernel. Then $$G/KG/K$$ is isomorphic to a subgroup of $$SpS_p$$, and so has order dividing $$p!p!$$. But it must also have order dividing $$|G||G|$$, and since $$pp$$ is the smallest prime that divides $$|G||G|$$, it follows that $$|G/K|=p|G/K|=p$$. Since $$|G/K|=[G:K]=[G:H][H:K]=p[H:K]|G/K| = [G:K]=[G:H][H:K] = p[H:K]$$, it follows that $$[H:K]=1[H:K]=1$$, so $$K=HK=H$$. Since $$KK$$ is normal, $$HH$$ was in fact normal.


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# Drag each label to the correct location on the image.Match each tool, material, or action to the correct

Drag each label to the correct location on the image. Match each tool, material, or action to the correct technique.

$Drag each label to the correct location on the image. Match each tool, material, or action to the$

milkshakegrande101

Explanation:

legend101xD

i dont understand everyone say image but yet again there are no images if you want me to help you put an image down

Explanation:

thx

sorry if that was rude but for real put an image bud

wardlawshaliyah

Relief: an artist uses wood or linoleum,  an artist uses a tool called a squeegee, and an artist uses a matrix made out of limestone

silkscreen: an artist uses a grease crayon, and an artist creates a raised surface that is then inked.  the rest go under lithography

Explanation:

1hannacarson

whats litter something

Explanation:


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Home » Ask Physics » Problem from laws of motion

# Problem from laws of motion

Share with:

both block and wedge are of mass m . Question is to find acceleration of wedge and block .

I can understand how to solve it taking wedge as frame of reference .

but when am trying to find solution taking ground as frame of reference and finding their acceleration different .

acceleration of wedge =[ gsin(theta)cos(theta) ] / ( 1 + sin^2 (theta) )
acceleration of block = big one

My approach to solve it wrt ground frame of reference :

For block  : gsin(theta ) is acceleration
Normal Force = mg cos (theta)

For wedge : g cos (theta ) sin (theta) = acceleration

But a should come as I said earlier . Where am I wrong ?

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Q : 1        ABCD is a quadrilateral in which P, Q, R and S are mid-points of the sides AB, BC, CD and DA (see Fig ). AC is a diagonal. Show that :                 (i)    and

Given :   ABCD is a quadrilateral in which P, Q, R and S are mid-points of the sides AB, BC, CD and DA (see Fig ). AC is a diagonal.

To prove :    and

Proof: In ACD,

S is the midpoint of DA.                (Given)

R  is the midpoint of DC.               (Given)

By midpoint theorem,

and

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#### Two dimensional device simulation and characterization of GaAs floated electron channel field effect transistor = 갈륨비소 부동전자채널 전계효과 트랜지스터의 2차원 소자 시뮬레이션 및 특성분석

Cited 0 time in Cited 0 time in
• Hit : 270
The transconductance compression near subthreshold regime observed in GaAs Floated Electron Channel Field Effect Transistor (FECFET) is investigated by the electron flux profile extracted from the drain current versus gate voltage measurement in the saturation region of operation and two-dimensional device simulation. PISCES-2B and MEDICI are used for the two dimensional simulation. The applicability of the simulators for the simulation of GaAs MESFET is studied taking into account the velocity enhancement effect with reducing gate length. It is demonstrated that PISCES-2B and MEDICI can be well utilized for both DC and AC device simulation of state-of-art GaAs MESFET although they have some limits in describing the effect of velocity overshoot in submicrometer gate length GaAs MESFET. It is shown that the transconductance compression of GaAs FECFET is caused by electron pile-up near the top vertex of void due to the non-uniform profile of transport parameters in the region. The drift mobility profile of GaAs active layer on the top vertex is obtained by the simple etch-and-measure experiment for confirming the non-uniform profile of transport parameters near the top vertex of void. The process sensitivity of the device performance of FECFET is analyzed by using the device simulation when key structure parameters of FECFET are changed. It is shown from the analysis that the optimal vertical distance between the top vertex and the ohmic front of FECFET is about 0.2 $\mu$m. When the metallized gate length becomes decreased, the current-gain cut-off frequency of FECFET becomes increased following the linear relationship with the inverse of the gate length. As the gate length becomes reduced down to 0.3 $\mu$m, the effect of gate fringe capacitance becomes evident and thus the increase of the cut-off frequency is saturated. In addition, the important DC parameters such as threshold voltage and transconductance are more sensitive to gate displacement relative to cen...
Kwon, Young-Seresearcher권영세researcher
Description
한국과학기술원 : 전기 및 전자공학과,
Publisher
한국과학기술원
Issue Date
1994
Identifier
69048/325007 / 000845234
Language
eng
Description

학위논문(박사) - 한국과학기술원 : 전기 및 전자공학과, 1994.2, [ iv, 129 p. ]

URI
http://hdl.handle.net/10203/36204


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## Final Exam

Take-home exam. Comprehensive, will cover all material learned in class.

The exam will be posted here on Fri, Apr 29, in the evening, and collected in MATH 610 at 10am on Wed, May 4. An envelope will be affixed to the door for early submissions.

### Final Score

Two schemes for calculating your final score will be used: with or without Final Exam.

Scheme I = (3/11)ME1 + (3/11)ME2 + (3/11)FE + (2/11)HW
Scheme II = (3/8)ME1 + (3/8)ME2 + (1/4)HW

where FE, MEi, HW are the scores (in %) for Final Exam, Midterm Exam i, Homework, respectively.

You will have to email me by 5:30pm on Fri, Apr 29 to tell whether you will be completing the Final Exam (Scheme I will be used) or not (Scheme II will be used). If you do not contact me by then, your score will be calculated according to Scheme I.

## Midterm Exam 2

Take-home exam. Deadline: 12:30pm on Fri, Apr 22


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## Friday, 6 December 2013

### MMSE Estimators for Speech Enhancement

MMSE (Minimum Mean Square Error) estimators are commonly used in speech enhancement, with many different estimators applied to different speech representations. In its most basic form, an MMSE estimator is used to find the most likely clean signal given a noisy signal. I will start by explaining a time domain MMSE estimator (which won't be very useful), and then we can move on to better ones.

Some notation: $$x_n$$ is our clean signal ($$n$$ is the sample index), $$y_n$$ is our noisy signal, and $$d_n$$ will be our noise. We'll assume our noise is additive i.e. $y_n = x_n + d_n$

We will now start applying some assumptions: we assume both $$x_n$$ and $$d_n$$ are gaussian variables, with different variances. As an assumption, this is not a very good one. Time domain speech is not gaussian, and most real noise is also not gaussian. One of the things this assumption has going for it though is that it results in a very simple estimator.

So I should now explain what an estimator actually is. Given a probability distribution function for a random variable, we want to pick the most likely value the random variable can take. There are two main ways of doing this: MMSE estimators, and MAP estimators. I'll explain MAP estimators in another post, they tend to be simpler to compute but they don't usually perform as well as MMSE estimators.

MMSE estimators are derived as follows: given a pdf $$p(x_n|y_n)$$ (this is the probability of the clean signal given the noisy signal) we wish to find $$\hat{x}_n$$ (this is called "x hat"),which is our estimate of the clean signal. To do this we find the expected value of the pdf $$p(x_n|y_n)$$. We use the following formula for expectation:

$\hat{x}_n = \int_{-\infty}^{\infty}x_n p(x_n|y_n)dx_n$

Lets look at the pdfs we know (remember we assumed both $$x$$ and $$d$$ are gaussian distributed):

$p(x_n) = \dfrac{1}{\sqrt{2\pi\sigma_x^2}} \exp \left( \dfrac{-x_n^2}{2\sigma_x^2} \right)$ $p(d_n) = \dfrac{1}{\sqrt{2\pi\sigma_d^2}} \exp \left( \dfrac{-d_n^2}{2\sigma_d^2} \right)$

Using the 2 pdfs above, we can derive the pdf for $$p(y_n|x_n)$$ (the pdf of the noisy signal given the clean signal). Since we assumed $$y_n = x_n + d_n$$, if we are given the value $$x_n$$, we know $$y_n$$ must be close to it with a variance of $$\sigma_d^2$$ i.e. the noise variance. This results in:

$p(y_n|x_n) = \dfrac{1}{\sqrt{2\pi\sigma_d^2}} \exp \left(\dfrac{ -(y_n-x_n)^2 }{ 2\sigma_d^2 } \right)$

Despite how easy it is to calculate $$p(y_n|x_n)$$, it is not the pdf we want. We can compute $$p(x_n|y_n)$$ using Bayes rule:

$p(x_n|y_n) = \dfrac{p(y_n|x_n)p(x_n)}{\int_{-\infty}^\infty p(y_n|x)p(x) dx}$

The value we actually want is:

$\hat{x}_n = \int x_n p(x_n|y_n) dx_n = \int \dfrac{x_n p(y_n|x_n)p(x_n)}{\int_{-\infty}^\infty p(y_n|x)p(x) dx} dx_n$

Now we substitute the expressions for the pdfs into our equations to get a bit of a monster:

$\hat{x}_n = \int{ \dfrac{x_n \dfrac{1}{\sqrt{2\pi\sigma_d^2}} \exp \left(\dfrac{ -(y_n-x_n)^2 }{ 2\sigma_d^2 } \right) \dfrac{1}{\sqrt{2\pi\sigma_x^2}} \exp \left( \dfrac{-x_n^2}{2\sigma_x^2} \right)}{\int_{-\infty}^\infty \dfrac{1}{\sqrt{2\pi\sigma_d^2}} \exp \left(\dfrac{ -(y_n-x)^2 }{ 2\sigma_d^2 } \right) \dfrac{1}{\sqrt{2\pi\sigma_x^2}} \exp \left( \dfrac{-x^2}{2\sigma_x^2} \right) dx} dx_n}$

If you want to do it by hand you can simplify the above expression a bit and use a book of integrals to solve it, but the easier way is just to use mathematica to do it for you (or this page which helps with integrals). The result is the following very elegant solution:

$\hat{x}_n = \dfrac{\sigma_x^2}{\sigma_x^2 + \sigma_d^2}y_n$

If $$x_n$$ and $$d_n$$ were actually gaussian random variables, then the estimator we derived above is optimal i.e. no better estimator of $$y_n$$ exists. In reality, we would also need to know $$\sigma_d^2$$ and $$\sigma_x^2$$, and these can be very tricky to estimate. In the next post I will use the same method but different probability distributions to derive other estimators.

In the next post I'll do a slightly better MMSE estimator that doesn't use gaussians. By using more complicated pdfs we can improve our estimator, but we will also get much more complicated integrals. In some cases the integrals won't be analytically solvable at all, so we'll have to be happy approximating them numerically. If this is not possible because of e.g. timing constraints, we'll have to settle on a MAP estimator.


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# Centroid of a 2D region integral explanation

I've been told that to find the center of mass for a region defined in the x-y plane I need to use this formula

Edit: A region defined parametrically by $x(t)$ and $y(t)$ where $t$ is $a \leq t \leq b$

$Y=\dfrac{\int_a^b y(t) x(t) y'(t)dt}{\int_a^b x(t) y'(t)dt}$

Where the denominator is the area of the region in question. I can almost see why it works but I'm hung up on the $y'(t)$

I can see how this

$Y=\lim\limits_{n \to \infty}\dfrac{\sum\limits_{i=0}^n y(t_i)x(t_i)}{\sum\limits_{i=0}^{n} x(t_i)}$

represents the y coordinate of the center of mass of the region but the problem is how it translates to integral form.

$\int_a^b y(t)x(t)y'(t)~dt = \lim\limits_{n \to \infty} \sum\limits_{x=0}^n y(t_i)x(t_i)(y(t_i +\Delta t)-y(t_i))$

and

$\int_a^b x(t)y'(t)~dt = \lim\limits_{n \to \infty} \sum\limits_{x=0}^n x(t_i)(y(t_i +\Delta t)-y(t_i))$

It looks like it would work just as long as the $y(t_i +\Delta t)-y(t_i)$ could cancel

When you divide the two you would get $$\dfrac{\int_a^b y(t)x(t)y'(t)~dt}{\int_a^b x(t)y'(t)~dt} \lim\limits_{n \to\infty} \dfrac{\sum\limits_{x=0}^n y(t_i)x(t_i)(y(t_i +\Delta t)-y(t_i))}{\sum\limits_{x=0}^n x(t_i)(y(t_i +\Delta t)-y(t_i))}$$

Where do I go from here to show that this is in fact the weighted average of the two functions?

-
You should explain what $x,y$ and $t$ are. – nbubis Apr 25 '12 at 13:45


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# Range Minimum Query

Given an array $A[1,n]$ of $n$ objects taken from a well-ordered set (such as numbers), a Range Minimum Query (or RMQ) from $i$ to $j$ asks for the position of a minimum element in the sub-array $A[i,j]$.

For example, when $A=[0,5,2,5,4,3,1,6,3]$, then the answer to the range minimum query for the $A[3,8]=[2,5,4,3,1,6]$ is $7$, as $A[7]=1$.

In a typical setting, the array $A$ is static, i.e., elements are not inserted or deleted during a series of queries, and the queries to be answered on-line (i.e., the whole set of queries are not known in advance to the algorithm). In this case a suitable preprocessing the array into a data structure (often called a preprocessing scheme) ensures faster query answering.

It is known that a $O(n)$-time preprocessing is sufficient to answer subsequent queries in $O(1)$ time. The space of the resulting scheme is actually very small, namely $O(n)$ bits (see Fischer & Heun (2007)).

RMQs can be used to solve the lowest common ancestor problem, and is used as a tool for many tasks in exact and approximate string matching.

## References

• Berkman, Omer; Vishkin, Uzi (1993), "Recursive Star-Tree Parallel Data Structure", SIAM Journal on Computing 22 (2): 221–242, doi:10.1137/0222017
• Bender, Michael; Farach-Colton, Martin (2000), "The LCA Problem Revisited", Lecture Notes in Computer Science 1776, Springer-Verlag, pp. 88–94, doi:10.1007/10719839_9
• Fischer, Johannes; Heun, Volker (2007), "A New Succinct Representation of RMQ-Information and Improvements in the Enhanced Suffix Array.", Proceedings of the International Symposium on Combinatorics, Algorithms, Probabilistic and Experimental Methodologies, Lecture Notes in Computer Science 4614, Springer-Verlag, pp. 459–470, doi:10.1007/978-3-540-74450-4_41


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Synopsis

# A New Phase for Superfluid Helium-3

Physics 14, s156
Confining helium-3 in an aerogel container and subjecting it to a magnetic field reveals a long-predicted superfluid phase.

Superfluidity is a familiar phenomenon in helium-4 ( ${}^{4}\text{He}$), whose individual atoms are bosons. It can also occur in helium-3 ( ${}^{3}\text{He}$), when individual (fermionic) atoms pair up to make bosonic complexes called Cooper pairs. Because in ${}^{3}\text{He}$ these Cooper pairs have nonzero total spin, superfluid ${}^{3}\text{He}$ has a richer phase space than superfluid ${}^{4}\text{He}$. Now, Vladimir Dmitriev at the P. L. Kapitza Institute for Physical Problems, Russia, and colleagues have added to that richness, observing the long-predicted, but never-before-seen, superfluid $𝛽$ phase of ${}^{3}\text{He}$ [1].

The main factors that determine the phase of superfluid ${}^{3}\text{He}$ are its temperature and pressure. But additional factors also play a role. One factor is the presence of a magnetic field, which has previously been shown to split the so-called A phase into two phases that contain Cooper pairs with different spin compositions. Another factor, which also changes the structure of the Cooper pairs, is the symmetry of the geometry in which the superfluid is confined. In a previous study, Dmitriev and colleagues confined superfluid ${}^{3}\text{He}$ in an aerogel whose pores were much longer than they were wide. In doing so, they were able to observe the superfluid’s so-called polar phase.

The researchers have now studied this anisotropically confined superfluid in a magnetic field. They find that the magnetic field changes the superfluid’s phase-transition path. Instead of reaching the polar phase via a single transition as the temperature drops, the superfluid goes through two phase transitions. In between these two transitions it forms a previously unseen $𝛽$ phase. The researchers say that their results could provide insight into the behaviors of other Cooper-pair-containing systems, such as unconventional superconductors, quantum gases, and, possibly, neutron stars.

–Marric Stephens

Marric Stephens is a Corresponding Editor for Physics Magazine based in Bristol, UK.

## References

1. V. V. Dmitriev et al., “Superfluid $𝛽$ phase of ${}^{3}\text{He}$,” Phys. Rev. Lett. 127, 265301 (2021).

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00-40 Heinz Hanßmann, Britta Sommer
A degenerate bifurcation in the H\'enon-Heiles family (172K, PostScript, gzipped and uuencoded) Jan 24, 00
Abstract , Paper (src), View paper (auto. generated ps), Index of related papers

Abstract. The normalised H\'enon-Heiles family exhibits a degenerate bifurcation when passing through the separable case $\beta = 0$''. We clarify the relation between this degeneracy and the integrability at $\beta = 0$~. Furthermore we show that the degenerate bifurcation carries over to the H\'enon-Heiles family itself.

Files: 00-40.uu


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# Current Affairs 6th Class

#### Notes - Sorting Materials into Groups

Sorting Materials into Groups   Synopsis   ·                     The process of the classification of objects based on some known criteria is called grouping or sorting.   ·                     We have a huge variety of objects present around us. Because of this, it is very important to classify objects into groups. If we know the properties of one member of the group, it would be very easy to predict the properties of the other members too. Some of the properties of materials are:   ·                     Appearance The various parameters governing the appearance of materials are colour/texture, roughness, shape, size, etc.   ·                      Lustre Some materials shine when light falls on them. This property is generally observed in objects that are grouped as metals, e.g. gold and silver. They are hard, strong, flexible and good conductors of heat and electricity. They can be drawn into wires (ductile) and sheets (malleable).   ·                     Hardness It is the measure of how resistant a material is to various kinds of shape changes when a force is applied. Hardness can also be measured by the ease with which a material can be compressed or stretched. Materials that can be easily stretched and compressed are called soft materials. Materials which are difficult to compress and stretch are called hard materials.   ·                     Solubility In general, solubility is defined as the ability of a substance to dissolve in water. In the process of dissolving/the substance which is being dissolved is called a solute and the substance in which the solute is dissolved is called a solvent. A mixture of solute and solvent is called a solution, e.g. if we add sugar to water, then, sugar is the solute and water is the solvent. The mixture of sugar and water is called sugar solution. Substances which are not soluble in water are called insoluble substances, e.g., sand, wood. If two liquids mix together completely then they are called misdble liquids, e.g. water and ink. If two liquids do not mix but form separate layers, then they are called immiscible liquids, e.g., water and oil.   ·                     Relative density or Density with respect to water $\text{Relative density =}\frac{\text{Mass of substance}}{\text{Mass of water}},\,\,\,\text{Density=}\frac{Mass}{Volume}$ The density of water at ${{4}^{o}}C$ is more...

#### Notes - Fun with Magnets

Fun With Magnets   Synopsis   ·                     Substances that can attract iron are called magnets.   ·                     Substances can be divided into magnetic and non-magnetic substances.   ·                     Substances that are attracted by a magnet are called magnetic substances. Usually things made up of iron, nickel or cobalt are magnetic.   ·                     Substances that are not attracted by a magnet are called nan-magnetic substances. Paper, plastic and wood are a few examples of non-magnetic substances.   ·                     Magnets are divided into two groups - natural and artificial magnets. Magnetite (lodestone) is a naturally occurring magnet.   ·                     Artificial magnets can be of many shapes and are commonly used in different articles like electric bells, radio, etc.   ·                     The magnetic force or the force of attraction in a magnet is concentrated at the two ends of a magnet. These ends are called poles. They are called the north pole and the south pole.   Properties of magnets (i) There are always two poles in a magnet placed at opposite ends. This holds true even if we break or cut the magnet into smaller pieces. Each piece will have two poles. (ii) When two different magnets are brought closer, like poles repel and unlike poles attract each other. (iii) A magnet, when suspended freely, always comes to rest in a line along the north-south line of that place.   ·                     The bar AB (of iron or steel) to be magnetized is stroked with a bar magnet from one end to the other using the same pole as shown in the figure. This process is repeated for about 50 times.   The end A from where the process starts develops the same polarity as the pole of the magnet stroking and the other end develops the opposite polarity.   ·                     Magnets lose their properties when they are heated, hammered or dropped and if they are not stored properly.   ·                     Magnetism is widely used to more...

#### Notes - Electricity and Circuits

Electricity and Circuits   Synopsis   ·                     Electricity is a form of energy widely used by man.   ·                     The simplest form of a circuit is when the two terminals of a cell are connected to the two terminals of a bulb through a switch.   ·                     In an electric circuit/the direction of current is taken from the positive terminal to the negative terminal of the source.   ·                     When a circuit is complete, current (charges) starts flowing. It stops flowing when the circuit breaks at a point.   ·                     A switch is used to break or complete a circuit.   ·                     Materials that allow electricity to pass through them are called conductors. Most of the metals and impure water are examples of conductors.   ·                     Materials that do not allow electricity to pass through them are called insulators. Dry wood, pure water, air and rubber are examples of insulators.   ·                     To represent an electric circuit certain symbols are being used. A list of few symbols are given below in the table.     more...

#### Notes - Light, Shadows and Reflections

Light, Shadows and Reflection   Synopsis   ?              Objects which emit light energy by themselves are called luminous bodies, e.g., the sun, the stars, and glow worms.   ·                     The bodies which do not have light energy of their own but reflect the light energy falling on them and hence are visible to us are called non-luminous bodies. e.g., the moon, objects around us, books, chairs, buildings, trees, etc.   ?              The moon appears bright due to the reflection of sunlight falling on it.   ?              Transparent bodies are substances through which light is propagated easily, e.g., glass/water/etc.   ?              Translucent bodies are substances through which light is propagated partially, e.g., oil, ground glass, etc.   ?              Opaque bodies are substances through which light is not propagated, e.g., wood, iron, etc.   ?              Light travels in straight lines.   ?              Smooth surfaces like mirrors form images.   ·                     The shadow of an object is formed because of the rectilinear propagation of light. A shadow is the area of darkness formed on the screen, when an opaque body is placed in between the screen and a source of light   ·                     A pin hole camera uses the principle of rectilinear propagation of light. It produces a real image which is much smaller than the object and is inverted (upside down).   ·                     The returning of a light ray passing through an optical medium into the same medium from the surface of the second medium is called the reflection of light.   ?              The surfaces which reflect light are called reflecting surfaces.   ?              Images are different from shadows.

#### Notes - Motion and Measurement of Distances

Motion and Measurement of Distances   Synopsis   ?                     Different modes of transport are used to go from one place to another.   ·                     In ancient times, people used the length of a foot, the width of a finger, the distance of a step as units of measurement. This caused variation and inaccuracy and a need to develop a uniform system of measurement arose.   ?                     Now, the International System of Units (S.I. unit) is followed all over the world.   ·                     A unit is a standardized quantity of a physical property, used as a factor to express quantities of that property.   ·                     A standard unit is the measurement value which remains the same even when it is measured by anybody at any place.   ·                     The standard units of length are millimetre (mm), centimetre (cm), metre (m) and kilometer (km).   ·                     The following is the relationship between the units. $10\text{ }mm=1\text{ }cm$ $100\,\,cm=1\text{ }m$ $1000\text{ }m=1\text{ }km$     ·                     The correct technique of measuring length is the eye being directly in line with the other end of the object as shown in the adjacent figure.   ·                     A thread or string and a ruler are used for measuring the length of a curved line and the circumference of a ball.   Types of motion   ?                     Motion in a straight line is called rectilinear motion.   ·                     A man walking and the motion of a cycle, a bus and that of a car on a straight path are all in translatory motion.   ·                     If the motions of points or parts of an object are along a circular path instead of a straight line path/then the motion of the body is said to be in rotatory motion.   ?                     more...

#### Notes - Fibre to Fabric

Fibre to Fabric   Synopsis   ·                     The thin strands of thread which are used to make yarns are called fibres.   ·                     Yarn is made by twisting the fibres.   ·                     Fibres are of two types - natural and synthetic.   ·                     Natural fibres are obtained either from plants or animals. e.g., cotton, jute, coconut, flax, silk, wool, etc.   ·                     Synthetic fibres are man-made fibres, which are made by using chemicals. e.g., polyester, acrylic, nylon, etc.   ·                     Separating seeds from cotton bolls is called ginning.   ·                     Yarn is spun from cotton fibre using takli or charkha.   ·                     Process of spinning yarn into a fabric is called weaving.   ·                     Jute fibre is made by immersing jute stem in water for a few days.   ·                     Woollen fabric is made by knitting. It is the criss-cross laying of yarns tightly using weaving machines or knitting rods. For example, a sweater is made of a fabric (wool) made of a single yarn (thread).   ·                     Woollen fibres are obtained from the fleece or fur of animals. Silk fibres are more...

#### Food

Food   Food Food is a substance eaten by the living organisms in order to remain alive. Fruits, vegetables, eggs, meat, milk, etc. are the examples of food.   Sources of Food Plants: Foods like cereals, pulses, fruits, vegetables, etc. are obtained from the plants. Animals: Food like meat, milk, eggs, honey, etc. are obtained from animals.   Producers and Consumers All living organisms has been broadly classified into two groups: producers and consumers.   Producers: All the green plants make their own food, therefore, green plants are called producers.                                               Consumers: Animals consume the food prepared by green plants, therefore, animals are called consumers.   Types of Consumers Consumers have been classified in the following groups on the basis of their food habits:   Herbivores The animals that eat only plants or plant products, are called herbivores. Cow, buffalo, goat, elephant, horse, rabbit, camel, etc. are the examples of herbivores animals.   Carnivores The animals that eat only other animals are called carnivores. Tiger, lion, cheetah, wolf, snake, etc. are the examples of carnivores animals. Omnivores The animals that eat both plants and animals are called omnivores. Man, bear, dog, crow etc. are the examples of omnivores animals.     Scavengers The animals that eat dead and decaying bodies of other animals are called scavengers. Vulture, hyena, etc. are the examples of scavengers.     Decomposers These are organisms such as bacteria and fungi that break down the dead remains of plants and animals.     Parasites The animals that live in or on the other animals for their survival are called parasites. For example, mosquitoes, flees, leeches, etc.     Components of Food The food that we eat is composed of carbohydrates, fats, proteins, vitamins, minerals, water, and roughage.     Functions and Sources of different components of food are given below:
 Circuit components Symbols Cell Battery Tap key Resistor
more...

#### Fibre to Fabric

Fibre to Fabric   Fibre to Fabric We use clothes for making variety of things like shirts, pants, skirts, sarees, uniforms, bed sheets, blankets, curtains, table cloths, towels, school bags, gunny bags, etc. So clothes are very important for us. In this chapter we will study about different types of fibres.   Clothes: Clothes are fabrics which are made up of yarns and in turn yarns are made up of fibres.   Fibres: Fibres are very thin, thread-like strand from which clothes are made up of.   Yarns: Yarns are made up of fibres. Fibres are turned into long twisted thread by the process of spinning called yarns.   Fabrics: Yarns are woven together to make fabrics. Then fabrics are used to prepare different types of clothes like shirts, pants, bags, etc.   Types of Fibres There are two types of fibres: natural fibres and synthetic fibres.   Natural Fibres The fibres which are obtained from natural sources are called natural fibres. Cotton, wool, silk, flax, and jute are the examples of natural fibres.   Cotton We obtain cotton fibres from cotton plants. Seeds of cotton plants are covered with white, soft cotton fibres. Cotton fibres are separated from seeds by hands or machines. The preparation of clothes or fabrics from cotton wools or cotton fibres involves mainly three processes which are as follows:   Ginning: The process of separating cotton wool from the cotton seeds is called ginning.   Spinning: The cotton fibres from cotton wool are drawn out and twisted to make yarns. This process is called spinning.   Weaving: In the process of weaving two distinct sets of yarns are interlaced at right angles to form a cloth. Cotton is mainly used to make clothes as it is soft and comfortable to wear. Cotton clothes let air in and can absorb sweat, therefore, very useful specially in hot and humid weather.   Wool Wool fibres are obtained from the hair of animals like sheep, goat, rabbit, yak and camel. Wool is cut off from the sheep with a thin layer of skin. The process is known as shearing. After shearing the wool is sent to the mills. Where it is cleaned, combed and spun to make fibre. This fibre is then woven or knitted to make wool clothes. Wool is used for knitting sweaters, shawls and other wool clothes. Wool is a fluffy material and can retain air inside it, therefore, woolen clothes are very useful in winter season.   Silk Silk fibres are obtained from silk worms. Caterpillars of silkworms cover themself from silk fibres called cocoon. The cocoon is boiled in water to separate the silk fibres from the silkworms. The process is known as reeling. Further the obtained silk fibres twisted to obtain silk yarns then silk yarns are woven to make silk clothes. The rearing of silkworm for the production of silk fibres is called sericulture.   Jute Jute fibres are obtained from more...

#### Grouping Materials and their Separation

Grouping Materials and their Separation   Sorting Materials into Groups All the objects that we see around us are made up of matter called materials. Like, animals, insects, birds, plants, trees, houses, machines, tools, tables, chairs, clothes, etc. are objects which are made up of matter. To understand the process of classification, first we have to know the basis of classification.   Basis of Classification The characteristics or properties, which are taken as the base for the classification of given objects, are called basis of classification. There are a number of characteristics which are taken as the basis for classification. Like 'living and non-living', 'natural and artificial’, 'solid, liquid and gas', ' shape', 'colour', etc. Note: You can also classify the materials by selecting the characteristics as per your own choice.   Properties of Matter Now let us understand some properties of matter.   Appearance Some objects have shiny appearance called lustre whereas some materials have dull appearance. For example, gold, silver, copper have shiny appearance whereas paper, wood, rubber have dull appearance.   Hardness and Softness The materials can be classified on the basis of hardness and softness. Candle, wax, rubber are the examples of soft materials whereas iron, diamond, glass are the examples of hard materials.   Solubility Some materials are soluble in water whereas others are not. Sugar, common salt, washing soda, lemon juice are soluble in water whereas glass, plastic, iron, mustard oil, coconut oil are not soluble in water.   Density Density is the mass per unit volume of a substance. The substances which sink in the water have higher density than water and the substances which float on the water have lower density than water. Substances like iron, copper, aluminium, silver, gold and glass have higher density than water whereas wood, plastic, ice, oil and petrol have lower density than water.   Transparency All the materials have been classified into three groups on the basis of transparency.
• Transparent: The materials through which light can pass are called transparent materials. For example, glass, water, air, alcohol, etc.
• Translucent: The materials through which light can pass partially are called translucent materials. For example. Butter paper, ground-glass, muddy water, etc.
• Opaque: The materials through which light cannot pass are called opaque materials.
For example, metals, stones, books, woods, etc.   Pure Substances The substances which are made up of only one kind of atoms or molecules are called pure substances. For example gold, silver, copper, etc.   Mixture When two or more than two different substances are mixed up together a mixture is formed. For example, air is a mixture of many gases like nitrogen, oxygen, carbon dioxide and some other gases.   Constituents of a Mixture The different substances which are present in a mixture are called constituents of the mixture. For example, nitrogen, oxygen, carbon dioxide and many other gases which are present in the air are constituents of air.

#### Changes Arorund Us

Changes Around Us   Changes Around Us We observe changes around us all the time. Changes may occur in shape, size, mass, density, colour, position, temperature, structure or in composition of a substance. So we can define a change as: ‘Transformation in one or more than one physical or chemical properties of a substance is called change’.   Types of Changes Types of changes on the basis of either the changes can be reversed to bring back the original substance or not:   Reversible Change A change which can be reversed to form the 'original substance' is called reversible change. For example, melting of ice, freezing of water, dissolution of salt in water, increase in temperature of a metal rod, etc.   Irreversible Changes A change which cannot be reversed to form the 'original substance' called irreversible change. For example, burning of wood, ripening of fruit, turning milk sour, etc. Types of changes on the basis of either a new substance is formed or not:   Physical Change The change, in which molecules of a substance do not undergo any change or no new substances are formed, are called physical changes. For example, melting of ice, freezing of water, evaporation of water, dissolution of salt in water.   Chemical Change The change, in which molecules of substance undergo change or new substances are formed, are called chemical changes. For example, burning of paper, rusting of iron, spoliation of food, etc. Types of changes on the basis of heat absorbed or evolved:   Exothermic The change in which heat is released. For example, burning of wood.   Endothermic The change in which heat is absorbed. For example, melting of ice.

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### I-0.4 - Mining Tax Act

32. An operator that sustains an annual loss in a fiscal year may claim, on or before the date on or before which the operator is required to file its return under section 36 for the fiscal year, an amount as a refundable duties credit for losses which must not exceed
(1)  for a fiscal year that ends before 31 March 2010, 12% of the lesser of
(a)  its adjusted annual loss for the fiscal year, and
(b)  the aggregate of the following amounts, without however exceeding the amount deducted by the operator under subparagraph e of paragraph 2 of section 8, as it read on 30 March 2010, in computing its annual profit for the fiscal year:
i.  the amount that is the amount by which the expenses in respect of exploration, mineral deposit evaluation and mine development work, incurred by the operator for the fiscal year in connection with mining operation, exceeds the amount of government assistance that the operator received or was entitled to receive for the fiscal year and that relates to those expenses, and provided that such expenses, despite section 16.2, have been declared by the operator to be deductible expenses, on or before the date on or before which the operator is required to file its return under section 36 for the fiscal year, and
ii.  the aggregate of all amounts each of which is the amount by which an amount referred to in subparagraph b.1 of paragraph 1 of section 16.1, as it read on 30 March 2010, that relates to expenses incurred by the operator during the fiscal year and declared by the operator to be deductible expenses, on or before the date provided in subparagraph i, exceeds the amount that is 25% of the government assistance that the operator received or was entitled to receive for the fiscal year and that relates to those expenses;
(2)  for a fiscal year that ends after 30 March 2010 and includes that date, if the operator is an eligible operator, the amount obtained by multiplying its tax rate for the fiscal year by the lesser of
(a)  its adjusted annual loss for the fiscal year, and
(b)  the aggregate of
i.  the expenses referred to in subparagraph b of subparagraph 1 of the second paragraph of section 16.1 that were incurred by the operator for the fiscal year, without exceeding the amount deducted by the operator for the fiscal year under subparagraph e of paragraph 2 of section 8, as it read on 30 March 2010,
ii.  50% of the amount of the expenses referred to in subparagraph a of subparagraph 1 of the second paragraph of section 16.9 that were incurred by the operator for the fiscal year, without exceeding the amount deducted by the operator for the fiscal year under subparagraph e.1 of paragraph 2 of section 8, as it read on 30 March 2010, and
iii.  the amount of the expenses referred to in subparagraph a of subparagraph 1 of the second paragraph of section 16.11 that were incurred by the operator for the fiscal year, without exceeding the amount deducted by the operator for the fiscal year under subparagraph e.2 of paragraph 2 of section 8, as it read on 30 March 2010;
(3)  for a fiscal year that ends after 30 March 2010 and includes that date, if the operator is not an eligible operator, the amount obtained by multiplying its tax rate for the fiscal year by the lesser of
(a)  its adjusted annual loss for the fiscal year, and
(b)  the aggregate of
i.  the expenses referred to in subparagraph b of subparagraph 1 of the second paragraph of section 16.1 that were incurred by the operator for the fiscal year, without exceeding the amount deducted by the operator for the fiscal year under subparagraph e of paragraph 2 of section 8, as it read on 30 March 2010, and
ii.  the amount of the expenses referred to in subparagraph a of subparagraph 1 of the second paragraph of section 16.11 that were incurred by the operator for the fiscal year, without exceeding the amount deducted by the operator for the fiscal year under subparagraph e.2 of paragraph 2 of section 8, as it read on 30 March 2010;
(4)  if the operator is an eligible operator, the amount obtained by multiplying, for a fiscal year that begins after 30 March 2010 but before 1 January 2014, its tax rate for that fiscal year and, for a fiscal year that begins after 31 December 2013, 16% by the lesser of;
(a)  its adjusted annual loss for the fiscal year, and
(b)  the aggregate of
i.  50% of the amount of the expenses referred to in subparagraph a of subparagraph 1 of the second paragraph of section 16.9 that were incurred by the operator for the fiscal year, without exceeding the amount deducted by the operator for the fiscal year under subparagraph d of subparagraph 2 of the second paragraph of section 8,
ii.  the amount of the expenses referred to in subparagraph a of subparagraph 1 of the second paragraph of section 16.11 that were incurred by the operator for the fiscal year, without exceeding the amount deducted by the operator for the fiscal year under subparagraph f of subparagraph 2 of the second paragraph of section 8;
iii.  the aggregate of all amounts each of which is the amount of the expenses referred to in subparagraph a of subparagraph 1 of the second paragraph of section 16.13.2 that were incurred by the operator for the fiscal year, without exceeding the amount deducted by the operator for the fiscal year under subparagraph h of subparagraph 2 of the second paragraph of section 8,
iv.  the aggregate of all amounts each of which is the amount of the expenses referred to in subparagraph a of subparagraph 1 of the second paragraph of section 16.13.4 that were incurred by the operator for the fiscal year, without exceeding the amount deducted by the operator for the fiscal year under subparagraph i of subparagraph 2 of the second paragraph of section 8; and
v.  the aggregate of all amounts each of which is the amount of the expenses referred to in subparagraph a of subparagraph 1 of the second paragraph of section 16.13.6 that were incurred by the operator for the fiscal year, without exceeding the amount deducted by the operator for the fiscal year under subparagraph j of subparagraph 2 of the second paragraph of section 8; and
(5)  if the operator is not an eligible operator, the amount obtained by multiplying, for a fiscal year that begins after 30 March 2010 but before 1 January 2014, its tax rate for that fiscal year and, for a fiscal year that begins after 31 December 2013, 16% by the lesser of
(a)  its adjusted annual loss for the fiscal year, and
(b)  the aggregate of
i.  the amount of the expenses referred to in subparagraph a of subparagraph 1 of the second paragraph of section 16.11 that were incurred by the operator for the fiscal year, without exceeding the amount deducted by the operator for the fiscal year under subparagraph f of subparagraph 2 of the second paragraph of section 8;
ii.  the aggregate of all amounts each of which is the amount of the expenses referred to in subparagraph a of subparagraph 1 of the second paragraph of section 16.13.2 that were incurred by the operator for the fiscal year, without exceeding the amount deducted by the operator for the fiscal year under subparagraph h of subparagraph 2 of the second paragraph of section 8,
iii.  the aggregate of all amounts each of which is the amount of the expenses referred to in subparagraph a of subparagraph 1 of the second paragraph of section 16.13.4 that were incurred by the operator for the fiscal year, without exceeding the amount deducted by the operator for the fiscal year under subparagraph i of subparagraph 2 of the second paragraph of section 8; and
iv.  the aggregate of all amounts each of which is the amount of the expenses referred to in subparagraph a of subparagraph 1 of the second paragraph of section 16.13.6 that were incurred by the operator for the fiscal year, without exceeding the amount deducted by the operator for the fiscal year under subparagraph j of subparagraph 2 of the second paragraph of section 8.
For the purpose of determining the amount of the expenses referred to in subparagraphs i and ii of subparagraph b of subparagraph 1 of the first paragraph, of the expenses referred to in subparagraphs i to iii of subparagraph b of subparagraph 2 of that paragraph, of the expenses referred to in subparagraphs i and ii of subparagraph b of subparagraph 3 of that paragraph, of the expenses referred to in subparagraphs i to v of subparagraph b of subparagraph 4 of that paragraph and of the expenses referred to in subparagraphs i to iv of subparagraph b of subparagraph 5 of that paragraph that were incurred by an operator for a fiscal year, the following rules apply:
(1)  the amount of the expenses is to be reduced by the amount of government assistance that the operator received or was entitled to receive for the fiscal year and that relates to the expenses; and
(2)  despite section 16.14, the expenses incurred by the operator in the fiscal year may be included in the expenses referred to in those subparagraphs only if the operator declares them as such to the Minister on or before the date on or before which the operator is required to file its return under section 36 for the fiscal year.
For the purposes of subparagraph i of subparagraph b of subparagraph 1 of the first paragraph, exploration , mine deposit evaluation , mine development , mineral deposit , mining operation and orebody have the meaning assigned by section 1, as it read on 30 March 2010.
1975, c. 30, s. 32; 1985, c. 39, s. 6; 1994, c. 47, s. 32; 1999, c. 83, s. 17; 2011, c. 6, s. 61; 2015, c. 21, s. 76; 2019, c. 14, s. 51; 2020, c. 16, s. 21.
32. An operator that sustains an annual loss in a fiscal year may claim, on or before the date on or before which the operator is required to file its return under section 36 for the fiscal year, an amount as a refundable duties credit for losses which must not exceed
(1)  for a fiscal year that ends before 31 March 2010, 12% of the lesser of
(a)  its adjusted annual loss for the fiscal year, and
(b)  the aggregate of the following amounts, without however exceeding the amount deducted by the operator under subparagraph e of paragraph 2 of section 8, as it read on 30 March 2010, in computing its annual profit for the fiscal year:
i.  the amount that is the amount by which the expenses in respect of exploration, mineral deposit evaluation and mine development work, incurred by the operator for the fiscal year in connection with mining operation, exceeds the amount of government assistance that the operator received or was entitled to receive for the fiscal year and that relates to those expenses, and provided that such expenses, despite section 16.2, have been declared by the operator to be deductible expenses, on or before the date on or before which the operator is required to file its return under section 36 for the fiscal year, and
ii.  the aggregate of all amounts each of which is the amount by which an amount referred to in subparagraph b.1 of paragraph 1 of section 16.1, as it read on 30 March 2010, that relates to expenses incurred by the operator during the fiscal year and declared by the operator to be deductible expenses, on or before the date provided in subparagraph i, exceeds the amount that is 25% of the government assistance that the operator received or was entitled to receive for the fiscal year and that relates to those expenses;
(2)  for a fiscal year that ends after 30 March 2010 and includes that date, if the operator is an eligible operator, the amount obtained by multiplying its tax rate for the fiscal year by the lesser of
(a)  its adjusted annual loss for the fiscal year, and
(b)  the aggregate of
i.  the expenses referred to in subparagraph b of subparagraph 1 of the second paragraph of section 16.1 that were incurred by the operator for the fiscal year, without exceeding the amount deducted by the operator for the fiscal year under subparagraph e of paragraph 2 of section 8, as it read on 30 March 2010,
ii.  50% of the amount of the expenses referred to in subparagraph a of subparagraph 1 of the second paragraph of section 16.9 that were incurred by the operator for the fiscal year, without exceeding the amount deducted by the operator for the fiscal year under subparagraph e.1 of paragraph 2 of section 8, as it read on 30 March 2010, and
iii.  the amount of the expenses referred to in subparagraph a of subparagraph 1 of the second paragraph of section 16.11 that were incurred by the operator for the fiscal year, without exceeding the amount deducted by the operator for the fiscal year under subparagraph e.2 of paragraph 2 of section 8, as it read on 30 March 2010;
(3)  for a fiscal year that ends after 30 March 2010 and includes that date, if the operator is not an eligible operator, the amount obtained by multiplying its tax rate for the fiscal year by the lesser of
(a)  its adjusted annual loss for the fiscal year, and
(b)  the aggregate of
i.  the expenses referred to in subparagraph b of subparagraph 1 of the second paragraph of section 16.1 that were incurred by the operator for the fiscal year, without exceeding the amount deducted by the operator for the fiscal year under subparagraph e of paragraph 2 of section 8, as it read on 30 March 2010, and
ii.  the amount of the expenses referred to in subparagraph a of subparagraph 1 of the second paragraph of section 16.11 that were incurred by the operator for the fiscal year, without exceeding the amount deducted by the operator for the fiscal year under subparagraph e.2 of paragraph 2 of section 8, as it read on 30 March 2010;
(4)  if the operator is an eligible operator, the amount obtained by multiplying, for a fiscal year that begins after 30 March 2010 but before 1 January 2014, its tax rate for that fiscal year and, for a fiscal year that begins after 31 December 2013, 16% by the lesser of;
(a)  its adjusted annual loss for the fiscal year, and
(b)  the aggregate of
i.  50% of the amount of the expenses referred to in subparagraph a of subparagraph 1 of the second paragraph of section 16.9 that were incurred by the operator for the fiscal year, without exceeding the amount deducted by the operator for the fiscal year under subparagraph d of subparagraph 2 of the second paragraph of section 8,
ii.  the amount of the expenses referred to in subparagraph a of subparagraph 1 of the second paragraph of section 16.11 that were incurred by the operator for the fiscal year, without exceeding the amount deducted by the operator for the fiscal year under subparagraph f of subparagraph 2 of the second paragraph of section 8;
iii.  the aggregate of all amounts each of which is the amount of the expenses referred to in subparagraph a of subparagraph 1 of the second paragraph of section 16.13.2 that were incurred by the operator for the fiscal year, without exceeding the amount deducted by the operator for the fiscal year under subparagraph h of subparagraph 2 of the second paragraph of section 8, and
iv.  the aggregate of all amounts each of which is the amount of the expenses referred to in subparagraph a of subparagraph 1 of the second paragraph of section 16.13.4 that were incurred by the operator for the fiscal year, without exceeding the amount deducted by the operator for the fiscal year under subparagraph i of subparagraph 2 of the second paragraph of section 8; and
(5)  if the operator is not an eligible operator, the amount obtained by multiplying, for a fiscal year that begins after 30 March 2010 but before 1 January 2014, its tax rate for that fiscal year and, for a fiscal year that begins after 31 December 2013, 16% by the lesser of
(a)  its adjusted annual loss for the fiscal year, and
(b)  the aggregate of
i.  the amount of the expenses referred to in subparagraph a of subparagraph 1 of the second paragraph of section 16.11 that were incurred by the operator for the fiscal year, without exceeding the amount deducted by the operator for the fiscal year under subparagraph f of subparagraph 2 of the second paragraph of section 8,
ii.  the aggregate of all amounts each of which is the amount of the expenses referred to in subparagraph a of subparagraph 1 of the second paragraph of section 16.13.2 that were incurred by the operator for the fiscal year, without exceeding the amount deducted by the operator for the fiscal year under subparagraph h of subparagraph 2 of the second paragraph of section 8, and
iii.  the aggregate of all amounts each of which is the amount of the expenses referred to in subparagraph a of subparagraph 1 of the second paragraph of section 16.13.4 that were incurred by the operator for the fiscal year, without exceeding the amount deducted by the operator for the fiscal year under subparagraph i of subparagraph 2 of the second paragraph of section 8.
For the purpose of determining the amount of the expenses referred to in subparagraphs i and ii of subparagraph b of subparagraph 1 of the first paragraph, of the expenses referred to in subparagraphs i to iii of subparagraph b of subparagraph 2 of that paragraph, of the expenses referred to in subparagraphs i and ii of subparagraph b of subparagraph 3 of that paragraph, of the expenses referred to in subparagraphs i to iv of subparagraph b of subparagraph 4 of that paragraph and of the expenses referred to in subparagraphs i to iii of subparagraph b of subparagraph 5 of that paragraph that were incurred by an operator for a fiscal year, the following rules apply:
(1)  the amount of the expenses is to be reduced by the amount of government assistance that the operator received or was entitled to receive for the fiscal year and that relates to the expenses; and
(2)  despite section 16.14, the expenses incurred by the operator in the fiscal year may be included in the expenses referred to in those subparagraphs only if the operator declares them as such to the Minister on or before the date on or before which the operator is required to file its return under section 36 for the fiscal year.
For the purposes of subparagraph i of subparagraph b of subparagraph 1 of the first paragraph, exploration , mine deposit evaluation , mine development , mineral deposit , mining operation and orebody have the meaning assigned by section 1, as it read on 30 March 2010.
1975, c. 30, s. 32; 1985, c. 39, s. 6; 1994, c. 47, s. 32; 1999, c. 83, s. 17; 2011, c. 6, s. 61; 2015, c. 21, s. 76; 2019, c. 14, s. 51.
32. An operator that sustains an annual loss in a fiscal year may claim, on or before the date on or before which the operator is required to file its return under section 36 for the fiscal year, an amount as a refundable duties credit for losses which must not exceed
(1)  for a fiscal year that ends before 31 March 2010, 12% of the lesser of
(a)  its adjusted annual loss for the fiscal year, and
(b)  the aggregate of the following amounts, without however exceeding the amount deducted by the operator under subparagraph e of paragraph 2 of section 8, as it read on 30 March 2010, in computing its annual profit for the fiscal year:
i.  the amount that is the amount by which the expenses in respect of exploration, mineral deposit evaluation and mine development work, incurred by the operator for the fiscal year in connection with mining operation, exceeds the amount of government assistance that the operator received or was entitled to receive for the fiscal year and that relates to those expenses, and provided that such expenses, despite section 16.2, have been declared by the operator to be deductible expenses, on or before the date on or before which the operator is required to file its return under section 36 for the fiscal year, and
ii.  the aggregate of all amounts each of which is the amount by which an amount referred to in subparagraph b.1 of paragraph 1 of section 16.1, as it read on 30 March 2010, that relates to expenses incurred by the operator during the fiscal year and declared by the operator to be deductible expenses, on or before the date provided in subparagraph i, exceeds the amount that is 25% of the government assistance that the operator received or was entitled to receive for the fiscal year and that relates to those expenses;
(2)  for a fiscal year that ends after 30 March 2010 and includes that date, if the operator is an eligible operator, the amount obtained by multiplying its tax rate for the fiscal year by the lesser of
(a)  its adjusted annual loss for the fiscal year, and
(b)  the aggregate of
i.  the expenses referred to in subparagraph b of subparagraph 1 of the second paragraph of section 16.1 that were incurred by the operator for the fiscal year, without exceeding the amount deducted by the operator for the fiscal year under subparagraph e of paragraph 2 of section 8, as it read on 30 March 2010,
ii.  50% of the amount of the expenses referred to in subparagraph a of subparagraph 1 of the second paragraph of section 16.9 that were incurred by the operator for the fiscal year, without exceeding the amount deducted by the operator for the fiscal year under subparagraph e.1 of paragraph 2 of section 8, as it read on 30 March 2010, and
iii.  the amount of the expenses referred to in subparagraph a of subparagraph 1 of the second paragraph of section 16.11 that were incurred by the operator for the fiscal year, without exceeding the amount deducted by the operator for the fiscal year under subparagraph e.2 of paragraph 2 of section 8, as it read on 30 March 2010;
(3)  for a fiscal year that ends after 30 March 2010 and includes that date, if the operator is not an eligible operator, the amount obtained by multiplying its tax rate for the fiscal year by the lesser of
(a)  its adjusted annual loss for the fiscal year, and
(b)  the aggregate of
i.  the expenses referred to in subparagraph b of subparagraph 1 of the second paragraph of section 16.1 that were incurred by the operator for the fiscal year, without exceeding the amount deducted by the operator for the fiscal year under subparagraph e of paragraph 2 of section 8, as it read on 30 March 2010, and
ii.  the amount of the expenses referred to in subparagraph a of subparagraph 1 of the second paragraph of section 16.11 that were incurred by the operator for the fiscal year, without exceeding the amount deducted by the operator for the fiscal year under subparagraph e.2 of paragraph 2 of section 8, as it read on 30 March 2010;
(4)  if the operator is an eligible operator, the amount obtained by multiplying, for a fiscal year that begins after 30 March 2010 but before 1 January 2014, its tax rate for that fiscal year and, for a fiscal year that begins after 31 December 2013, 16% by the lesser of;
(a)  its adjusted annual loss for the fiscal year, and
(b)  the aggregate of
i.  50% of the amount of the expenses referred to in subparagraph a of subparagraph 1 of the second paragraph of section 16.9 that were incurred by the operator for the fiscal year, without exceeding the amount deducted by the operator for the fiscal year under subparagraph d of subparagraph 2 of the second paragraph of section 8, and
ii.  the amount of the expenses referred to in subparagraph a of subparagraph 1 of the second paragraph of section 16.11 that were incurred by the operator for the fiscal year, without exceeding the amount deducted by the operator for the fiscal year under subparagraph f of subparagraph 2 of the second paragraph of section 8; and
(5)  if the operator is not an eligible operator, the amount obtained by multiplying, for a fiscal year that begins after 30 March 2010 but before 1 January 2014, its tax rate for that fiscal year and, for a fiscal year that begins after 31 December 2013, 16% by the lesser of
(a)  its adjusted annual loss for the fiscal year, and
(b)  the amount of the expenses referred to in subparagraph a of subparagraph 1 of the second paragraph of section 16.11 that were incurred by the operator for the fiscal year, without exceeding the amount deducted by the operator for the fiscal year under subparagraph f of subparagraph 2 of the second paragraph of section 8.
For the purpose of determining the amount of the expenses referred to in subparagraphs i and ii of subparagraph b of subparagraph 1 of the first paragraph, of the expenses referred to in subparagraphs i to iii of subparagraph b of subparagraph 2 of that paragraph, of the expenses referred to in subparagraphs i and ii of subparagraph b of subparagraphs 3 and 4 of that paragraph and of the expenses referred to in subparagraph b of subparagraph 5 of that paragraph that were incurred by an operator for a fiscal year, the following rules apply:
(1)  the amount of the expenses is to be reduced by the amount of government assistance that the operator received or was entitled to receive for the fiscal year and that relates to the expenses; and
(2)  despite section 16.14, the expenses incurred by the operator in the fiscal year may be included in the expenses referred to in those subparagraphs only if the operator declares them as such to the Minister on or before the date on or before which the operator is required to file its return under section 36 for the fiscal year.
For the purposes of subparagraph i of subparagraph b of subparagraph 1 of the first paragraph, exploration , mine deposit evaluation , mine development , mineral deposit , mining operation and orebody have the meaning assigned by section 1, as it read on 30 March 2010.
1975, c. 30, s. 32; 1985, c. 39, s. 6; 1994, c. 47, s. 32; 1999, c. 83, s. 17; 2011, c. 6, s. 61; 2015, c. 21, s. 76.
32. An operator that sustains an annual loss in a fiscal year may claim, on or before the date on or before which the operator is required to file its return under section 36 for the fiscal year, an amount as a credit on duties refundable for losses which must not exceed
(1)  for a fiscal year that ends before 31 March 2010, 12% of the lesser of
(a)  its adjusted annual loss for the fiscal year, and
(b)  the aggregate of the following amounts, without however exceeding the amount deducted by the operator under subparagraph e of paragraph 2 of section 8, as it read on 30 March 2010, in computing its annual profit for the fiscal year:
i.  the amount that is the amount by which the expenses in respect of exploration, mineral deposit evaluation and mine development work, incurred by the operator for the fiscal year in connection with mining operation, exceeds the amount of government assistance that the operator received or was entitled to receive for the fiscal year and that relates to those expenses, and provided that such expenses, despite section 16.2, have been declared by the operator to be deductible expenses, on or before the date on or before which the operator is required to file its return under section 36 for the fiscal year, and
ii.  the aggregate of all amounts each of which is the amount by which an amount referred to in subparagraph b.1 of paragraph 1 of section 16.1, as it read on 30 March 2010, that relates to expenses incurred by the operator during the fiscal year and declared by the operator to be deductible expenses, on or before the date provided in subparagraph i, exceeds the amount that is 25% of the government assistance that the operator received or was entitled to receive for the fiscal year and that relates to those expenses;
(2)  for a fiscal year that ends after 30 March 2010 and includes that date, if the operator is an eligible operator, the amount obtained by multiplying its tax rate for the fiscal year by the lesser of
(a)  its adjusted annual loss for the fiscal year, and
(b)  the aggregate of
i.  the expenses referred to in subparagraph b of subparagraph 1 of the second paragraph of section 16.1 that were incurred by the operator for the fiscal year, without exceeding the amount deducted by the operator for the fiscal year under subparagraph e of paragraph 2 of section 8, as it read on 30 March 2010,
ii.  50% of the amount of the expenses referred to in subparagraph a of subparagraph 1 of the second paragraph of section 16.9 that were incurred by the operator for the fiscal year, without exceeding the amount deducted by the operator for the fiscal year under subparagraph e.1 of paragraph 2 of section 8, as it read on 30 March 2010, and
iii.  the amount of the expenses referred to in subparagraph a of subparagraph 1 of the second paragraph of section 16.11 that were incurred by the operator for the fiscal year, without exceeding the amount deducted by the operator for the fiscal year under subparagraph e.2 of paragraph 2 of section 8, as it read on 30 March 2010;
(3)  for a fiscal year that ends after 30 March 2010 and includes that date, if the operator is not an eligible operator, the amount obtained by multiplying its tax rate for the fiscal year by the lesser of
(a)  its adjusted annual loss for the fiscal year, and
(b)  the aggregate of
i.  the expenses referred to in subparagraph b of subparagraph 1 of the second paragraph of section 16.1 that were incurred by the operator for the fiscal year, without exceeding the amount deducted by the operator for the fiscal year under subparagraph e of paragraph 2 of section 8, as it read on 30 March 2010, and
ii.  the amount of the expenses referred to in subparagraph a of subparagraph 1 of the second paragraph of section 16.11 that were incurred by the operator for the fiscal year, without exceeding the amount deducted by the operator for the fiscal year under subparagraph e.2 of paragraph 2 of section 8, as it read on 30 March 2010;
(4)  for a fiscal year that begins after 30 March 2010, if the operator is an eligible operator, the amount obtained by multiplying its tax rate for the fiscal year by the lesser of
(a)  its adjusted annual loss for the fiscal year, and
(b)  the aggregate of
i.  50% of the amount of the expenses referred to in subparagraph a of subparagraph 1 of the second paragraph of section 16.9 that were incurred by the operator for the fiscal year, without exceeding the amount deducted by the operator for the fiscal year under subparagraph d of subparagraph 2 of the second paragraph of section 8, and
ii.  the amount of the expenses referred to in subparagraph a of subparagraph 1 of the second paragraph of section 16.11 that were incurred by the operator for the fiscal year, without exceeding the amount deducted by the operator for the fiscal year under subparagraph f of subparagraph 2 of the second paragraph of section 8; and
(5)  for a fiscal year that begins after 30 March 2010, if the operator is not an eligible operator, the amount obtained by multiplying its tax rate for the fiscal year by the lesser of
(a)  its adjusted annual loss for the fiscal year, and
(b)  the amount of the expenses referred to in subparagraph a of subparagraph 1 of the second paragraph of section 16.11 that were incurred by the operator for the fiscal year, without exceeding the amount deducted by the operator for the fiscal year under subparagraph f of subparagraph 2 of the second paragraph of section 8.
For the purpose of determining the amount of the expenses referred to in subparagraphs i and ii of subparagraph b of subparagraph 1 of the first paragraph, of the expenses referred to in subparagraphs i to iii of subparagraph b of subparagraph 2 of that paragraph, of the expenses referred to in subparagraphs i and ii of subparagraph b of subparagraphs 3 and 4 of that paragraph and of the expenses referred to in subparagraph b of subparagraph 5 of that paragraph that were incurred by an operator for a fiscal year, the following rules apply:
(1)  the amount of the expenses is to be reduced by the amount of government assistance that the operator received or was entitled to receive for the fiscal year and that relates to the expenses; and
(2)  despite section 16.14, the expenses incurred by the operator in the fiscal year may be included in the expenses referred to in those subparagraphs only if the operator declares them as such to the Minister on or before the date on or before which the operator is required to file its return under section 36 for the fiscal year.
For the purposes of subparagraph i of subparagraph b of subparagraph 1 of the first paragraph, exploration , mine deposit evaluation , mine development , mineral deposit , mining operation and orebody have the meaning assigned by section 1, as it read on 30 March 2010.
1975, c. 30, s. 32; 1985, c. 39, s. 6; 1994, c. 47, s. 32; 1999, c. 83, s. 17; 2011, c. 6, s. 61.
32. An operator who sustains an annual loss in a fiscal year may claim, on or before the date on or before which he is required to file his return under section 36 for that fiscal year, an amount as a credit on duties refundable for losses which shall not exceed
(1)  for a fiscal year ending before 13 May 1994, 18% of the lesser of
(a)  the annual loss for that fiscal year; and
(b)  the aggregate of
i.  subject to paragraph c of section 27, as it read before 13 May 1994, the total of the amounts each of which is an amount deducted for that fiscal year, under paragraph m or n of section 8, as it read before 13 May 1994, as mining exploration and development expenses incurred after 23 April 1985 by an operator for work carried out in Québec, except expenses incurred for work carried out in respect of the mining of surface mineral substances, as defined in section 1 of the Mining Act (chapter M-13.1), and the mining of mineral substances the rights in or over which have been surrendered to the owner of the soil as provided in section 5 of that Act; and
ii.  the amount deducted by the operator for that fiscal year in respect of property actually used by him in Québec, as a depreciation allowance under paragraph o of section 8, as it read before 13 May 1994, from which amount shall be deducted the part thereof relating to property acquired before 24 April 1985;
(2)  for a fiscal year ending after 12 May 1994, 12% of the lesser of
(a)  the adjusted annual loss for that fiscal year; and
(b)  the total of the following amounts, without however exceeding the amount deducted by the operator under subparagraph e of paragraph 2 of section 8 in computing the operator’s annual profit for that fiscal year :
i.  the amount that is the amount by which the expenses in respect of exploration, mineral deposit evaluation and mine development work, incurred by the operator for the fiscal year in connection with mining operation, exceeds the amount of government assistance that the operator received or was entitled to receive for that fiscal year and that relates to those expenses, and provided that such expenses, notwithstanding section 16.2, have been declared by the operator to be deductible expenses, on or before the date on or before which the operator is required to file the operator’s return, in accordance with section 36, for that fiscal year, and
ii.  the total of all amounts each of which is the amount by which an amount referred to in subparagraph b.1 of paragraph 1 of section 16.1 that relates to expenses incurred by the operator during that fiscal year and declared by the operator to be deductible expenses, on or before the date provided in subparagraph i, exceeds the amount that is 25% of the government assistance that the operator received or was entitled to receive for that fiscal year and that relates to those expenses.
1975, c. 30, s. 32; 1985, c. 39, s. 6; 1994, c. 47, s. 32; 1999, c. 83, s. 17.
32. An operator who sustains an annual loss in a fiscal year may claim, on or before the date on or before which he is required to file his return under section 36 for that fiscal year, an amount as a credit on duties refundable for losses which shall not exceed
(1)  for a fiscal year ending before 13 May 1994, 18 % of the lesser of
(a)  the annual loss for that fiscal year; and
(b)  the aggregate of
i.  subject to paragraph c of section 27, as it read before 13 May 1994, the total of the amounts each of which is an amount deducted for that fiscal year, under paragraph m or n of section 8, as it read before 13 May 1994, as mining exploration and development expenses incurred after 23 April 1985 by an operator for work carried out in Québec, except expenses incurred for work carried out in respect of the mining of surface mineral substances, as defined in section 1 of the Mining Act (chapter M-13.1), and the mining of mineral substances the rights in or over which have been surrendered to the owner of the soil as provided in section 5 of that Act; and
ii.  the amount deducted by the operator for that fiscal year in respect of property actually used by him in Québec, as a depreciation allowance under paragraph o of section 8, as it read before 13 May 1994, from which amount shall be deducted the part thereof relating to property acquired before 24 April 1985;
(2)  for a fiscal year ending after 12 May 1994, 12 % of the lesser of
(a)  the adjusted annual loss for that fiscal year; and
(b)  the total of the amounts, without however exceeding the amount deducted by the operator under subparagraph e of paragraph 2 of section 8 in computing his annual profit for that fiscal year, each of which is the amount by which the expenses in respect of exploration, mineral deposit evaluation and mine development work, incurred by the operator for the fiscal year in connection with mining operation, exceeds the amount of government assistance that the operator received or was entitled to receive for that fiscal year and that relates to those expenses, and provided that such expenses, notwithstanding section 16.2, have been declared by the operator to be deductible expenses, on or before the date on or before which he is required to file his return, in accordance with section 36, for that fiscal year.
1975, c. 30, s. 32; 1985, c. 39, s. 6; 1994, c. 47, s. 32.
32. An operator who sustains an annual loss in a fiscal year may claim, as a refundable credit on duties, an amount equal to the lesser of 18% of
(a)  the annual loss, and
(b)  the allowable amount.
1975, c. 30, s. 32; 1985, c. 39, s. 6.
32. Notwithstanding section 30, where several operators have been related to each other at any time during a fiscal year, the total duties payable by them for the fiscal year ending in the same calendar year shall be computed on the aggregate annual profit of each of such operators, less a single exemption referred to in section 30, and the duties payable by each shall be the proportion of such total duties that his annual profit is of the aggregate annual profit of such operators.
Nevertheless, if before the expiry of the delay for filing his return under section 36, each of such operators sends to the office of the Minister an agreement respecting the apportionment among them of their total profit for the purposes of computing the duties, the total duties payable shall be apportioned according to such agreement.
1975, c. 30, s. 32.