text:
+0

1. factorize 9(2a-b)^2 - 4(2a-b)^2 -13

0
765
5
+297

1.  factorize 9(2a-b)^2 - 4(2a-b)^2 -13

2. if p = 2-a, then find the value of a^3 + 6ap + p^3 -8

3. if the parallel sides of trapezium are 25cm and 11cm, while its non- parallel sides are 15cm and 13 cm . find the area of the trapezium...

kes1968  Jul 12, 2015

#5
+20680
+5

2. if p = 2-a, then find the value of a^3 + 6ap + p^3 -8

$$\small{\text{ \begin{array}{rclcl} a^3 + 6ap + p^3 -8 \\ &=& a^3 + p^3 + 6ap -8 \qquad &|& \qquad (a^3+p^3) = (a+p)(a^2-ap+p^2)\\ &=& (a+p)(a^2-ap+p^2)+ 6ap -8 \qquad &|& \qquad p = 2-a \\ &=& (a+2-a)(a^2-ap+p^2)+ 6ap -8 \\ &=& 2(a^2-ap+p^2)+ 6ap -8 \\ &=& 2\left[ a^2-ap+p^2+ 3ap -4\right] \\ &=& 2( a^2+2ap+p^2 -4) \qquad &|& \qquad (a^2+2ap+p^2) = (a+p)^2 \\ &=& 2\left[ (a+p)^2 -4 \right] \qquad &|& \qquad p = 2-a \\ &=& 2\left[ (a+2-a)^2 -4) \right]\\ &=& 2( 2^2 -4)\\ &=& 2( 4 -4)\\ &=& 2\cdot 0\\ &=& 0 \end{array} }}$$

heureka  Jul 13, 2015
#1
+94120
+5

Hi Kes,

$$\\9(2a-b)^2 - 4(2a-b)^2 -13\\ 5(2a-b)^2-13\\ 5(4a^2-4ab+b^2)-13\\ 20a^2-20ab+5b^2-13\\$$

Mmm     I don't think that this can be factorized...

I suppose that it is the difference of 2 squares so it is equal to

$$(\sqrt5(2a-b)-\sqrt{13})(\sqrt5(2a-b)+\sqrt{13})$$

but I don't think this looks very nice....

Melody  Jul 12, 2015
#2
+94120
+5

2. if p = 2-a, then find the value of a^3 + 6ap + p^3 -8

$$\\a^3 + 6ap + p^3 -8 \\ = (6ap+p^3)-(8-a^3)\\ =p(6a+p^2)-(2-a)(4+2a+a^2)\\ sub\;\; 2-a for p\\ =p(6a+p^2)-p(4+2a+a^2)\\  factor out the p\\ =p(6a+p^2-4-2a-a^2)\\ =p(p^2-a^2+4a-4)\\ =p((2-a)^2-a^2+4a-4)\\ =p(4+a^2-4a-a^2+4a-4)\\ =p(0)\\ =0$$

Melody  Jul 12, 2015
#3
+94120
+5

3. if the parallel sides of trapezium are 25cm and 11cm, while its non- parallel sides are 15cm and 13 cm . find the area of the trapezium...

I am let the long parallel side 25cm be the bottom.

15+13=28  Only 3cm more than 25cm.

So it is obvious to me that the two angles at the bottom MUST both be acute angles.

Here is the trapezium.

Now I am going to cut the two triangles of the end and push them up together so that I have a triangle with side lengths   14cm on the bottom and 15cm and 13cm on the other two sides

The height of the triangle is also the height of the trapezium.

Here is a pic

Using pythagoras's theorem

$$\\15^2=(14-x)^2+h^2\\ 225=196+x^2-28x+h^2\\ 29=x^2-28x+h^2\qquad(1)\\\\ 169=x^2+h^2\qquad (2)\\ (2)-(1)\\ 140=28x\\ x=5\\ so\\ 169=25+h^2\\ 144=h^2\\ h=12cm\\\\ Area of trapezium  = \frac{(25+11)}{2}\times 12=216cm^2$$

Melody  Jul 12, 2015
#4
+20680
+5

1.  factorize 9(2a-b)^2 - 4(2a-b)^2 -13

$$\small{\text{ \begin{array}{rcl} 9x^2 - 4x^2 -13 &=& 5x^2 - 13 \qquad | \qquad x= 2a-b \\\\ 5x^2 - 13 &=& 0\\ 5x^2 &=& 13 \\\\ x^2 &=& \dfrac{13}{5}\\\\ x^2 &=& 2.6 \\ \mathbf{x_{1,2} } & \mathbf{=} & \mathbf{\pm\sqrt{2.6}}\\\\ 5x^2 - 13 &=& 5\cdot(x-\sqrt{2.6})(x+\sqrt{2.6})\qquad | \qquad x= 2a-b \\\\ 9(2a-b)^2 - 4(2a-b)^2 -13 = 5(2a-b)^2 - 13 &=& 5\cdot\left[(2a-b)-\sqrt{2.6}\right] \left[(2a-b)+\sqrt{2.6} \right] \\\\ \end{array} }}$$

heureka  Jul 13, 2015
#5
+20680
+5

2. if p = 2-a, then find the value of a^3 + 6ap + p^3 -8

$$\small{\text{ \begin{array}{rclcl} a^3 + 6ap + p^3 -8 \\ &=& a^3 + p^3 + 6ap -8 \qquad &|& \qquad (a^3+p^3) = (a+p)(a^2-ap+p^2)\\ &=& (a+p)(a^2-ap+p^2)+ 6ap -8 \qquad &|& \qquad p = 2-a \\ &=& (a+2-a)(a^2-ap+p^2)+ 6ap -8 \\ &=& 2(a^2-ap+p^2)+ 6ap -8 \\ &=& 2\left[ a^2-ap+p^2+ 3ap -4\right] \\ &=& 2( a^2+2ap+p^2 -4) \qquad &|& \qquad (a^2+2ap+p^2) = (a+p)^2 \\ &=& 2\left[ (a+p)^2 -4 \right] \qquad &|& \qquad p = 2-a \\ &=& 2\left[ (a+2-a)^2 -4) \right]\\ &=& 2( 2^2 -4)\\ &=& 2( 4 -4)\\ &=& 2\cdot 0\\ &=& 0 \end{array} }}$$

heureka  Jul 13, 2015


text:
## can an argument containing a contradiction be valid argument

1

I know that validity has nothing with truth of the conclusion or with how good argument is in general, and an argument is valid iff the truth of its premises guarantees the truth of its conclusion.

However, my teacher gave us an argument such that

Grass is green.

Grass is not green.

Therefore Cows bark.

and he said that it is a valid argument, because it is impossible for this example when all premises are true ,the conclusion is false. However, I think that this example contradicts with the main idea of validity. Because the premises don't guarantee the truth of the conclusion.

Can you enlighten me about whether it is a valid or not.

3

Well, in what way do you think the truth of the premises does not guarantee the truth of the conclusion? In which situations is the promise "If the premises are true, the conclusion will be true" broken?

The definition of validity is:

For all interpretations it holds that if all premises are true under that interpretation, then the conclusion is true under that interpretation as well.

The negation of this is

Not for all interpretations it holds that if all premises are true under that interpretation, then the conclusion is true under that interpretation as well.

which is equivalent to

There is an interpretation for which it does not hold that if all premises are true under that interpretation, then the conclusion is true under that interpretation as well.

which is in turn equivalent to

There is an interpretation such that all premises are true but the conclusion is false under that interpretation.

That is, an argument being invalid amounts to saying that there is a concrete counter interpretation which makes all premises true but the conclusion false. If there is no interpretation which can make all premises true to begin with, then in particular there can be no such counter interpretation. If there arises no situation in which the condition on the truth preservance guarantee (the truth of the premises) takes effect, then there is no situation in which this promise can be broken.

So yes, the argument is valid, precisely for the reason cited by your teacher. An argument that is valid because the premises are contradictory is called vacuously valid.

You may be interested in the notion of a sound argument: A sound argument is one which is valid and where in addition the premises are true in the real world. Premises which are contradictory can obviously not be true in the real world, so the above argument is unsound. This may be closer to your intuition of a "correct" argument than the notion of validity.

Is this not a counter interpretation? "Grass is green. Grass is not green. Therefore cows do not bark". I used the same premises, but reached a different conclusion. Shouldn't that not happen for valid arguments? – Cell – 2020-08-07T22:02:33.597

Actually to be more precise I reached the opposite or contradictory conclusion, but I can no longer edit my comment. – Cell – 2020-08-07T22:30:21.220

@Cell An interpretation is an assignment of truth values to sentences. A counter interpretation to an invalid argument is an interpretation that makes all premises true but the conclusion false. There can be no interpretation that satisfies both "Grass is green" and "Grass is not green", hence there can be no counter interpretation. Your argument is valid for the same reason as stated in the answer. – lemontree – 2020-08-07T22:45:43.860

And no, reaching different conclusions from the same premises is perfectly possible. For instance, from the premises if p then q and r; p we can conclude both the conclusions q and r. The scenario where we can draw two conclusions that are contradictory (e.g. Cows bark and Cows do not bark) happens precisely when the premises are contradictory. This doesn't make the argument invalid, however, it just shows that we must reject at least one of the premises. – lemontree – 2020-08-07T22:48:47.023

... And it also shows that the arguments can not be sound: At least one of the premises must be false in the real world if we can draw contradictory conclusions from them. – lemontree – 2020-08-07T22:56:09.120

2

You state (slightly paraphrased):

the truth of the premises doesn't guarentee the truth of the conclusion.

But the stance of classical logic (re: this, see below) is that in fact the truth of the premises does guarantee the truth of the conclusion. There is no conceivable situation where the premises are true but the conclusion is false, since there is no conceivable situation where the premises are true, full stop. The principle at play here is called "ex falso quodlibet" or "the principle of explosion."

And in fact this isn't special to classical logic. Intuitionistic and modal logics also have this principle by default (there are e.g. such things as "relevant modal logics" - see below - but the usual modal logics are simpler than that). Basically, this situation occurs in any logical system in which (i) a deduction is thought of as valid iff it has no countermodel and (ii) the semantics for the system does not permit "impossible" models.

That said, not every logical framework accepts this sort of reasoning. In particular, we might want to use a framework where "From A we can deduce p" only happens if A somehow "causes" p - the deduction above would intuitively not be acceptable in such a situation since impossible grass has no bearing on cow-nature. The terms "relevance logic" and (more generally) "paraconsistent logic" are key here.

-2

The example argument is not valid.

(1) The terms of the conclusion (cows, bark) appear nowhere in the premises. I believe this fallacy is irrelevancy.

(2) A syllogism has exactly three terms; all three appear in the premises. The premises in the example have only two terms (grass, green). In effect, there is no middle term.

(3) A syllogism having a negative premise must have a negative conclusion. Here, one premise is negative but the conclusion is positive.

(4) If a term is distributed in the conclusion, it must be distributed in the premises. Here, "cows" in "Cows bark" is distributed. The term "cow" is not distributed in either premise.

That the terms of a conclusion do not appear in the premises does not make an argument invalid in the logical sense of validity. – lemontree – 2020-08-07T17:54:09.223

format: plain


text:
# Graphing linear inequalities in two variables

### Graphing linear inequalities in two variables

In this lesson, we will talk about how to solve linear inequalities as well as how to graph the solutions on number lines and the xy plane.

#### Lessons

• Introduction
Introduction to Inequalities
a)
Graph the solution on a number line:
$x = 3$
$x \ge 3$
$x$ > $3$
$x \le 3$
$x$ < $3$

b)
Graph the solution on the xy-plane:
$x = 3$
$x \ge 3$
$x$ > $3$
$x \le 3$
$x$ < $3$

c)
Graph the solution on the xy-plane:
$y = -2$
$y \ge -2$
$y$ > $-2$
$y \le -2$
$y$ < $-2$

• 1.
Graphing Linear Inequalities in Two Variables
Graph:
i)
$y = x - 1$
ii)
$y \ge x - 1$
iii)
$y$ < $x - 1$

• 2.
Graphing Linear Inequalities in Two Variables
Graph:
i)
$6x + 5y = 15$
ii)
$6x + 5y \ge 15$
iii)
$6x$ $+ 5y$ < $15$


text:
# The Lorentz Force Law

## The Prompt

“Write down the Lorentz force law.”

## Context

This SWBQ is a useful activity for helping students recall what the force is experienced by a charged particle subject to an electric and magnetic field, or

$$\vec{F}=q\left(\vec{E}+\vec{v} \times \vec{B}\right) \; \; .$$

## Wrap Up

This SWBQ can be incorporated before or after any activity where students must calculate the force exerted on a loop of current in a magnetic field.

##### Views

New Users

Curriculum

Pedagogy

Institutional Change

Publications


text:
We explore the possibility of using a single monocular camera to forecast the time to collision between a suitcase-shaped robot being pushed by its user and other nearby pedestrians. We develop a purely image-based deep learning approach that directly estimates the time to collision without the need of relying on explicit geometric depth estimates or velocity information to predict future collisions. While previous work has focused on detecting immediate collision in the context of navigating Unmanned Aerial Vehicles, the detection was limited to a binary variable (i.e., collision or no collision). We propose a more fine-grained approach to collision forecasting by predicting the exact time to collision in terms of milliseconds, which is more helpful for collision avoidance in the context of dynamic path planning. To evaluate our method, we have collected a novel large-scale dataset of over 13,000 indoor video segments each showing a trajectory of at least one person ending in a close proximity (a near collision) with the camera mounted on a mobile suitcase-shaped platform. Using this dataset, we do extensive experimentation on different temporal windows as input using an exhaustive list of state-of-the-art convolutional neural networks (CNNs). Our results show that our proposed multi-stream CNN is the best model for predicting time to near-collision. The average prediction error of our time to near collision is 0.75 seconds across our test environments.

One of the largest obstacles facing scanning probe microscopy is the constant need to correct flaws in the scanning probe in situ. This is currently a manual, time-consuming process that would benefit greatly from automation. Here we introduce a convolutional neural network protocol that enables automated recognition of a variety of desirable and undesirable scanning probe tip states on both metal and non-metal surfaces. By combining the best performing models into majority voting ensembles, we find that the desirable states of H:Si(100) can be distinguished with a mean precision of 0.89 and an average receiver-operator-characteristic curve area of 0.95. More generally, high and low-quality tips can be distinguished with a mean precision of 0.96 and near perfect area-under-curve of 0.98. With trivial modifications, we also successfully automatically identify undesirable, non-surface-specific states on surfaces of Au(111) and Cu(111). In these cases we find mean precisions of 0.95 and 0.75 and area-under-curves of 0.98 and 0.94, respectively.

Neural network-based methods have recently demonstrated state-of-the-art results on image synthesis and super-resolution tasks, in particular by using variants of generative adversarial networks (GANs) with supervised feature losses. Nevertheless, previous feature loss formulations rely on the availability of large auxiliary classifier networks, and labeled datasets that enable such classifiers to be trained. Furthermore, there has been comparatively little work to explore the applicability of GAN-based methods to domains other than images and video. In this work we explore a GAN-based method for audio processing, and develop a convolutional neural network architecture to perform audio super-resolution. In addition to several new architectural building blocks for audio processing, a key component of our approach is the use of an autoencoder-based loss that enables training in the GAN framework, with feature losses derived from unlabeled data. We explore the impact of our architectural choices, and demonstrate significant improvements over previous works in terms of both objective and perceptual quality.

This paper proposes an efficient unsupervised method for detecting relevant changes between two temporally different images of the same scene. A convolutional neural network (CNN) for semantic segmentation is implemented to extract compressed image features, as well as to classify the detected changes into the correct semantic classes. A difference image is created using the feature map information generated by the CNN, without explicitly training on target difference images. Thus, the proposed change detection method is unsupervised, and can be performed using any CNN model pre-trained for semantic segmentation.

We exploit altered patterns in brain functional connectivity as features for automatic discriminative analysis of neuropsychiatric patients. Deep learning methods have been introduced to functional network classification only very recently for fMRI, and the proposed architectures essentially focused on a single type of connectivity measure. We propose a deep convolutional neural network (CNN) framework for classification of electroencephalogram (EEG)-derived brain connectome in schizophrenia (SZ). To capture complementary aspects of disrupted connectivity in SZ, we explore combination of various connectivity features consisting of time and frequency-domain metrics of effective connectivity based on vector autoregressive model and partial directed coherence, and complex network measures of network topology. We design a novel multi-domain connectome CNN (MDC-CNN) based on a parallel ensemble of 1D and 2D CNNs to integrate the features from various domains and dimensions using different fusion strategies. Hierarchical latent representations learned by the multiple convolutional layers from EEG connectivity reveal apparent group differences between SZ and healthy controls (HC). Results on a large resting-state EEG dataset show that the proposed CNNs significantly outperform traditional support vector machine classifiers. The MDC-CNN with combined connectivity features further improves performance over single-domain CNNs using individual features, achieving remarkable accuracy of $93.06\%$ with a decision-level fusion. The proposed MDC-CNN by integrating information from diverse brain connectivity descriptors is able to accurately discriminate SZ from HC. The new framework is potentially useful for developing diagnostic tools for SZ and other disorders.

The matrix-based Renyi's \alpha-entropy functional and its multivariate extension were recently developed in terms of the normalized eigenspectrum of a Hermitian matrix of the projected data in a reproducing kernel Hilbert space (RKHS). However, the utility and possible applications of these new estimators are rather new and mostly unknown to practitioners. In this paper, we first show that our estimators enable straightforward measurement of information flow in realistic convolutional neural networks (CNN) without any approximation. Then, we introduce the partial information decomposition (PID) framework and develop three quantities to analyze the synergy and redundancy in convolutional layer representations. Our results validate two fundamental data processing inequalities and reveal some fundamental properties concerning the training of CNN.

It is challenging to detect curve texts due to their irregular shapes and varying sizes. In this paper, we first investigate the deficiency of the existing curve detection methods and then propose a novel Conditional Spatial Expansion (CSE) mechanism to improve the performance of curve text detection. Instead of regarding the curve text detection as a polygon regression or a segmentation problem, we treat it as a region expansion process. Our CSE starts with a seed arbitrarily initialized within a text region and progressively merges neighborhood regions based on the extracted local features by a CNN and contextual information of merged regions. The CSE is highly parameterized and can be seamlessly integrated into existing object detection frameworks. Enhanced by the data-dependent CSE mechanism, our curve text detection system provides robust instance-level text region extraction with minimal post-processing. The analysis experiment shows that our CSE can handle texts with various shapes, sizes, and orientations, and can effectively suppress the false-positives coming from text-like textures or unexpected texts included in the same RoI. Compared with the existing curve text detection algorithms, our method is more robust and enjoys a simpler processing flow. It also creates a new state-of-art performance on curve text benchmarks with F-score of up to 78.4$\%$.

We developed a convolution neural network (CNN) on semi-regular triangulated meshes whose vertices have 6 neighbours. The key blocks of the proposed CNN, including convolution and down-sampling, are directly defined in a vertex domain. By exploiting the ordering property of semi-regular meshes, the convolution is defined on a vertex domain with strong motivation from the spatial definition of classic convolution. Moreover, the down-sampling of a semi-regular mesh embedded in a 3D Euclidean space can achieve a down-sampling rate of 4, 16, 64, etc. We demonstrated the use of this vertex-based graph CNN for the classification of mild cognitive impairment (MCI) and Alzheimer's disease (AD) based on 3169 MRI scans of the Alzheimer's Disease Neuroimaging Initiative (ADNI). We compared the performance of the vertex-based graph CNN with that of the spectral graph CNN.

Gaze tracking is an important technology in many domains. Techniques such as Convolutional Neural Networks (CNN) has allowed the invention of gaze tracking method that relies only on commodity hardware such as the camera on a personal computer. It has been shown that the full-face region for gaze estimation can provide better performance than from an eye image alone. However, a problem with using the full-face image is the heavy computation due to the larger image size. This study tackles this problem through compression of the input full-face image by removing redundant information using a novel learnable pooling module. The module can be trained end-to-end by backpropagation to learn the size of the grid in the pooling filter. The learnable pooling module keeps the resolution of valuable regions high and vice versa. This proposed method preserved the gaze estimation accuracy at a certain level when the image was reduced to a smaller size.

The convolutional neural network (CNN) architecture is increasingly being applied to new domains, such as malware detection, where it is able to learn malicious behavior from raw bytes extracted from executables. These architectures reach impressive performance with no feature engineering effort involved, but their robustness against active attackers is yet to be understood. Such malware detectors could face a new attack vector in the form of adversarial interference with the classification model. Existing evasion attacks intended to cause misclassification on test-time instances, which have been extensively studied for image classifiers, are not applicable because of the input semantics that prevents arbitrary changes to the binaries. This paper explores the area of adversarial examples for malware detection. By training an existing model on a production-scale dataset, we show that some previous attacks are less effective than initially reported, while simultaneously highlighting architectural weaknesses that facilitate new attack strategies for malware classification. Finally, we explore how generalizable different attack strategies are, the trade-offs when aiming to increase their effectiveness, and the transferability of single-step attacks.

This letter describes a network that is able to capture spatiotemporal correlations over arbitrary timestamps. The proposed scheme operates as a complementary, extended network over spatiotemporal regions. Recently, multimodal fusion has been extensively researched in deep learning. For action recognition, the spatial and temporal streams are vital components of deep Convolutional Neural Network (CNNs), but reducing the occurrence of overfitting and fusing these two streams remain open problems. The existing fusion approach is to average the two streams. To this end, we propose a correlation network with a Shannon fusion to learn a CNN that has already been trained. Long-range video may consist of spatiotemporal correlation over arbitrary times. This correlation can be captured using simple fully connected layers to form the correlation network. This is found to be complementary to the existing network fusion methods. We evaluate our approach on the UCF-101 and HMDB-51 datasets, and the resulting improvement in accuracy demonstrates the importance of multimodal correlation.

Compact convolutional neural networks gain efficiency mainly through depthwise convolutions, expanded channels and complex topologies, which contrarily aggravate the training efforts. In this work, we identify the shift problem occurs in even-sized kernel (2x2, 4x4) convolutions, and eliminate it by proposing symmetric padding on each side of the feature maps (C2sp, C4sp). Symmetric padding enlarges the receptive fields of even-sized kernels with little computational cost. In classification tasks, C2sp outperforms the conventional 3x3 convolution and obtains comparable accuracies to existing compact convolution blocks, but consumes less memory and time during training. In generation tasks, C2sp and C4sp both achieve improved image qualities and stabilized training. Symmetric padding coupled with even-sized convolution is easy to be implemented into deep learning frameworks, providing promising building units for architecture designs that emphasize training efforts on online and continual learning occasions.

The Zone of Avoidance makes it difficult for astronomers to catalogue galaxies at low latitudes to our galactic plane due to high star densities and extinction. However, having a complete sky map of galaxies is important in a number of fields of research in astronomy. There are many unclassified sources of light in the Zone of Avoidance and it is therefore important that there exists an accurate automated system to identify and classify galaxies in this region. This study aims to evaluate the efficiency and accuracy of using an evolutionary algorithm to evolve the topology and configuration of Convolutional Neural Network (CNNs) to automatically identify galaxies in the Zone of Avoidance. A supervised learning method is used with data containing near-infrared images. Input image resolution and number of near-infrared passbands needed by the evolutionary algorithm is also analyzed while the accuracy of the best evolved CNN is compared to other CNN variants.

Recent advancements have led to a proliferation of machine learning systems used to assist humans in a wide range of tasks. However, we are still far from accurate, reliable, and resource-efficient operations of these systems. For robot perception, convolutional neural networks (CNNs) for object detection and pose estimation are recently coming into widespread use. However, neural networks are known to suffer overfitting during training process and are less robust within unseen conditions, which are especially vulnerable to {\em adversarial scenarios}. In this work, we propose {\em Generative Robust Inference and Perception (GRIP)} as a two-stage object detection and pose estimation system that aims to combine relative strengths of discriminative CNNs and generative inference methods to achieve robust estimation. Our results show that a second stage of sample-based generative inference is able to recover from false object detection by CNNs, and produce robust estimations in adversarial conditions. We demonstrate the efficacy of {\em GRIP} robustness through comparison with state-of-the-art learning-based pose estimators and pick-and-place manipulation in dark and cluttered environments.

We present a novel learning framework for vehicle recognition from a single RGB image. Unlike existing methods which only use attention mechanisms to locate 2D discriminative information, our unified framework learns a joint representation of the 2D global texture and 3D-bounding-box in a mutually correlated and reinforced way. These two kinds of feature representation are combined by a novel fusion network, which predicts the vehicle's category. The 2D global feature is extracted using an off-the-shelf detection network, where the estimated 2D bounding box assists in finding the region of interest (RoI). With the assistance of the RoI, the 3D bounding box and its corresponding features are generated in a geometrically correct way using a novel \textit{3D perspective Network} (3DPN). The 3DPN consists of a convolutional neural network (CNN), a vanishing point loss, and RoI perspective layers. The CNN regresses the 3D bounding box under the guidance of the proposed vanishing point loss, which provides a perspective geometry constraint. Thanks to the proposed RoI perspective layer, the variation caused by viewpoint changes is corrected via the estimated geometry, enhancing the feature representation. We present qualitative and quantitative results for our approach on the vehicle classification and verification tasks in the BoxCars dataset. The results demonstrate that, by learning how to extract features from the 3D bounding box, we can achieve comparable or superior performance to methods that only use 2D information.

Top


text:
# Water Demand | Definition, types and calculations

The community, society, and industry need water for different uses. Wate demand is the accurate estimation of total water. The unit of Water demand is LPCD ( liters per person per day). While planning the water supply scheme for an area, it is essential to determine the total water required for a different purpose.

## Various types of water Demand

There are many variable factors affecting water consumption. So it is tough to assess the public water demand precisely. To determine the quantity of water certain thumb rules and empirical formulas are available. Each formula is available for a particular use. Various types of demand are

• Domestic water demand: Water require for different household activities.
• Industrial water demand: represents the industry’s need.
• Other water demand: Institution and commercial use.
• Public Demand: Demand for public uses
• Urgent Demand: Fire demand
• Water requires to compensate for losses in waste and thefts.

## Domestic water demand

Domestic Water demand includes the water required for drinking, cooking, bathing, lawn sprinkling, gardening, sanitation purpose, etc. It depends upon habits, social status, and climatic conditions of people. As per IS 1172-1993, the domestic consumption varies from 135 lpcd- 200 lpcd depending upon the economic situation and full flushing system. Minimum Domestic water consumption for Indian towns and cities with full flushing system and the Weaker section as per IS 1172-1993

## Industrial Water Demand

It represents the need of industries, either existing or likely to start in the future. This quantity will thus vary with the number and types of industries present in the city. For small-scale industries, in general, it is 50 lpcd. In industrial cities, the per-capita water requirement is 450 lpcd.

## Institutional and Commercial Water Demand

It includes the use of institutions, such as hospitals, hotels, restaurants, schools and colleges, railway stations, offices, etc. This quantity will certainly vary with the nature of the city and with the number and types of commercial establishments and institutions present in it. It varies from 20 lpcd to 50 lpcd.

## Demand for Public Uses

This includes Watering in a public park, gardening, washing, sprinkling on roads, use in a public fountain, etc. In India, generally, public demand is zero. It is less than $5%$ of total consumption. Demand for public use is 10 lpcd

## Fire Demand

It is the amount of water required for firefighting purpose in case of a fire break out in an area. This water is required to be available at a pressure of about 100 to 150 $text{kN/m}^2$or 10 to 15m head of water. If the population is less than 50000 fire demand is not calculated. For larger city fire demand was calculated. there are various formulas available to calculate the fire demand

1. Kuichiling’s Formula $Q=3182\times \sqrt{P}$ where, Q= water required in lit/min P= Population in 1000s
2. Freeman Formula $Q=1136\times [p/10 +10]$
3. National Board of Fire Under Writers Formulas 1. when the population is less than or equal to 200000, $Q = 4637 \times \sqrt{P} \times [1 – (0.01\sqrt{P})]$
4. Buston’s Formula $Q = 5663 \times \sqrt{P}$

## Water required to compensate for losses in waste and thefts

This includes water loss in leakage due to bad plumbing or damaged meters, stolen water, and other losses and wastes. It is taken as 15% of the total consumption.

## Conclusions

Water demand is represented as per capita. per capita water demand= (Total yearly requirement of the city in liters/ (365 times Design population)). The per capita water demand is 335 lit/ hour/ day.

• Domestic water demand: Water required for different household activities is approx 200 lpcd.
• Industrial water demand: this represents the need of the industry, and it is 50 lpcd.
• Other water demands: Institution and commercial use is 20 lpcd.
• Public Demand: Demand for public uses is 10 lpcd
• Urgent Demand: Fire demand depends upon the size and population of the city.
• Theft: Water requires to compensate losses in wastes and thefts is 55 lpcd..

## Android Apps

At eigenplus, our goal is to teach civil engineering students about structural analysis and design starting from the fundamental principles. We do this with the help of interactive android applications and accompanying web articles and videos.

Our apps have helped more than 400 thousand students across the world to understand and learn the concepts of structural engineering. Check out our apps on the google play store.

This article was crafted by a group of experts at eigenplus to ensure it adheres to our strict quality standards. The individuals who contributed to this article are:

# Author

## Suprabha Panda

M.Tech

She is an assistant professor with masters in Environmental Engineering.

PhD

PhD


text:
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Question of the Week - 41 (If x and y are non-zero integers and...)

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Question of the Week - 41 (If x and y are non-zero integers and...)  [#permalink]

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30 Mar 2019, 02:36
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Question of the Week #41

If x and y are non-zero integers and |x – 3| ≤ 2 and -1 ≤ y ≤ 8 then the least value of $$\frac{x}{y}$$ lies in which of the following range?

A. Between -5.5 and -4.5
B. Between -4.5 and -3.5
C. Between -3.5 and -2.5
D. Between -2.5 and -1
E. Greater than -1

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Re: Question of the Week - 41 (If x and y are non-zero integers and...)  [#permalink]

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30 Mar 2019, 02:59
If x and y are non-zero integers and |x – 3| ≤ 2 and -1 ≤ y ≤ 8 then the least value of $$\frac{x}{y}$$ lies in which of the following range?

We know the range of y, and now we will work on range of x by solving |x – 3| ≤ 2..
Two cases for |x – 3| ≤ 2 : (I) x-3≤ 2...x≤ 5, and (II) -(x-3)≤ 2...3-x≤ 2......1≤ x
Range of x = > x-3≤ 2........1≤x≤ 5

Now, for the least value of $$\frac{x}{y}$$, we will take y as negative, that is -1. The denominator should have the largest possible value, so 5..
$$\frac{x}{y}$$=>$$\frac{5}{-1}=-5$$.

-5 is between -5.5 and -4.5, and so the answer is A
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Re: Question of the Week - 41 (If x and y are non-zero integers and...)  [#permalink]

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30 Mar 2019, 05:38
solve for
|x – 3| ≤ 2
x<=5
and -x<=1 ; x>=1
so 1>=x<=5
and
-1 ≤ y ≤ 8
so to get least value of x/y ; x = 5 and y = -1
-5
IMO A

EgmatQuantExpert wrote:
Question of the Week #41

If x and y are non-zero integers and |x – 3| ≤ 2 and -1 ≤ y ≤ 8 then the least value of $$\frac{x}{y}$$ lies in which of the following range?

A. Between -5.5 and -4.5
B. Between -4.5 and -3.5
C. Between -3.5 and -2.5
D. Between -2.5 and -1
E. Greater than -1

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Question of the Week - 41 (If x and y are non-zero integers and...)  [#permalink]

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02 Apr 2019, 21:41

Solution

Given:
In this question, we are given that
• The numbers x and y are non-zero integers.
• Also, |x – 3| ≤ 2 and -1 ≤ y ≤ 8.

To find:
We need to determine
• The least value of $$\frac{x}{y}$$ lies in which of the given ranges.

Approach and Working:
For the non-zero integer x,
• |x – 3| ≤ 2
Or, -2 ≤ x – 3 ≤ 2
Or, 1 ≤ x ≤ 5

Therefore, possible values of x = 1, 2, 3, 4, 5

Similarly, for the non-zero integer y,
• -1 ≤ y ≤ 8

Therefore, possible values of y = -1, 1, 2, 3, 4, 5, 6, 7, 8

Now, for $$\frac{x}{y}$$ to be least, we should take one of them as positive and the other one as negative.
• As we can see from the derived values of x and y, x can be positive only whereas y can be negative.
• Hence, we should take negative value of y, with maximum possible magnitude, and maximum value of x.

Negative value of y, with maximum magnitude = -1
Maximum value of x = 5
• Therefore, the minimum value of $$\frac{x}{y}$$ = $$\frac{5}{-1}$$ = -5
• This value lies between -5.5 and -4.5

Hence, the correct answer is option A.

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Question of the Week - 41 (If x and y are non-zero integers and...)   [#permalink] 02 Apr 2019, 21:41
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text:
# [OS X TeX] Suddenly, I'm Italian

Themis Matsoukas tmatsoukas at me.com
Fri Dec 2 07:56:54 EST 2011

On Dec 1, 2011, at 6:46 PM, Herbert Schulz wrote:
> Looks like you are using a combination of TeXShop's Command Completion (e.g., \be+TAB --- you appear to have set TAB as the trigger) and Apple's Completion (e.g., definit+ESC).

In TS's preferences you can choose either ESC or TAB to trigger command autocompletion. No matter which one you choose, def+ESC will open Apple's dictionary but def+TAB will not. This does not seem consistent. By the way, I only got the english dictionary to show up.

Themis


text:
### DISPLAY

Displays a one- or two-dimensional NDF

#### Description:

This application displays a one- or two-dimensional NDF  as an image on the current graphics device. The minimum and maximum data values to be displayed can be selected in several ways (see Parameter MODE). Data values outside these limits are displayed with the colour of the nearest limit. A key showing the relationship between colour and data value can be displayed (see Parameter KEY).

Annotated axes or a simple border can be drawn around the image (see Parameters AXES and BORDER). The appearance of these may be controlled in detail (see Parameters STYLE and BORSTYLE).

A specified colour lookup table may optionally be loaded prior to displaying the image (see Parameter LUT). For devices which reset the colour table when opened (such as PostScript files), this may be the only way of controlling the colour table.

The image is produced within the current graphics database picture. The co-ordinates at the centre of the image, and the scale of the image can be controlled using Parameters CENTRE, XMAGN and YMAGN. Only the parts of the image that lie within the current picture are visible; the rest is clipped. The image is padded with bad pixels if necessary.

#### Usage:

display in [comp] clear [device] mode [centre] [xmagn] [ymagn] [out]
$\left\{\right\$
low=? high=? percentiles=? sigmas=?

mode

#### Parameters:

TRUE if labelled and annotated axes are to be drawn around the image. These display co-ordinates in the current co-ordinate Frame  of the supplied NDF, and may be changed using application WCSFRAME (see also Parameter USEAXIS). The width of the margins left for the annotation may be controlled using Parameter MARGIN. The appearance of the axes (colours, founts, etc.) can be controlled using the STYLE Parameter. [current value]
The colour with which to mark any bad (i.e. missing) pixels in the display. There are a number of options described below.
• "MAX" –- The maximum colour index used for the display of the image.

• "MIN" –- The minimum colour index used for the display of the image.

• An integer –- The actual colour index. It is constrained between 0 and the maximum colour index available on the device.

• A named colour –- Uses the named colour from the palette, and if it is not present, the nearest colour from the palette is selected.

• An HTML colour code such as #ff002d.

If the colour is to remain unaltered as the lookup table is manipulated choose an integer between 0 and 15, or a named colour. The suggested default is the current value. [current value]

TRUE if a border is to be drawn around the regions of the displayed image containing valid co-ordinates in the current co-ordinate Frame of the NDF. For instance, if the NDF contains an Aitoff all-sky map, then an elliptical border will be drawn if the current co-ordinate Frame is galactic longitude and latitude. This is because pixels outside this ellipse have undefined positions in galactic co-ordinates. If, instead, the current co-ordinate Frame had been pixel co-ordinates, then a simple box would have been drawn containing the whole image. This is because every pixel has a defined position in pixel co-ordinates. The appearance of the border (colour, width, etc.) can be controlled using Parameter BORSTYLE. [current value]
A group of attribute settings describing the plotting style to use for the border (see Parameter BORDER).

A comma-separated list of strings should be given in which each string is either an attribute setting, or the name of a text file preceded by an up-arrow character "^". Such text files should contain further comma-separated lists which will be read and interpreted in the same manner. Attribute settings are applied in the order in which they occur within the list, with later settings overriding any earlier settings given for the same attribute.

Each individual attribute setting should be of the form:

$<$name$>$=$<$value$>$

where $<$name$>$ is the name of a plotting attribute, and $<$value$>$ is the value to assign to the attribute. Default values will be used for any unspecified attributes. All attributes will be defaulted if a null value (!)–-the initial default–-is supplied. To apply changes of style to only the current invocation, begin these attributes with a plus sign. A mixture of persistent and temporary style changes is achieved by listing all the persistent attributes followed by a plus sign then the list of temporary attributes.

See Section E for a description of the available attributes. Any unrecognised attributes are ignored (no error is reported). [current value]

The co-ordinates of the data pixel to be placed at the centre of the image, in the current co-ordinate Frame of the NDF (supplying a colon ":" will display details of the current co-ordinate Frame). The position should be supplied as a list of formatted axis values separated by spaces or commas. See also Parameter USEAXIS. A null (!) value causes the centre of the image to be used. [!]
TRUE if the current picture is to be cleared before the image is displayed. [current value]
The NDF array component to be displayed. It may be "Data", "Quality", "Variance", or "Error" (where "Error" is an alternative to "Variance" and causes the square root of the variance values to be displayed). If "Quality" is specified, then the quality values are treated as numerical values (in the range 0 to 255). ["Data"]
The name of the graphics device used to display the image. The device must have at least 24 colour indices or grey-scale intensities. [current graphics device]
If FILL is set to TRUE, then the image will be ‘stretched’ to fill the current picture in both directions. This can be useful when displaying images with markedly different dimensions, such as two-dimensional spectra. The dynamic default is TRUE if the array being displayed is one-dimensional, and FALSE otherwise. []
The data value corresponding to the highest pen in the colour table. All larger data values are set to the highest colour index when HIGH is greater than LOW, otherwise all data values greater than HIGH are set to the lowest colour index. The dynamic default is the maximum data value. There is an efficiency gain when both LOW and HIGH are given on the command line, because the extreme values need not be computed. (Scale mode)
The input NDF structure containing the data to be displayed.
TRUE if a key to the colour table is to be produced to the right of the display. This can take the form of a colour ramp, a coloured histogram of pen indices, or graphs of RGB intensities, all annotated with data value. The form and appearance of this key can be controlled using Parameter KEYSTYLE, and its horizontal position can be controlled using Parameter KEYPOS. If the key is required in a different location, set KEY=NO and use application LUTVIEW after displaying the image. [TRUE]
##### KEYPOS( 2 ) = _REAL (Read)
The first element gives the gap between the right-hand edge of the display and the left-hand edge of the key, as a fraction of the width of the current picture. If a key is produced, then the right-hand margin specified by Parameter MARGIN is ignored, and the value supplied for KEYPOS is used instead.

The second element gives the vertical position of the key as a fractional value in the range zero to one: zero puts the key as low as possible, one puts it as high as possible. A negative value (no lower than -1) causes the key to match the height of the display image. This may mean any text, like a label, for the horizontal axis may not appear, though if AXES is TRUE there is usually room. [current value]

A group of attribute settings describing the plotting style to use for the key (see Parameter KEY).

A comma-separated list of strings should be given in which each string is either an attribute setting, or the name of a text file preceded by an up-arrow character "^". Such text files should contain further comma-separated lists which will be read and interpreted in the same manner. Attribute settings are applied in the order in which they occur within the list, with later settings overriding any earlier settings given for the same attribute.

Each individual attribute setting should be of the form:

$<$name$>$=$<$value$>$

where $<$name$>$ is the name of a plotting attribute, and $<$value$>$ is the value to assign to the attribute. Default values will be used for any unspecified attributes. All attributes will be defaulted if a null value (!)–-the initial default–-is supplied. To apply changes of style to only the current invocation, begin these attributes with a plus sign. A mixture of persistent and temporary style changes is achieved by listing all the persistent attributes followed by a plus sign then the list of temporary attributes.

See Section E for a description of the available attributes. Any unrecognised attributes are ignored (no error is reported).

Axis 1 is always the data value axis. So for instance, to set the label for the data-value axis, assign a value to "Label(1)" in the supplied style.

To get a ramp key (the default), specify "form=ramp". To get a histogram key (a coloured histogram of pen indices), specify "form=histogram". To get a graph key (three curves of RGB intensities), specify "form=graph". If a histogram key is produced, the population axis can be either logarithmic or linear. To get a logarithmic population axis, specify "logpop=1". To get a linear population axis, specify "logpop=0" (the default). To annotate the long axis with pen numbers instead of pixel value, specify "pennums=1" (the default, "pennums=0", shows pixel values). [current value]

The data value corresponding to the lowest pen in the colour table. All smaller data values are set to the lowest colour index when LOW is less than HIGH, otherwise all data values smaller than LOW are set to the highest colour index. The dynamic default is the minimum data value. There is an efficiency gain when both LOW and HIGH are given on the command line, because the extreme values need not be computed. (Scale mode)
Name of the NDF containing a colour lookup table in its Data array; the lookup table is written to the graphics device’s colour table. The purpose of this parameter is to provide a means of controlling the appearance of the image on certain devices, such as colour printers, that do not have a dynamic colour table (i.e. the colour table is reset when the device is opened). If used with dynamic devices (such as X-windows), the new colour table remains after this application has completed. A null value (!) causes the existing colour table to be used.

The LUT must be two-dimensional, the dimension of the first axis being 3, and the second being arbitrary. The method used to compress or expand the colour table if the second dimension is different from the number of unreserved colour indices is controlled by Parameter NN. Also the LUT’s values must lie in the range 0.0–1.0. [!]

##### MARGIN( 4 ) = _REAL (Read)
The widths of the margins to leave around the image for axis annotations, given as fractions of the corresponding dimension of the current picture. The actual margins used may be increased to preserve the aspect ratio of the data. Four values may be given, in the order: bottom, right, top, left. If fewer than four values are given, extra values are used equal to the first supplied value. If these margins are too narrow any axis annotation may be clipped. If a null (!) value is supplied, the value used is (for all edges); 0.15 if annotated axes are being produced; 0.04, if a simple border is being produced; and 0.0 if neither border nor axes are being produced. [current value]
The method by which the maximum and minimum data values to be displayed are chosen. The options are as follows.
• "Current" –- The image is scaled between the upper and lower limits that were used by the previous invocation of DISPLAY. If the previous scaling limits cannot be determined, the MODE value reverts to "Scale".

• "Faint" –- The image is scaled between the mean data value minus one standard deviation and the mean data value plus seven standard deviations. The scaling values are reported so that the faster Scale mode may be utilised later.

• "Flash" –- The image is flashed on to the screen without any scaling at all. This is the fastest option.

• "Percentiles" –- The image is scaled between the data values corresponding to two percentiles. The scaling values are reported so that the faster Scale mode may be used later.

• "Range" –- The image is scaled between the minimum and maximum data values.

• "Scale" –- You define the upper and lower limits between which the image is to be scaled. The application reports the maximum and the minimum data values for reference and makes these the suggested defaults.

• "Sigmas" –- The image is scaled between two standard-deviation limits. The scaling values used are reported so that the faster Scale mode may be utilised later.

If TRUE the input lookup table is mapped to the colour table by using the nearest-neighbour method. This preserves sharp edges and is better for lookup tables with blocks of colour. If NN is FALSE, linear interpolation is used, and this is suitable for smoothly varying colour tables. NN is ignored unless LUT is not null. [FALSE]
The number of histogram bins used to compute percentiles for scaling. (Percentiles mode) [2048]
##### OUT = NDF (Write)
A scaled copy of the displayed section of the image. Values in this output image are integer colour indices shifted to exclude the indices reserved for the palette (i.e. the value zero refers to the first colour index following the palette). The output NDF is intended to be used as the input data in conjunction with SCALE=FALSE. If a null value (!) is supplied, no output NDF will be created. This parameter is not accessed when SCALE=FALSE. [!]
##### PENRANGE( 2 ) = _REAL (Read)
The range of colour indices (“pens”) to use. The supplied values are fractional values where zero corresponds to the lowest available colour index and 1.0 corresponds to the highest available colour index. The default value of [0.0,1.0] thus causes the full range of colour indices to be used. Note, if Parameter LUT is null (!) or Parameter SCALE is FALSE then this parameter is ignored and the fill range of pens is used. [0.0,1.0]
##### PERCENTILES( 2 ) = _REAL (Read)
The percentiles that define the scaling limits. For example, [25,75] would scale between the quartile values. (Percentile mode)
If TRUE the input data are to be scaled according to the value of Parameter MODE. If it is FALSE, MODE is ignored, and the input data are displayed as is (i.e. the data values are simply converted to integer type and used as indices into the colour table). A value of zero refers to the first pen following the palette. A FALSE value is intended to be used with data previously scaled by this or similar applications which have already performed the required scaling (see Parameter OUT). It provides the quickest method of image display within this application. [TRUE]
##### SIGMAS( 2 ) = _REAL (Read)
The standard-deviation bounds that define the scaling limits. To obtain values either side of the mean both a negative and a positive value are required. Thus [-2,3] would scale between the mean minus two and the mean plus three standard deviations. [3,-2] would give the negative of that.
If TRUE, then the default value for YMAGN equals the value supplied for XMAGN, resulting in all pixels being displayed as squares on the display surface. If a FALSE value is supplied for SQRPIX, then the default value for YMAGN is chosen to retain the pixels original aspect ratio at the centre of the image. [current value]
A group of attribute settings describing the plotting style to use for the annotated axes (see Parameter AXES).

A comma-separated list of strings should be given in which each string is either an attribute setting, or the name of a text file preceded by an up-arrow character "^". Such text files should contain further comma-separated lists which will be read and interpreted in the same manner. Attribute settings are applied in the order in which they occur within the list, with later settings overriding any earlier settings given for the same attribute.

Each individual attribute setting should be of the form:

$<$name$>$=$<$value$>$

where $<$name$>$ is the name of a plotting attribute, and $<$value$>$ is the value to assign to the attribute. Default values will be used for any unspecified attributes. All attributes will be defaulted if a null value (!)–-the initial default–-is supplied. To apply changes of style to only the current invocation, begin these attributes with a plus sign. A mixture of persistent and temporary style changes is achieved by listing all the persistent attributes followed by a plus sign then the list of temporary attributes.

See Section E for a description of the available attributes. Any unrecognised attributes are ignored (no error is reported). [current value]

USEAXIS is only accessed if the current co-ordinate Frame of the NDF has more than two axes. A group of two strings should be supplied specifying the two axes which are to be used when annotating the image, and when supplying a value for Parameter CENTRE. Each axis can be specified using one of the following options.
• Its integer index within the current Frame of the input NDF (in the range 1 to the number of axes in the current Frame).

• Its Symbol  string such as "RA" or "VRAD".

• A generic option where "SPEC" requests the spectral axis, "TIME" selects the time axis, "SKYLON" and "SKYLAT" picks the sky longitude and latitude axes respectively. Only those axis domains present are available as options.

A list of acceptable values is displayed if an illegal value is supplied. If a null (!) value is supplied, the axes with the same indices as the two used pixel axes within the NDF are used. [!]

The horizontal magnification for the image. The default value of 1.0 corresponds to ’normal’ magnification in which the the image fills the available space in at least one dimension. A value larger than 1.0 makes each data pixel wider. If this results in the image being wider than the available space then the image will be clipped to display fewer pixels. See also Parameters YMAGN, CENTRE, SQRPIX, and FILL. [1.0]
The vertical magnification for the image. A value of 1.0 corresponds to ’normal’ magnification in which the image fills the available space in at least one dimension. A value larger than 1.0 makes each data pixel taller. If this results in the image being taller than the available space then the image will be clipped to display fewer pixels. See also Parameters XMAGN, CENTRE, and FILL. If a null ((!) value is supplied, the default value used depends on Parameter SQRPIX. If SQRPIX is TRUE, the default YMAGN value used is the value supplied for XMAGN. This will result in each pixel occupying a square area on the screen. If SQRPIX is FALSE, then the default value for YMAGN is chosen so that each pixel occupies a rectangular area on the screen matching the pixel aspect ratio at the centre of the image, determined within the current WCS Frame.  [!]

#### Results Parameters

##### SCAHIGH = _DOUBLE (Write)
On exit, this holds the data value which corresponds to the maximum colour index in the displayed image. In Flash mode or when there is no scaling the highest colour index is returned.
##### SCALOW = _DOUBLE (Write)
The data value scaled to the minimum colour index for display. In Flash mode or when there is no scaling the lowest colour index is used. The current display linear-scaling minimum is set to this value.

#### Examples:

display ngc6872 mode=p percentiles=[10,90] noaxes
Displays the NDF called ngc6872 on the current graphics device device. The scaling is between the 10 and 90 per cent percentiles of the image. No annotated axes are produced.
display vv256 mode=flash noaxes border borstyle="colour=blue,style=2"
Displays the NDF called vv256 on the current graphics device device. There is no scaling of the data; instead the modulus of each pixel with respect to the number of colour-table indices is shown. No annotated axes are drawn, but a blue border is drawn around the image using PGPLOT line style number 2 (i.e. dashed lines).
display mode=fa axes style="^sty,grid=1" margin=0.2 clear out=video $\setminus$
Displays the current NDF DATA component with annotated axes after clearing the current picture on the current graphics device device. The appearance of the axes is specified in the text file sty, but this is modified by setting the Grid  attribute to 1 so that a co-ordinate grid is drawn across the plot. The margins around the image containing the axes are made slightly wider than normal. The scaling is between the $-$1 and $+$7 standard deviations of the image around its mean. The scaled data are stored in an NDF called video.
display kn26 axes key keypos=[0.0,-1.0] keystyle=^key.sty $\setminus$
Displays the NDF called kn26 using the current scaling, surrounded by axes. It adds a colour-table key to the right that abuts the data picture and is aligned vertically with the image. The plot attributes set in the text file key.sty controls the appearance of the key.
display video noscale $\setminus$
Displays the DATA component of the NDF called video (created in the previous example) without scaling within the current picture on the current graphics device.
display in=cgs4a comp=v mode=sc low=1 high=5.2 device=xwindows
Displays the VARIANCE component of NDF cgs4a on the xwindows device, scaling between 1 and 5.2.
display mydata centre="12:23:34 -22:12:23" xmagn=2 badcol="red" $\setminus$
Displays the NDF called mydata centred on the position RA=12:23:34, DEC=-22:12:23. This assumes that the current co-ordinate Frame in the NDF is an equatorial (RA/DEC) Frame. The image is displayed with a magnification of 2 so that each data pixel appears twice as large (on each axis) as normal. Fewer data pixels may be displayed to ensure the image fits within the available space in the current picture. The current scaling is used, and bad pixels are shown in red.
display ngc6872 mode=ra device=lj250 lut=pizza
Displays the NDF called ngc6872 on the LJ250 device. The lookup table in the NDF called pizza is mapped on the LJ250’s colour table. The scaling is between the minimum and maximum of the image.

#### Notes:

• For large images the resolution of the graphics device may allow only a fraction of the detail in the data to be plotted. Therefore, large images will be compressed by block averaging when this can be done without loss of resolution in the displayed image. This saves time scaling the data and transmitting them to the graphics device. Note that the default values for Parameters LOW and HIGH are the minimum and maximum values in the compressed floating-point data.

• If no Title  is specified via the STYLE parameter, then the TITLE component in the NDF is used as the default title for the annotated axes. Should the NDF not have a TITLE component, then the default title is instead taken from current co-ordinate Frame in the NDF, unless this attribute has not been set explicitly, whereupon the name of the NDF is used as the default title.

• The application stores a number of pictures in the graphics database in the following order: a FRAME picture containing the annotated axes, the image area, and the border; if there is a key, a KEY picture encompassing the key and its annotations; and a DATA picture containing just the image area. Note, the FRAME picture is only created if annotated axes or a border have been drawn, or if non-zero margins were specified using Parameter MARGIN. The world co-ordinates in the DATA picture will be pixel co-ordinates. A reference to the supplied NDF, together with a copy of the WCS information in the NDF are stored in the DATA picture. On exit the current database picture for the chosen device reverts to the input picture.

• The data type of the output NDF depends on the number of colour indices: _UBYTE for no more than 256, _UWORD for 257 to 65535, and _INTEGER otherwise. The output NDF will not contain any extensions, UNITS, QUALITY, and VARIANCE; but LABEL, TITLE, WCS and AXIS information are propagated from the input NDF. The output NDF does not become the new current data array. It is a Simple NDF (because the bad-pixel flag  is set to false in order to access the maximum colour index, and to handle sections), therefore only NDF-compliant applications can process it.

#### Related Applications

KAPPA: WCSFRAME, PICDEF, LUTVIEW; FIGARO: IGREY, IMAGE, MOVIE.

#### Implementation Status:

• This routine correctly processes the AXIS, DATA, QUALITY, VARIANCE, LABEL, TITLE, WCS, and UNITS  components of the input NDF.


text:
# The BriefA Blog about the LSAT, Law School and Beyond

A histogram looks a lot like a bar graph and, indeed, you can think of it as a special kind of bar graph. But instead of having just any kind of category (as a bar graph does), histograms have, as their categories, certain ranges of values. So, for example, suppose the students in your class score the following scores on their test:

Now, those numbers are kind of unwieldy. So to get a sense of how many students are acing your class (getting an A) or failing your class (getting an F), you might categorize their test scores according to certain ranges. So put all the 90-100 scores together in one category, all the 80 - 89 scores in the same category, and so on. Graphing this, we would get the following:

This graph tells us that 4 students scored between 90 and 100; 3 students between 80 and 89; and so on.

Example 1

In the following histogram, approximately how many students scored a 170 or higher?

Example 2

In the below histogram, what if anything can we conclude about the median of the data?

Example 3

In the following chart, what can we conclude about the range of the data?

Practice Problems

1. In the following histogram, approximately how many students scored a 170 or lower?
2. In the below histogram, what if anything can we conclude about the range of the data?

3. In the below histogram, what is the minimum possible number of students who scored higher than 75? What about the maximum?

Comment on this

A bar graph tells you how many objects in a given category you have. For example, in the following bar graph:

Each bar tells you how many children fall into that height range. So, for example, there are 2 children between 0 and 4 feet; there are 7 between 4 feet 1 inch and 4 feet 6 inches, and so on.

Example 1

How many children are between 4 feet 7 inches and 6 feet? How many children are taller than 6 feet?

We can also have segmented bar graphs like the following:

These bar graphs allow us to compare different groups, in this case the group of 1 year olds against the group of 2 year olds, 3 year olds, and so on. We can thus see how their preferences shift over time.

To read the graph, the horizontal signs tell us which categories we have (here, the different ages 1 to 5) and then the vertical component tell us how many members of that category (e.g. how many one year-olds) in accordance with the key to the right. Thus, the left-most blue bar tells us how many 1 year-olds have "Mighty Man" as their favorite superhero. The right-most yellow bar tells us how many 5 year-olds have "Valiant Vanessa" as their favorite hero and so on.

Example 2

Which superhero gains the most fans as the children mature from being 1 years-old to 5 years-old?

Example 3

Which superhero has the most supporters overall (i.e. across all the age groups)?

Example 4

Which of the following tables goes with the given bar graph:

A.

B.

C.

D.

Practice Problems

1. Which of the following tables is compatible with the given graph?

A.

B.

C.

D.

2. A new kid, Richard, joins the class. Richard is 4'8. Given that the graph below accurately depicts the heights of Richard's classmates, what is the maximum possible number of classmates that are taller than Richard? What is the minimum possible number?
3. In the below chart, what percentage of three-year olds take Tornado Terry to be their favorite hero?

4. In the below chart, which superhero loses the most fans as children age from 1 to 5?

5. In the below chart, what if anything can you conclude about the median height of the class?

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In this post, we’ll define the mean, median, and mode. Along the way, we’ll work through an example of how to find each of them. We’ll also talk about why the median is responds less to outliers than the mean does, a fact which can sometimes be crucial in solving a GRE problem.

Finding the Mean

The mean (or average) of a variable is found by adding up all of the values of that variable and then dividing by the number of observations. So to go back to our lemonade example:

Finding the Median

Now, finding the median is a little trickier. Imagine lining up all of the values for your variable from least to greatest. Then, the median is the one in the middle. So for example, if we have the data:

We order it from least to greatest to get:

And then we pick the middle value, namely .

Sometimes, if there is a long string of values, it will be hard to see which value is the middle one. So take the data:

We order it from least to greatest:

And then we simply cross off the numbers at the end, both the left-most and right-most observation, to get:

1, 1, 1, 1, 2, 2, 3, 3, 3, 3, 4, 6, 7, 8, 9

And we repeat:

1, 1, 1, 1, 2, 2, 3, 3, 3, 3, 4, 6, 7, 8, 9

Until we get:

1, 1, 1, 1, 2, 2, 3, 3, 3, 3, 4, 6, 7, 8, 9

And thus the median is 3.

Sometimes we will have an even number of observations. So suppose we have the data:

Crossing through the numbers on the end, we get:

1, 2, 3, 4

But now, it seems, we are stuck. For the median is a single number - not two numbers! And yet if we cross out any more, we will eliminate all of our numbers.

In these cases, we say that the median is the average of the middle two numbers. Thus, the median in the above case is

So in summary, the median is either the middle number or, if you have two middle numbers, the average of those two middle numbers. Now, we try to find the median for our lemonade example:

Finding the Mode

Finally, the mode is the value that occurs most often. So, if you have the data:

Then the mode is .

Now, a set of numbers can have more than one mode, as in this example:

Here, since both and occur three times (and no other number occurs three or more times), they are both modes.

Often, to find the mode it can help to put the numbers in ascending order (so you can see how often certain values are repeated). To return to our lemonade example:

Median is More Resilient to Outliers

Finally, one property of the median is that it is "more resilient to outliers." What does this mean? First, let's be clear on what an outlier is: an outlier is a data value that is very far from many of the other observations. So suppose you have some data on how many days in a month it rains:

But one month, you have constant downpours and so you get as a new observation. Your new data is:

But the is quite far from all of the other observations. If we were to graph this data on a boxplot:

We would see that sticks out like a sore thumb. That's how you know it is an outlier. (There are more precise definitions of an outlier, but you won't need them for the GRE).

Now, when we say that the median is "more resilient" to outliers than the mean, we are saying that if we add an outlier to our data, the median is affected less than the mean. In other words, the change to the median will be less than the change to the mean.

Let's confirm that this is the case in our above example. To do so, we will compute the old mean/median, the new mean/median and compare them.

The old mean is:
The old median is: since both 14 and 15 are in the middle of

The new mean is: .
The new median is: .

So we can see that the new mean is higher than the old mean, whereas the median only increased by . Thus, the median changed less than the mean did, exactly as predicted.

Why is the Median More Resilient?

Now, you might wonder why the median is more resilient than the mean to outliers. You won't need to know this for the GRE, but it might help in remembering which one is more resilient.

Here's why. When you add a new value to the data set, the median doesn't really care how large/small the new number is. All that matters is whether it is larger than the previous median or smaller. If it is larger than the previous median, then the new median moves a number to the right. If it is smaller, the new median moves a number to the left. But the mean does care about how large/small the new number is. So if the new number is an outlier, then it is, by definition, really large or really small compared to the other numbers. So the mean responds a lot to this new number whereas the median just moves a number to the right/left. That's why outliers affect the mean more than they do the median.

Practice Problems

1. Suppose we have the following heights of a kindergarten class (in inches): 29, 31, 33, 33, 37, 28, 29, 30. Find the mean, median, and mode of this data.
2. Suppose the ruler we used to measure everyone's height was mislabeled: in fact, everyone is five inches taller than the ruler suggests. Take the data in question 1, adjust for this fact, and find the new mean, median, and mode.
3. A new kid who is remarkably tall (60 inches) joins the class. Which statistic is more affected by this change, the median or the mean?

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In this post, we'll talk about ranges, quartiles, and percentiles. These are all ways of getting a sense of what the overall distribution looks like and what the possible outcomes look like.

Let's start with the range. The range is the difference between the largest and the smallest value in your data. So suppose you have the following data:

Then, the range is . Thus, the range tells you the interval over which your data is distributed.

Quartiles are more complicated. As the name suggests, quartiles are a way of dividing up the data in to four parts. So, if we have the following data:

The quartiles are the points that divide up the data in four segments, each with the same number of observations. For the above data, we would get:

You can see how in each section, we have exactly three points which is one fourth of our total data (made up of 12 points).

More formally, the second quartile is the median. Then, the first quartile is the median of all the values that are less than the median (or second quartile). The third quartile is the median of all the values that are greater than the median.

Now how can we find the quartiles?

Well, we already know how to find the second quartile since that's just the median. After finding the median, separate the data into two halves: the lower half is made up of all the values smaller than our median, whereas the higher half is all the values greater than our median.

Then, find the median of both halves. The median of the lower half will be the first quartile, and the median of the upper half will be the second quartile value.

This makes some intuitive sense since 25% is exactly at the middle between 0 and 50%, which is the data that makes up our lower half, whereas 75% is the middle of 50% and 100%, which is the data that makes up our upper half. So if we were looking for a process to split the data into four parts, each containing 25% of our observations, this method (to find the quartiles) would be an excellent candidate for that task.

Finally, the interquartile range is just the value of the third quartile minus the value of the first quartile.

Example 1

Find the first, second, and third quartiles of: 21, 13, 37, 45, 5, 1, 9, 17, 33, 41, 25, 29. Then find the interquartile range.

Practice Problems

1. Find the first, second, and third quartiles of: 30, 22, 90, 43, 28, 65, 2, 12. Then, find the interquartile range. Also, find the range.
2. Find the first, second, and third quartiles for: 111, 210, 291, 240, 287, 534, 323, 222, 401. Then, find the interquartile range. Also, find the range.
3. For the following numbers, is the interquartile range larger than the range? 22, 12, 32, 53, 46, 23, 21, 19, 35, 42, 31, 21, 24

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In the next few posts, we will introduce different kinds of graphs. These graphs allow you to visually represent data in ways that emphasize different aspects of the data. In preparation, let's look at some data represented with different graphs:

$$\begin{center} \begin{tabular}{ |c|c| } \hline Date & Lemonades Sold\\ \hline 7/2/19 & 1 \\ \hline 7/3/19 & 2 \\ \hline 7/4/19 & 5 \\ \hline 7/5/19 & 2 \\ \hline 7/6/19 & 3\\ \hline 7/7/19 & 2\\ \hline 7/8/19 & 5\\ \hline 7/9/19 & 4 \\ \hline 7/10/19 & 3\\ \hline 7/11/19 & 3\\ \hline \end{tabular} \end{center}$$

Here are some of the difference kinds of graphs you may encounter:

Bar graph:

Circle graph:

Box plot:

Scatterplot:

In subsequent posts, we will talk more about how to read each of these graphs.

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Talk of "data" is ubiquitous -- what does that word mean in the context of the GRE? We will think about data as observations of given variables. Now what does that mean?

Well suppose we are trying to help our child increase her earnings from a lemonade stand. Some days, she sells a lot and some days she sells almost nothing. To figure out why, we might start keeping track of how much she sells on a given day. So we would start making a table that looks like:

Now, each of the rows in the table is an observation. And we have our variables at the top of the columns: the date and the number of lemonades sold. And more generally: variables are just the characteristics that we keep track of, while an observation is some particular occasion when we record the values of our variables.

Now, a very simple way to keep track of data is via a frequency distribution. This is a table that records, on the left-hand side, possible values of a given variable, and on the right-hand side, it records how often those values appeared. Applied to the above data, we would get:

This table answers questions like "How often did my child sell three lemonades?" To find the answer, we go to the "Lemonades Sold" column and look for the row with three lemonades sold. In that row, the right hand column (corresponding to the number of days) says one. So there was one day where the child sold three lemonades.

And, in addition to a frequency distribution, we can also create a relative frequency distribution which records, on the left-hand side, possible values of a given variable, and on the right hand-side, the percentage of all observations where that value occurred. So, in the above example, we would get:

since the total number of days is and 20% and 40% and so on through the table.

Now, to really get a handle on what is driving her lemonade sales, we should probably add some more variables (e.g. daily temperature, day of week) and collect some more observations:

Here are some practice problems on the above concepts:

Practice Problems:

1. Using the above data, construct a frequency table that tells you how often a certain number of lemonades was sold:
2. Using the above data, how many lemonades were sold on Tuesdays?
3. Using the above data, how many days had more than 3 lemonades sales?
4. On days where the temperature was at least 73 degrees, how many lemonades did she sell on average?
5. On days where the temperature was less than 73 degrees, how many lemonades did she sell on average?

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In the next few posts, we’ll unpack some of the statistical terms that the GRE throws around. These terms can be seen as ways to answer two central questions:

1. What’s Typical?
2. What’s Possible?

In response to the question of "What's Typical?" we can find the median, mean, or mode of the data. These give you a sense of what the average value is, or what the most common value is; in short, they describe what a typical value might be.

And in answering "What's Possible?" we can find the range, quartiles, or percentiles of the data. These give you a sense of the possibilities and how the possibilities are distributed.

In later posts, we will give actual definitions of these concepts. But first, some overarching advice about learning these concepts.

There are four main ways you’ll get tested on these concepts.

1. Given some data, find the value of one of these concepts (e.g. the mean, median, range, etc.).
2. Given some data, alter the data in some way (e.g. by adding 10 to every value), and then find the value of one of these concepts.
3. Given some graph/chart, estimate the value of one of these concepts.
4. Answer some question about some concept's general properties (e.g. that the interquartile range is always less than or equal to the range).

Thus, you want to understand the formal definition well enough to do questions of the first and second types. But you also want to have some intuitive sense for the concepts, so that you can do questions that fall in the third and fourth categories.

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Around 20% of GRE questions involve "data analysis." These questions typically involve looking at data/graphs/tables in order to find certain values, like the median or percent change. Other questions ask about important statistical concepts, like the normal distribution.

All the posts in this series can be found here:

Content for other parts of the math GRE can be found here:

For reasons known only to ETS, number line problems are categorized with geometry. Such problems are basically algebra problems: they present you with some diagram or some facts about certain variables and then ask whether some equations or inequalities could be true, given certain relationships among the variables. Here is a straightforward example:

Example 1

x                         y
A. is greater than
B. is greater than
C. The two quantities are equal
D. It cannot be determined which, if any, is greater.

Now, a number line simply gives all the numbers from least to greatest. So the numbers to the right of the number line are greater than those to the left. And if there are markings on the number line, then you may (provided the question says nothing to the contrary) assume that the markings are evenly spaced. What does this mean? Well in the following diagram:

you know that the distance between and is only half of the distance between and . Using algebra, we could express this as:

Now, how do we solve number line problems? It will depend on what the question asks for. Some questions ask for which of the following could be true, whereas others ask about which of the following must be true.

If the questions asks which equations could be true, then we are looking for either a set of numbers that makes the equation true and is compatible with the diagram/given information, or we want to show that the equation is somehow inconsistent with our diagram/given information.

If the questions asks which equations must be true, then we are looking for either a set of numbers that is compatible with the diagram/given information but makes our equation false, or we want to show that the equation somehow follows from the diagram/given information.

Whether you should look for a specific example or some proof of the consistency/inconsistency of the given equation is a judgment call that you need to make in the moment. It's hard to give general principles about when to do which, but after doing some practice problems and just thinking about the equation at hand, you should develop some sense of how to decide if an equation is consistent or not. For example:

Example 2
Let and . Must the following be true?

After doing some practice problems, you get a sense of which numbers to consider when you come to one of these problems. And by running through those numbers quickly, you can often show that some equation either can be satisfied (which answers whether the equation could be true) or can be violated (which answers whether the equation must be true).

The only real piece of advice I have here is two-fold: First, always try easy, concrete numbers. Don't just think about "one negative and one positive value." It's a lot more general to think that way, but also a lot harder to evaluate. Try, instead, or and . Nice, easy numbers to calculate. And if you want a small number, try or Second, it is easy to think about what happens if both are normal, natural numbers (e.g. 1 and 2). But also think about what happens if they are small, positive numbers (e.g.  and ) or only one is positive (e.g. 1 and -1) or both are negative (e.g. -1 and -2). If the equation works (or is always violated) in all of those cases, then probably it's always true (or always false).

When it comes to proving that some equation is either always true/false given the diagram/information in the question, look for easy inferences. And if you can't see a way to prove it quickly, consider just flagging it and coming back. If you've gone through a few diverse examples and they all come out the same way, probably the equation actually is always true (or always false) and the time spent confirming that might be better spent on other problems.

Practice Problems:

1. The markings on the below number line are evenly spaced:

Which of the following must be true (select all that apply):
A.
B.
C.

2. Suppose we know and  Which of the following must be true (select all that apply):
A.
B.
C.

3. Suppose we know Which of the following must be true (select all that apply):
A.
B.
C.
D.
E.

4. The markings in the below number line are equally spaced. Which of the following could be true:

A.
B.
C.

5. The markings in the below number line are equally spaced. Which of the following must be true:

A.
B.
C.

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Geometry questions sometimes involve various graphs. For example:

Example 1
Find the area of the following square:

Now, these problems do not differ substantially from what we have before. Essentially, graphing problems give you the same information in a slightly different way. So instead of telling you how long the sides of a particular shape are, they might give you the coordinates of the endpoints. But of course, we know how to infer the length of that side, for we know the Pythagorean Theorem:

Example 2
A rectangle has a side whose vertices are at (1, 3) and (2, 2). Find the length of that side.

In this vein, a more realistic problem might look like:

Example 3
A triangle has vertices at (1, 4), (2,3), and (0,0). Find its perimeter.

Similarly, graphing problems can involve angles:

Example 4
A triangle has vertices at and . Find the degree of angle .

In short, if you have a geometry problem that involves certain points or a diagram, you can generally use those points or that diagram to infer aspects like the lengths of certain sides or the degrees of various angles, and then use that information as you normally would.

Practice Problems:

1. A rectangle (shown below) has vertices at (0,1), (2, 0), and (2,5). Where is its fourth vertex?

2. A circle has its center at point and its circumference includes the point . Find its radius.

3. A triangle has vertices at (1, 4), (2,3), and (0,0). Find its area.

4. The square in the following diagram has points at (2, 1) and (1, 4) as labelled. What is the area of the square?

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text:
# Convergence of the exponential martingale

How can we show that this martingale $$e^{aW_{t} - \frac{1}{2}a^2t}$$ converges to $0$ as $t \rightarrow \infty$ using law of iterated logarithm, for $a \neq 0$.

-
You could try the following steps: (1) translate the desired result in terms of $aW_t-\frac12a^2t$, (2) deduce from the law of iterated logarithm an upper bound on $aW_t$ for $t$ large enough, (3) conclude. – Did Feb 18 '12 at 0:54

Assume that $a\neq0$. Note that
$$\exp\left(aW_t-\frac{1}{2}a^2t\right) =\exp\left(t\left(\frac{aW_t}{t}-\frac{1}{2}a^2\right)\right).$$
From the law of the iterated logarithm, it follows as a special case that $W_t/t$ converges almost surely to zero as $t\to\infty$. Therefore, as $a\neq0$, $\frac{aW_t}{t}-\frac{1}{2}a^2$ converges almost surely to $-\frac{1}{2}a^2$, which is negative. As a consequence, the argument to the exponential in the above converges to minus infinity, and so the exponential itself converges to zero.
The law of the iterated logarithm is overkill here: the fact that $W_t/t \to 0$ a.s. is just the strong law of large numbers. – Nate Eldredge Apr 13 '12 at 1:16


text:
## Stream: general

### Topic: inhabited and nonempty instances

#### Sebastien Gouezel (May 19 2020 at 07:28):

It came up in #2720 that we don't have an instance saying that an add_comm_group is nonempty. Inhabiting basic classes like that can create problems: for instance, if you inhabit monoids by picking 1, and add_monoids by picking 0, then a ring would be inhabited both by 0 and 1, which can be bad. I would argue that, when there is a zero, it's always this one we should pick (although this is not completely obvious for groups with zeros). What do you think of adding the instances

instance has_zero.to_inhabited (α : Type*) [has_zero α] : inhabited α := ⟨0⟩
instance has_one.to_nonempty (α : Type*) [has_one α] : nonempty α := ⟨1⟩

Or should we replace the first one also with nonempty? Or drop this idea because it would lead to performance problems?

#### Gabriel Ebner (May 19 2020 at 07:33):

Conflicting nonempty instances are completely fine since nonempty is in Prop. I don't see any immediate problems with the inhabited by zero proposal either. Performance could be an issue, yes.

#### Gabriel Ebner (May 19 2020 at 07:34):

Why does #2720 require nonempty at all?

#### Sebastien Gouezel (May 19 2020 at 07:38):

In #2720, @Joseph Myers defines affine spaces, which are spaces on which a vector space acts freely and transitively. The empty space is endowed with a free and transitive action of any space, but it is not what you want, so requiring that an affine space is nonempty is natural. Then, when you want to show that a vector space is also an affine space for its action by translation on itself, you need to have available the instance saying that the vector space is nonempty.

#### Mario Carneiro (May 19 2020 at 07:40):

Conflicting inhabited instances can definitely be a problem. I don't think it is always guaranteed to be the case that 0 is the default element of every type that has a zero

#### Mario Carneiro (May 19 2020 at 07:42):

I recommend putting haveI : inhabited α := ⟨0⟩ whenever you need it in a proof

#### Mario Carneiro (May 19 2020 at 07:43):

or using a local instance

#### Sebastien Gouezel (May 19 2020 at 07:55):

I have prepared two branches, one using has_zero.to_inhabited and the other one has_zero.to_nonempty, to see if it breaks something or if compilation time increases. Let's wait for CI.

#### Sebastien Gouezel (May 19 2020 at 07:56):

Mario Carneiro said:

I don't think it is always guaranteed to be the case that 0 is the default element of every type that has a zero

In theory yes, but in practice I can't think of any example. If it works in 99,9% of the cases, we could enforce it as a rule.

#### Mario Carneiro (May 19 2020 at 07:56):

It seems like having these as local instances will solve most of the problems. Not all uses of inhabited have to do with algebra

#### Mario Carneiro (May 19 2020 at 07:58):

this looks like a very domain specific thing to do and I'd rather not add a global rule about it

#### Mario Carneiro (May 19 2020 at 07:59):

what is the default element of with_bot nat?

#### Rob Lewis (May 19 2020 at 08:00):

Another option if you have nonempty T and only need inhabited T within a proof, the tactic inhabit T will upgrade the instance locally.

#### Mario Carneiro (May 19 2020 at 08:00):

oh, does that tactic exist?

#### Mario Carneiro (May 19 2020 at 08:01):

If so it could do the same thing if it can prove T is a monoid or add_monoid

#### Rob Lewis (May 19 2020 at 08:02):

https://leanprover-community.github.io/mathlib_docs/tactics.html#inhabit

tactic#inhabit

tactic#inhabit

lean#200

#### Sebastien Gouezel (May 19 2020 at 08:07):

The with_bot nat is a good example:

example : (has_zero.zero : with_top nat) ≠ default (with_top nat):= with_top.zero_ne_top

I am convinced that we should not have this as a global instance. In fact, I only care about nonempty, which can not create any theoretical problem. My only concern is that it could create performance problems: if you want to prove that a type is nonempty, and start looking for a zero, then there are so many ways to have a zero that it could lead to exponential blow-up very quickly. I guess these instances should be marked with pretty low priority to get a reasonable behavior.

#### Gabriel Ebner (May 19 2020 at 08:11):

You get the same performance issue with inhabited. And yes, they should have a low priority. (I believe we even have a linter that tells you that.)

#### Yury G. Kudryashov (May 19 2020 at 09:34):

Probably a better way to deal with the issue in #2720 is to require [nonempty] only where it is needed.

#### Yury G. Kudryashov (May 19 2020 at 09:35):

I mean, in most cases it doesn't matter if we call the empty set an affine space or not.

#### Kevin Buzzard (May 19 2020 at 09:49):

It comes up in the fundamental theorem relating torsors to group cohomology

#### Yury G. Kudryashov (May 19 2020 at 09:54):

Is it hard to add [nonempty] in this theorem?

I don't think so

#### Sebastien Gouezel (May 19 2020 at 10:01):

It feels strange that an affine space could be not in bijection with the underlying vector space, but I agree it is probably a good idea to remove it from the definition, and add it back when needed.

#### Kevin Buzzard (May 19 2020 at 10:04):

It feels strange that you can divide by 0, but we have to get used to these things ;-)

#### Sebastien Gouezel (May 19 2020 at 10:12):

Anyway, I have added the nonempty instances to the library, and it doesn't create bad performance problems: in a file importing a lot of things

lemma zou (α : Type*) [nondiscrete_normed_field α] : nonempty α :=
by apply_instance -- elaboration: tactic execution took 0ms

lemma zou2 (α : Type*) : nonempty α :=
by apply_instance -- fails, elaboration: tactic execution took 230ms

Since users could be surprised that Lean is unable to prove a ring is nonempty, I think we should include these instances. PR at #2743

#### Sebastien Gouezel (May 19 2020 at 10:14):

(In the case of zou2, if I activate the trace for class instances, the trace is too long, truncated at 262144 characters, as should be expected).

#### Reid Barton (May 19 2020 at 10:36):

Oh yes. Torsors are nonempty of course. They should extend nonempty.

#### Reid Barton (May 19 2020 at 10:39):

Well maybe not extend nonempty if that's problematic, just add it as a field I guess.

#### Reid Barton (May 19 2020 at 10:44):

Oh I see, it is there already :thumbs_up:

#### Reid Barton (May 19 2020 at 10:50):

I commented on the PR, but actually I think the best way to deal with #2720 is just to add nonempty P as a field of the structure.

#### Joseph Myers (May 19 2020 at 10:53):

If you allow affine spaces to be empty, and then probably allow affine subspaces of a nonempty affine space to be empty, you get things like equality for affine subspaces (given the bundled vector subspace, which seems natural to have) no longer being the same as equality for the set of points. Anything to do with dimension also gets complicated by having to allow dimension -1, which is the natural dimension for an empty space but no longer the same as the dimension of the underlying vector space. (You get complications either way, but I suspect "the intersection of two affine subspaces is only an affine subspace if nonempty" may be easier to deal with than the complications from allowing empty spaces and subspaces.)

#### Reid Barton (May 19 2020 at 11:05):

Hmm, so you are viewing an affine subspace as a "sub-(affine space)", that is, a subset which is a torsor for a subgroup of the original group, right? Then right, if the subset is empty (and if we allowed that), then the group could be anything.

#### Sebastien Gouezel (May 19 2020 at 11:45):

I agree that including nonemptyness as a field of the structure would solve everything, as it would avoid the typeclass inference issues.

#### David Wärn (May 19 2020 at 12:36):

Maybe if you want the empty space then you should define affine spaces in terms of "linear combinations where the coefficients sum to 1" rather than torsors

#### Sebastien Gouezel (May 19 2020 at 12:37):

No one wants the empty space, the question is rather the best way to avoid it :)

#### Reid Barton (May 19 2020 at 13:07):

I like this affine linear combination definition, but I'm not sure how it relates to additional structure on the vector space like a norm.

#### David Wärn (May 19 2020 at 14:55):

How about this? Define the vector space C (C for cone) spanned by an affine combination space A, left adjoint to the forgetful functor. Consider the kernel D (D for differences) of the linear map C -> k corresponding to the constant 1 map A -> k. Any point of A should naturally induce an affine isomorphism between D and A. Thus norms on D correspond to "Euclidean" distance functions on A. Presumably these distance functions can also be characterised by some sort of homogeneity and the parallelogram law

Last updated: May 16 2021 at 20:13 UTC