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# Generalized quasi-perfect numbers

A number $n \in \mathbb{N}$ is called quasi-perfect if $\sigma(n) = 2n+1$, where $\sigma$ is the sum of divisors function. It is known that if $n$ is quasi-perfect, then it must be the square of an odd integer (an oddly beautiful Putnam problem from the 1970s). To date no number is known to be quasi-perfect.

My question concerns the existence of 'generalized' quasi-perfect numbers, or rather, let $a,b \in \mathbb{N} \cup \{0\}$ be fixed integers, and call an integer $n$ $(a,b)$ quasi-perfect if it satisfies $\sigma(n) = an+b$.

Are there any known values of $a,b$ for which the number of $(a,b)$ quasi-perfect numbers are known to be infinite?

-
I wondered about this a while ago and dblues had some nice preliminary observations in the very last post of this AoPS thread: artofproblemsolving.com/Forum/viewtopic.php?f=57&t=83696 –  Daniel m3 Sep 6 '13 at 16:42
Primes. You might consider negative b, otherwise the answer may turn out to be no. –  The Masked Avenger Sep 6 '13 at 16:44

Clearly, if $n=p$ whith $p$ being prime answers the question. $\sigma(p)=p+1$ which proves the case for $a=b=1$

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Or, more generally, if you take any $m$, then $n=pm$ with $p$ coprime to $m$ gives $\sigma(pm)=(p+1)\sigma(m)$, which means $a=b=\sigma(m)$ works. –  James Cranch Sep 7 '13 at 2:38
Sorry James, but that doesn't work. (Try m=5 and various p.) –  The Masked Avenger Sep 7 '13 at 3:13
@JamesCranch Can you give a numeric example? You need $\sigma(pm)=a p m + b$ and your example doesn't appear of this form. –  joro Sep 7 '13 at 12:06
Sorry: you're right, I'm wrong! –  James Cranch Sep 7 '13 at 13:06

Unfortunately, I can only give some non-answers to this question.

In the case $b=0$, I believe the answer is still unknown. There are some references in Sandor, et al, The Handbook of Number Theory starting on page 105. In particular, the best that appears to be known is the following upper bound: $$\#\{n \le x | \sigma(n) = a n\} \le c x^{c' \log \log \log x/ \log \log x}$$ for all $a \in \mathbb{Q}$. ($c,c'$ are independent of $a$.)

I believe Konstantinos's example could be extended slightly: if $a=1$ and $b\neq 1$, then the number of solutions should be finite.

-

For similar questions searching OEIS might help.

Trying $\sigma(n) = 2 n + 2$ returns A088831 Numbers n whose abundance sigma(n)-2n=+2

A comment in the sequence:

If $2^n-3$ is prime (n is a term of A050414) then $2^{n-1}(2^n-3)$ is in the sequence.

I supposes it plausible that there infinitely many primes of the form $2^n-3$.

There are provable solutions for $a=2$ and for $a=k$ when k-multiperfect numbers $\sigma(m) = k m$ exist.

Let $m$ be perfect number ($\sigma(m)=2m$) and $p$ a prime coprime to $m$.

$\sigma( m p ) = 2 m ( p + 1 ) = 2 m p + 2 m$

$a=2, b=2 m$.

For k-multiperfect numbers $m$, $a=k, b= k m$.

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That seems to differ from what I was trying to do, insofar as it's actually correct. –  James Cranch Sep 7 '13 at 13:57
@JamesCranch maybe your method will work for prime $p$ and $a$ rational (depending on sigma), yet the question is about integer $a$. –  joro Sep 7 '13 at 14:08


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# Perl Weekly Challenge 148: Cardano Triplets

by Abigail

## Challenge

Write a script to generate first 5 Cardano Triplets.

A triplet of positive integers (a,b,c) is called a Cardano Triplet if it satisfies the below condition.

$\sqrt[3]{a + b \cdot \sqrt{c}} + \sqrt[3]{a - b \cdot \sqrt{c}} = 1$

### Example

(2,1,5) is the first Cardano Triplets.

## Discussion

First point of business is, what defines the first 5 triplets? Do we order them on smallest a, then smallest b? Do we order them on the minimum of a, b, and c (breaking ties with the second smallest)? Do we order them on the sum of a, b and c? Something else?

We decide to order them on the sum of a, b, and c.

#### Project Euler

Cordano triplets are the subject of task 251 of Project Euler.

### Solution

We start off by manipulating the formula above to eliminate the cube and square roots ([U3]):

$\begin{array}{lcl} \sqrt[3]{a + b \sqrt{c}} + \sqrt[3]{a - b \sqrt{c}} = 1 & \implies & \\ \sqrt[3]{a + b \sqrt{c}} = 1 - \sqrt[3]{a - b \sqrt{c}} & \implies & \text{(cubing both sides)} \\ a + b \sqrt{c} = 1 - 3 \left(\sqrt[3]{a - b \sqrt{c}}\right) \left(1 - \sqrt[3]{a - b \sqrt{c}}\right) - a + b \sqrt{c} & \implies & \\ 2 a - 1 = -3 \left(\sqrt[3]{a - b \sqrt{c}}\right) \left(1 - \sqrt[3]{a - b \sqrt{c}}\right) & \implies & \text{(identity on second line)} \\ 2 a - 1 = -3 \left(\sqrt[3]{a - b \sqrt{c}}\right) \left(\sqrt[3]{a + b \sqrt{c}}\right) & \implies & \text{(cubing both sides)} \\ 8 a^3 - 12 a^2 + 6 a - 1 = -27 \left(a^2 - b^2 c\right) & \implies & \\ 8 a^3 + 15 a^2 + 6 a - 27 b^2 c = 1 \end{array} \tag{1}$

Given the formula $$8 a^3 + 15 a^2 + 6 a - 27 b^2 c = 1$$, we can think of a parametrization.

Let's take this modulo $$3$$ [JM]:

$\begin{array}{lcl} 8 a^3 + 15 a^2 + 6 a - 27 b^2 c \equiv 1 \text{ mod } 3 & \implies & \text{(}15 a^2 \text{, } 6 a \text{, and } 27 b^2 c \text{ are all divisible by } 3 \text{)} \\ 2 a^3 \equiv 1 \text{ mod } 3 & \implies & \text{(multiply both sides by 2)} \\ 4 a^3 \equiv 2 \text{ mod } 3 & \implies & \\ a^3 \equiv 2 \text{ mod } 3 & \implies & \\ a \equiv 2 \text{ mod } 3 & \implies & \\ a = 3 k + 2 \text{ for some integer } k \geq 0 & & \end{array} \tag{2}$

Now, if we plug in the result of (2) into the result of (1), we get:

$\begin{array}{lcl} 8 a^3 + 15 a^2 + 6 a - 27 b^2 c = 1 & \implies & \\ 27 b^2 c = 8 a^3 + 15 a^2 + 6 a - 1 & \implies & \text{(result of (2))} \\ 27 b^2 c = 8 (3 k + 2)^3 + 15 (3k + 2)^2 + 6 (3k + 2) - 1 & \implies & \text{(Mathematica)} \\ 27 b^2 c = 27 (k + 1)^2 (8k + 5) & \implies & \\ b^2 c = (k + 1)^2 (8k + 5) & & \end{array}$

So, for each $$k \geq 0$$ we find an $$a$$, and a $$b^2 c$$. But there can be multiple solutions for $$b$$ and $$c$$:

1. Each divisor $$d_1$$ of $$k + 1$$ will be a solution for $$b$$.
2. For each divisor $$d_2$$ of $$8k + 5$$, if $$d_2$$ is a perfect square, $$\sqrt{d_2}$$ will be a solution for $$b$$.
3. For each divisor $$d_1$$ found in step 1, and each divisor $$\sqrt{d_2}$$ found in step 2, $$d_1 \sqrt{d_2}$$ will be a solution for $$b$$.

Once we have $$k$$ and $$b$$, we can calculate $c = \frac{(k + 1)^2 (8 k + 5)}{b^2}$

#### Stopping criterium

How do we know we can stop generating triangles? We will keep an array of the best 5 triangles found so far. If $$3 k + 2 = a$$ is larger than the largest of the sums of the best 5 triangles found so far, we know we cannot do better with larger $$k$$, so we can stop.

### Perl

First, we start off with some initialization. We keep the best triples found so far in an array @out — for each triple, we also store its sum. We initialize @out with high numbers, much larger than any triple we will report. We also keep track of the index of the largest triple in @out:

use Math::Prime::Util qw [divisors];
use List::Util        qw [sum];

my $COUNT = 5; my$A     = 0;
my $B = 1; my$C     = 2;
my $SUM = 3; my @out; foreach my$i (0 .. $COUNT - 1) {$out [$i] = [(999999) x 3];$out [$i] [$SUM] = sum @{$out [$i]};
}

my $max_index = 0;  We then start a loop, incrementing k. Given a k, we find a, and store the value k + 1 and 8 * k + 5. for (my$k = 0; 3 * $k + 2 <=$out [$max_index] [$SUM]; $k ++) { my$a  = 3 * $k + 2; my$f1 =     $k + 1; my$f2 = 8 * $k + 5;  We then find the divisors of f1, and the square roots of the divisors of f2, discarding the ones which are integers: my @d1 = divisors ($f1);
my @d2 = grep {$_ == int ($_)} map {sqrt $_} divisors ($f2);

We know multiply each element of @d1 and @d2, assign this to b, and calculate c. If the sum of a, b and c is better than the fifth best triple found so far, we discard the fifth best triple, replacing it with a, b and c. But we only do this if a, b and c are not yet in @out: we don't need duplicates:

foreach my $d1 (@d1) { D2: foreach my$d2 (@d2) {
my $b =$d1 * $d2; my$c = $f1 *$f1 * $f2 / ($b * $b); if ($a + $b +$c < $out [$max_index] [$SUM]) { # # Avoid duplicate entries # foreach my$info (@out) {
next D2 if $$info [A] == a &&$$info [$B] ==$b;
}

#
# Put triple in the output structure
#
$out [$max_index] = [$a,$b, $c,$a + $b +$c];

We now have to find the index of the newest fifth best triple:

#
# Find the index of the new highest value
#
$max_index = 0; my$max_sum = $out [$max_index] [$SUM]; for (my$i = 1; $i <$COUNT; $i ++) { if ($out [$i] [$SUM] > $max_sum) {$max_index = $i;$max_sum   = $out [$i] [$SUM]; } }  Once we have found all triples, we can print them: say "@$_[$A,$B, \$C]" for @out;

Find the full program on GitHub.

### Python

Most languages don't have a divisor method (or they may have in some module, but I generally don't go searching for them), so we have to find the divisors ourselves. Python for instance.

We'll skip the initialization, and dive directly into the body of the main loop, where we have to find the divisors of f1 and f2:

a  = 3 * k + 2
f1 =     k + 1
f2 = 8 * k + 5

d1 = []
for i in range (1, f1 + 1):
if i * i > f1:
break
if f1 % i == 0:
d1 . append (i)
if i != f1 // i:
d1 . append (f1 // i)

d2 = []
for i in range (1, f2 + 1):
if i * i > f2:
break
if f2 % i == 0:
s1 = math . isqrt (i)
s2 = math . isqrt (f2 // i)
if s1 * s1 == i:
d2 . append (s1)
if s2 * s2 == f2 // i and s1 != s2:
d2 . append (s2)

Finding the possibilities for b and c isn't very different from the Perl solution:

for d1v in d1:
for d2v in d2:
b = d1v * d2v
c = f1 * f1 * f2 // (b * b)

if a + b + c < out [max_index] [SUM]:
seen = False
for i in range (COUNT):
if out [i] [A] == a and out [i] [B] == b:
seen = true
break
if seen:
break

out [max_index] = [a, b, c, a + b + c]

max_index = 0
max_sum   = out [max_index] [SUM]
for i in range (1, COUNT):
if max_sum < out [i] [SUM]:
max_sum   = out [i] [SUM]
max_index = i

Find the full program on GitHub.

### Other languages

We also have implementation in some other languages, all using a very similar algorithm:

AWK, bc, C, Go, Java, Lua, Node.js, Pascal, R, Ruby, and Tcl.

format: plain


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# Animation in beamer?

I was wondering if it is possible to make an animation in beamer. I normally work on graph theory. By a graph I mean a set of vertices and some edges joining them.

I wish to make an animation showing an isomorphism from one graph to the other. I could draw graphs using tikz package. But I don't know how to make the motion. So what I really want is to make an animation showing one point smoothly move to another point.

For example, I want something as below happen:

I don't use beamer but one can start with that :

\documentclass[margin=5pt]{standalone}
\usepackage{tikz,animate}
\usepackage[active,tightpage]{preview}
\usetikzlibrary{calc}

\xdef\C{8}
\xdef\H{4.5}

\begin{document}

\foreach \nxb in {0,.05,...,1,1,.95,...,0,0} {%
\begin{preview}
%\begin{animateinline}[controls]{10}
%   \multiframe{11}{nxb=0+.1}{%

\begin{tikzpicture}

\draw (-1,-.5) rectangle (2*\H,2*\H-.5) ;

\coordinate (C6) at (0,0) ;
\coordinate (C4) at (\C,0) ;
\coordinate (C1) at (0,\C) ;
\coordinate (C3) at (\C,\C) ;
\coordinate (C2) at ($(C1)!.5!(C3)$) ;
\coordinate (C5) at ($(C6)!.5!(C4)$) ;
\coordinate (C7) at ($(C6)!.5!(C3)$) ;

\coordinate (H5) at ($(C1)!.5!(C4)$) ;
\begin{scope}[shift=(H5)]
\coordinate (H6) at (0:\H) ;
\coordinate (H7) at (60:\H) ;
\coordinate (H4) at (120:\H) ;
\coordinate (H3) at (180:\H) ;
\coordinate (H2) at (240:\H) ;
\coordinate (H1) at (300:\H) ;
\end{scope}

\foreach \i in {1,...,7} {%
\fill ($(H\i)!\nxb!(C\i)$) circle (2pt) coordinate (M\i) ; }
\draw (M1)--(M2)--(M3)--(M4)--(M5)--(M6)--(M7)--(M4)
(M6)--(M1)--(M5) ;

\end{tikzpicture}
%}
%\end{animateinline}
\end{preview}}
\end{document}

• two things I am not sure. 1. How can you convert it to the gif file? When I compiled your codes I got a pdf file and it shows me the consecutive slides. 2. How do you embed this into beamer? Thanks – Easy May 25 '14 at 13:29
• To made the animated gif you see an this page, I used a step of .01 instead of .05 and imported the pdf file in gimp as calques not as images with 300 ppp, then exported it as gif, et voilà. I don't use beamer at all, maybe I will soon, but now I can't help you with this. – Tarass May 25 '14 at 14:50
• @Easy: It's not possible to include animated gif files in beamer. You can include animations but not gif. – Ignasi May 25 '14 at 21:04


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# Cite in the caption of a graphic but not in \listoffigures?

Sometimes I'd like to cite the reference in the caption of a graphic. But I want to avoid that this citation is displayed in the \listoffigures. How can I achieve that?

Fore example with this code:

\begin{figure}[htb]
\centering
\includegraphics{overview_pyramid.png}
\caption{Overview pyramid \cite{LM2010}}\label{fg:overview_pyramid}
\end{figure}

-

\caption[Overview pyramid]{Overview pyramid \cite{LM2010}}\label{fg:overview_pyramid}


text:
# 10.1 - Role of the Covariate

Printer-friendly version

To illustrate the role the covariate has in the ANCOVA, let’s look at a hypothetical situation wherein investigators are comparing salaries of male vs. female college graduates. A random sample of 5 individuals for each gender is compiled, and a simple one-way ANOVA is performed:

 Males Females 78 80 43 50 103 30 48 20 80 60

$H_0: \mu_{\text{Males}}=\mu_{\text{Females}}$

SAS coding for the One-way ANOVA (ancova_example_sascode.txt)

Here is the output we get:

To perform one-way ANOVA test in Minitab, you can first open the data (ancova_example_data.txt) and enter this into a Minitab worksheet.

Go to Stat > ANOVA > One Way…

In the pop-up window that appears, select salary as the Response and gender into Factor as shown below.

Click OK, and then here is the Minitab output that you get.

Because the p-value > α (.05), they can’t reject the H0.

A plot of the data shows the situation:

However, they recognize that the length of time that someone has been out of college is likely to influence how much money they are making. So they also included a question asking how many years they have been out of college (ranging from 1 to 5 years for this sample):

 Females Males Salary years Salary years 80 5 78 3 50 3 43 1 30 2 103 5 20 1 48 2 60 4 80 4

We can see that indeed, there is a general trend for people to earn more the longer they are out of college. The fundamental idea of including a covariate is to take this trending into account and effectively ‘control for’ the number of years they have been out of college. In other words, we hope to include the covariate in the ANOVA so that the comparison between Males and Females can be made without the complicating factor of years out of college.


text:
Journal cover Journal topic
Proceedings of the International Association of Hydrological Sciences An open-access publication for refereed proceedings in hydrology
Journal topic
Proc. IAHS, 380, 29-35, 2018
https://doi.org/10.5194/piahs-380-29-2018
Proc. IAHS, 380, 29-35, 2018
https://doi.org/10.5194/piahs-380-29-2018

Post-conference publication 18 Dec 2018

Post-conference publication | 18 Dec 2018

# Remote sensing-based soil water balance for irrigation water accounting at the Spanish Iberian Peninsula

Remote sensing-based soil water balance
Jesús Garrido-Rubio1, Alfonso Calera Belmonte1, Lorena Fraile Enguita1, Irene Arellano Alcázar1, Mario Belmonte Mancebo1, Isidro Campos Rodríguez1, and Raquel Bravo Rubio2 Jesús Garrido-Rubio et al.
• 1Remote Sensing & GIS Group, Institute for Regional Development (IDR), University of Castilla-La Mancha (UCLM), Albacete, 02071, Spain
• 2Subdirección General de Regadíos y Economía del Agua (SGRyEA), Ministerio de Agricultura y Pesca, Alimentación y Medio Ambiente (MAPAMA), Madrid, 28005, Spain
Abstract

Temporal series maps of irrigated areas, and the corresponding irrigation water requirements based on remote sensing, is a recognized tool contributing to water governance at different scales, from water user associations to whole river basin districts. These thematic cartographies offer a first estimation of the crop irrigation requirements, and a biophysical based approach of the temporal and spatial distribution of the crop water use in the cultivated areas. This work describes the operational application of these methodologies, providing valuable information for water governance and management purposes. The basic products obtained in the whole Spanish part of the Iberian Peninsula during the period 2014–2017 were: (i) annual maps of irrigated crops based on time series of multispectral satellite imagery; and (ii) the direct remote sensing-based water accounting, by quantifying agricultural water flows (e.g. rainfall, irrigation, evapotranspiration, drainage and recharge), through a remote sensing-based soil water balance. Hence this paper provides a remote sensing based water accounting approach, which relies on dense time series of multispectral imagery acquired by the multisensor constellation arranged by Landsat 8 and Sentinel-2 satellites, jointly with meteorological data and agronomic knowledge. Then, based on these purpose and approach, annual and monthly maps of net irrigation water requirements have been elaborated at the most practical spatial and temporal scales for water governance purposes over big areas such river basin districts. This work summarizes the methodologies used and discuss the technical and non-technical feasibility of the proposed approach.

1 Introduction

The monitoring of irrigated areas and the accounting of their water use are essential instruments for European policies, in order to promote an efficient and sustainable use of water, according to the Water Framework Directive, WFD (2000/60/EC, 2000), and transposed into each country member legislation.

Monitoring of crops and their water requirements are possible through data derived from the Earth Observation, EO, combined with meteorological information. The Copernicus program of the European Union, through the European Space Agency (ESA), operates a constellation of Sentinel satellites, which provide data from the Earth's surface at a spatial resolution and a temporal frequency unprecedented in the history of remote sensing. Consequently the Copernicus programs exhibit a great potential to perform the monitoring required by the WFD.

Aware of the great socio-economic and environmental importance of irrigation in Spain, the Ministry of Agriculture and Fisheries, Food and Environment (MAPAMA) pushes actions to improve the efficiency of water use on irrigation. Among other, through the Deputy Directorate General for Irrigation and Water Economy (SGRyEA), it makes available freely to users the information collected by the network of agrometeorological automatic stations, known as the SIAR network (Agroclimatic Information System for Irrigation). A step further is the SPIDER-SIAR 2014–2017 project, which joins the potential provided by the EO programs for the spatial and temporal description of the vegetation cover, to the agrometeorological information provided by the SIAR network, all for the knowledge of irrigated areas and their water use in the area of mainland Spain.

Table 1SPIDER-SIAR products and scope of its services regarding different water user types.

2 Objective, products and stakeholders

The main objectives of the SPIDER-SIAR project are, (1) improving water efficiency, and (2) better managing water resources by public authorities through mapping irrigated areas, estimating surface cover, and determining their net irrigation water requirements (NIWR) in the mainland Spain (2014–2017 period). Spanish main irrigation features are summarized as a surface that covers around 3.7 Mha, with wheat, barley, maize, vineyard, olive, fruit trees, and citric as main crops, and mainly irrigated by drip and sprinkle systems (around 75 %). Regarding such spatial extent, SPIDER-SIAR mainly uses a time series of satellite images of the constellation Sentinel-2A (S2A), Sentinel-2B (S2B) and Landsat 8 (L8), together with agrometeorological information of the SIAR network and ancillary information.

Methodology and materials used allow obtaining results from a minimum area of 0.1 ha, based on a daily temporal scale, and able to monitor all typologies of agricultural surfaces. These primary results are suitable to aggregate into other scales of spatial and temporal interest.

The beneficiaries of the project are at the same time water users and water managers, who can share the data at their different administrative scales, like irrigation communities, river basin authorities, regions, nation-wide and European levels, which deal with the planning of water resources in the medium and long term, through the execution and monitoring of river basin management plans (RBMPs). All the products can be accessed freely through a WebGIS platform called SPIDER-SIAR. Table 1 details the products and relates them to their potential users.

3 Materials and methodology

## 3.1 Time series of satellite images

The images used are those acquired, in the area of the mainland Spain, along the period 2014 to 2017, by the multispectral sensors on board the S2A, S2B and L8 platforms. The European Space Agency (ESA) and the National Aeronautics and Space Administration (NASA) from USA operate these platforms and distribute freely the images. Once those high cloud cover images have been discarded, a total number of 13 006 images were processed, of which 2437 scenes correspond to L8, 7957 granules to S2A, and 2612 granules to S2B. These images are then converted into the three basic SPIDER-SIAR products: (i) Vegetation Index (VI) time series of NDVI (Rouse et al., 1973); (ii) an appropriate colour composition images (RGB) to monitor irrigated agriculture; and iii) the biophysical crop parameter Kcb. These time series products (unless RGB) are atmospherically corrected by absolute normalization (Campos et al., 2011). Additionally, cloudy and shadows pixels were removed using L8 Quality Band and Sen2Cor algorithms for S2A and S2B. Consequently, a virtual multisensor constellation was built to provide time series of NDVI able to describe accurately the temporal evolution of the vegetation cover.

## 3.2 Thematic maps of irrigated land use

For mapping the irrigated crops, the approach uses the vegetation temporal evolution pattern as a basic criterion, which is well described by the temporal trajectory of the NDVI values of each pixel. Moreover, to guide spatially the classification analysis, the approach integrates multiple ancillary official sources, especially the layer of SIGPAC enclosures (Spanish national layer to attend Common Agricultural Policy System Information).

The methodology assumes that: (1) the water supply produces differences in size and duration of irrigated vegetation cover with respect to the one not; (2) time series of VI images are able to record these differences; and hence (3), decisions-tree rules can be applied to segment between irrigated and non-irrigated crops. Consequently, the knowledge about the temporal evolution of the different crop canopy cover is the essential instrument in the irrigated crops identification. It is necessary to highlight that crops that present a similar evolution both in their phenology and in soil fraction cover, will show a similar evolution in their NDVI time trajectory. In those crops that exhibits a high proportion of bare soil, especially woody crops, and in which the irrigation reaches only a supplementary value, the NDVI time series among irrigated and non-irrigated exhibits small differences.

SPIDER-SIAR annual thematic maps of irrigated areas present 9 classes in which the irrigated crops are grouped: four of these classes group the annual herbaceous crops depending on the season when the crop growing cycle occurs (spring, summer, spring-summer, and autumn-winter); four more classes are used to group woody crops (vines, olives, citrus and deciduous fruit trees); and finally, one class for crops developed under greenhouses.

## 3.3 Thematic maps of Net Irrigation Water Requirements (NIWR)

Thematic maps of net irrigation water requirements (NIWR) of those irrigated areas (excluding greenhouses), are estimated by carrying out a water balance in the root soil layer. This soil water balance (SWB) follows the dual coefficient approach described in the FAO56 manual (Allen et al., 1998). For each pixel, we use the time series of NDVI values to estimate the value of the Kcb, what it is called “reflectance-based crop coefficient approach” (Neale et al., 1989). We select a unique crop-independent NDVI-Kcb relationship using Eq. (1) (Campos et al., 2010), according with the real time and operational purpose at large scales (Calera et al., 2017), like other projects are dealing with, i.e. the case of California State at USA (Melton et al., 2012).

$\begin{array}{}\text{(1)}& \mathrm{Kcb}=\mathrm{1.44}\cdot \left(\mathrm{NDVI}\right)-\mathrm{0.10}\end{array}$

Furthermore, the use of a single equation provided some benefits for the operational monitoring of NIWR. For applications at large scales, our actual knowledge of the crop biophysical variables such as LAI and canopy resistance must be considered unknown. Note that the relationship between LAI and VI is crop dependent and strongly affected by the canopy architecture (Campos et al., 2017; Anderson et al., 2004) while the available information about land uses is restricted to general crop types (see Sect. 3.2) and difficulty improved. In addition, although the proposed relationship was originally developed for irrigated-row structured vineyard, the parameterization of the function does not differ from other relationships obtained for herbaceous crops like corn (Bausch and Neale, 1987; Gonzalez-Piqueras et al., 2004), or biomass sorghum (López-Urrea et al., 2016). In the same way, the used equation had been evaluated in bush-vines in Spain (Campos et al., 2016), vineyards in Australia (Hornbuckle, 2014), natural vegetation (Campos et al., 2013), and table grapes trained in overhead trellises system (Balbontín et al., 2017).

Figure 1Daily evolution of the different components estimated by the RS-SWB computed by HidroMORE® tool.

Figure 2Inputs databases used in HidroMORE® to calculate the RS-SWB at pixel scale and spatially distributed.

In parallel, and in order to obtain NIWR, daily data from meteorological stations of the dense SIAR network, which placed strategically on the irrigated areas, enable to obtain reference evapotranspiration and precipitation data maps by interpolating the raw point data. And finally in the same way, the soil water storage is a required input in the SWB, described by the Total Available Water (TAW) parameter defined as the amount of water that a crop can extract from its root zone (Allen et al., 1998). Mapping TAW utilizes soil hydraulic properties from the European Soil Database (Panagos et al., 2012), and maximum root depth, which depends in crop type and maximum allowed depth till impervious rock.

## 3.4 HidroMORE®, a tool for remote sensing based soil water balance (RS-SWB) over large scales

As described previously, the soil water balance turns into a RS-SWB regarding the use of dense NDVI time series, in order to monitor crop biophysical conditions. However, as the RS-SWB is perform at daily time scale, NDVI synthetic images are created, by means of linear temporal interpolation for those dates where no image is available. Furthermore, the dual crop coefficient approach applied, deals with real vegetation conditions after taking into account daily soil water content available for crops. Consequently, a daily monitoring on vegetation evolution development is afforded per each pixel, and thus, achieving local crop biophysical conditions into the soil water balance.

The tool used, able to compute such RS-SWB for the whole Spanish part of Iberian Peninsula, at pixel size scale of 10 m × 10 m, is the HidroMORE® software (Moreno et al., 2017; Torres et al., 2010). HidroMORE® estimates the NIWR that maintains the crop in optimum conditions. Crop optimum conditions means that soil water content is not allowed to be lower than a predetermined threshold, avoiding water stress, or maintaining this water stress on a controlled way, like is the case i.e. for vine grapes. Besides, computation estimates as well the other SWB components like adjusted crop evapotranspiration (ETcadj), run-off (RO), and deep percolation (DP) at same space and time scales. Figure 1 shows the computation of SWB components, for one pixel.

Inputs for HidroMORE® come from different databases (Fig. 2) like satellite images from L8, S2A and S2B platforms, meteorological daily records from the SIAR network and soil types from the European Soil Database.

Figure 3Monthly evolution of the time series of net irrigation water requirements (red chart) and precipitation (green chart), jointly with the NDVI time series (cyan chart) along 2014–2017.

## 3.5 Products accessibility via WebGIS platform

Thematic maps of irrigated areas and its NIWR, are generated at pixel scale, what means a spatial resolution of 0.1 ha, and for NIWR, at daily step. For operational purposes, NIWR data are aggregated at monthly scale. All these products can be online accessed through the SPIDERwebGIS® platform. This platform allows the visualization and display of results, especially as regards the time series. The user must only include the login demo and password demo through the following link: http://maps.spiderwebgis.org/login/?custom=spider-siar (last access: 13 September 2018).

4 Results of remote sensing-based Irrigation Water Accounting for the mainland Spain

Regarding operational and water management purposes, results from the RS-SWB are aggregated to monthly and annual frequency with the sum of the results estimated at daily time scale. Selected frequency is the reference one used by water managers for governance purpose at the river basin scales. In consequence, 48 monthly thematic maps of NIRW (and other previous cited SWB components), are shown in SPIDERwebGIS® platform.

Figure 3 shows a graph taken from the WebGIS platform that plot the time series of NDVI, Precipitation and NIWR over a single pixel herbaceous irrigated class along the four study years. Therefore, water managers can study the components of the RS-SWB over their territory with temporal and spatially distributed information.

Figure 4Comparison among the RS-based NIWR and NIWR calculated from official RBMPs for the Spanish mainland along 2014–2017.

Furthermore, a first evaluation of the remote sensing based irrigation water accounting (RS-IWA) computed in the SPIDER-SIAR project is shown in Fig. 4.

After computing the annual NIWR of the four study years, a zonal statistics calculation was performed per year using the mainland Spain river basin districts perimeters extracted from RBMPs official sources. Among the statistics parameters calculated, annual average NIWR value was then converted into the total volume of NIWR per year, using the irrigated area within each river basin. Finally, the four years average value was compared with the annual gross water demand presented for the river basin water managers in the RBMPs.

Those values were collected from the monitoring and evaluation document about the called by WFD, 2nd cycle River Basin Management Plans (2015–2021), developed by the Spanish Ministry of Agriculture and Fisheries, Food and Environment (MAPAMA and CEDEX, 2017). Then, in order to compare the same irrigation water parameter, the annual gross water demand was converted in NIWR by a global coefficient of 0.65 regarding the water that finally reaches irrigated plots fields, after consider the transport, distribution and application efficiency. The global coefficient assumed, due to the lack of specific data, was selected after consulting the twelve main RBMPs, e.g. Jucar River Basin 0.55, 0.62 in case of Duero River Basin, or 0.76 for the Guadalquivir River Basin.

Figure 4 shows the result for the comparison RS-based NIWR against NIWR calculated from currently official RBMPs. In despite of the encouraging agreement on this comparison, where R2 value is around 0.95 and RMSE value is 480 hm3, further analysis will be needed to pave the way to use this technique as an instrument of water planning.

5 Conclusions

The described approach to estimate a remote sensing-based irrigation water accounting (RS-IWA), produces a thematic cartography about irrigated land use and their net irrigation water requirements, spatial and temporally distributed on suitable scales for water governance purposes. The provided information aims to increase water efficiency, help on water governance and management purposes at river basin, regional and national scale, according to the WFD development. The approach requires further analysis about the accuracy and temporal stability of the results.

Data availability
Data availability.

Author contributions
Author contributions.

JGR, ACB, LFR, IAA and MBM conceived and conducted the work; JGR and ACB analysed the data; JGR, ACB and ICM wrote and review the article; and RBR and her team at The MAPAMA supervised the project and results provided for the whole period 2014–2017.

Competing interests
Competing interests.

The authors declare that they have no conflict of interest.

Special issue statement
Special issue statement.

This article is part of the special issue “Earth Observation for Integrated Water and Basin Management: New possibilities and challenges for adaptation to a changing environment”. It is a result of The Remote Sensing & Hydrology Symposium, Cordoba, Spain, 8–10 May 2018.

Acknowledgements
Acknowledgements.

The authors would like to thank both reviewers for their effort to read and provide comments and suggestions, that led us to improve the content and meaning of the paper.

Edited by: Maria P. González-Dugo
Reviewed by: two anonymous referees

References

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Calera, A., Campos, I., Osann, A., D'Urso, G., and Menenti, M.: Remote Sensing for Crop Water Management: From ET Modelling to Services for the End Users, Sensors, 17, 1104, https://doi.org/10.3390/s17051104, 2017.

Campos, I., Neale, C. M. U., Calera, A., Balbontín, C., and González-Piqueras, J.: Assessing satellite-based basal crop coefficients for irrigated grapes (Vitis vinifera L.), Agr. Water Manage., 98, 45–54, https://doi.org/10.1016/j.agwat.2010.07.011, 2010.

Campos, I., Odi, M., Belmonte, M., Martínez-Beltrán, C., and Calera, A.: Obtención de series multitemporales y multisensor de índices de vegetación mediante un procedimiento de normalización absoluta, XIV Congreso de la Asociación Española de Teledetección, 4, 2011.

Campos, I., Villodre, J., Carrara, A., and Calera, A.: Remote sensing-based soil water balance to estimate Mediterranean holm oak savanna (dehesa) evapotranspiration under water stress conditions, J. Hydrol., 494, 1–9, https://doi.org/10.1016/j.jhydrol.2013.04.033, 2013.

Campos, I., Balbontín, C., González-Piqueras, J., González-Dugo, M. P., Neale, C. M. U., and Calera, A.: Combining a water balance model with evapotranspiration measurements to estimate total available soil water in irrigated and rainfed vineyards, Agr. Water Manage., 165, 141–152, https://doi.org/10.1016/j.agwat.2015.11.018, 2016.

Campos, I., Neale, C. M. U., Suyker, A. E., Arkebauer, T. J., and Gonçalves, I. Z.: Reflectance-based crop coefficients REDUX: For operational evapotranspiration estimates in the age of high producing hybrid varieties, Agr. Water Manage., 187, 140–153, https://doi.org/10.1016/j.agwat.2017.03.022, 2017.

Gonzalez-Piqueras, J., Calera, A., Gilabert, M. A., Cuesta, A., and Tercero, F. D. l. C.: Estimation of crop coefficients by means of optimized vegetation indices for corn, Proc. SPIE 5232, Remote Sensing for Agriculture, Ecosystems, and Hydrology V, https://doi.org/10.1117/12.511317, 24 February 2004.

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# Mathematics for Technology I (Math 1131)

Durham College, Mathematics
Free
• 0 lessons
• 0 quizzes
• 10 week duration
• ##### Numerical Computation

Here you'll be introduced to the bare basics of mathematics. Topics include commonly used words and phrases, symbols, and how to follow the order of operations.

• ##### Measurements

An introduction to numerical computation. Emphasis is placed on scientific and engineering notation, the rule of significant figures, and converting between SI and Imperial units.

• ##### Trigonometry with Right Triangles

Here we focus on right angle triangles within quadrant I of an x-y plane. None of the angles we evaluate here are greater than 90°. A unit on trigonometry with oblique triangles is covered later.

• ##### Trigonometry with Oblique Triangles

This unit is a continuation of trigonometry with right triangles except we'll extend our understanding to deal with angles *greater* than 90°. Resolving and combining vectors will be covered at the end of this unit.

• ##### Geometry

This unit focuses on analyzing and understand the characteristics of various shapes, both 2D and 3D.

## Mathematics for Technology I (Math 1131)

### Number Types and Symbols

In mathematics, you will have trouble understanding the material unless you clearly understand the language of the words that are being used. Here are some commonly used words and phrases that you need to know, and that will be repeated throughout the duration of this course.

## Integers

The values:

… −4,   −3,   −2,   −1,   0,   1,   2,   3,   4 …

are called integers because integers are positive or negative whole numbers, including zero. The three dots on the ends indicate that the sequence of numbers continues indefinitely in both directions.

## Rational Numbers

The rational numbers include the integers and all other numbers that can be expressed as the quotient (the result of division) of two integers. In other words, a rational number is any number that can be expressed as a fraction. Some rational numbers are:

7 is a rational number because all whole numbers can be expressed as the number over 1:

$\frac{7}{1}$

## Irrational Numbers

These are numbers that cannot be expressed as a quotient. For example:

All of these equations can never be the quotient of a fraction because they’re never ending numbers. Evaluating the √2 gives you 1.414213562 …

## Symbols of Equality and Inequality

Several symbols are used to show the relative positions of two quantities on the number line.

x = y   means that x equals y, and that x and y occupy the same position on the number line.

x ≠ y   means that x and y are not equal and have different locations on the number line.

x > y   means that x is greater than y, and x lies to the right of y on the number line.

x < y   means that x is less than y, and x lies to the left of y on the number line.

x ≈ y   means that x is approximately equal to y, and that x and y are near each other on the number line. Other symbols sometimes used for approximately equal to are ≅ and ≃.

## Reciprocal

The reciprocal is a function that takes any number and flips it (reciprocates). For example, the reciprocal of 5 is 1/5. Generally, this is summarized as:

More examples:

## Exponents

Given the mathematical expression:

52

This statement is the same as taking the base (positive 5) and multiply it in itself 2 times (exponent of 2). This can be read as 5 raised to the power of 2 or 5-squared.

More examples:

Notice the significance of the brackets in #2: by not enclosing the base in brackets, the sign of the answer is changes from being positive in #1 to negative in #2.

## Absolute Value

The absolute value of a number is its magnitude regardless of its sign. It is written |n|, where n is any number. It is the distance between n and zero on the number line, without regard to direction. For example:

Note: you must treat the absolute a symbol of grouping, much like parentheses.


text:
# How do you factor 4x^3 + 32?

Aug 25, 2016

$4 \left(x + 2\right) \left({x}^{2} - 2 x + 4\right)$

#### Explanation:

First step is to take out a $\textcolor{b l u e}{\text{common factor}}$ of 4.

$\Rightarrow 4 {x}^{3} + 32 = 4 \left({x}^{3} + 8\right) \ldots \ldots . . \left(A\right)$

Now, ${x}^{3} + 8 \textcolor{b l u e}{\text{ is a sum of cubes}}$ and in general is factorised as follows.

$\textcolor{red}{| \overline{\underline{\textcolor{w h i t e}{\frac{a}{a}} \textcolor{b l a c k}{{a}^{3} + {b}^{3} = \left(a + b\right) \left({a}^{2} - a b + {b}^{2}\right)} \textcolor{w h i t e}{\frac{a}{a}} |}}} \ldots \ldots . . \left(B\right)$

now, ${\left(x\right)}^{3} = {x}^{3} \text{ and " (2)^3=8rArra=x" and } b = 2$

substitute these values for a and b into (B)

$\Rightarrow {x}^{3} + 8 = \left(x + 2\right) \left({x}^{2} - 2 x + 4\right)$

substitute back into (A)

$\Rightarrow 4 {x}^{3} + 32 = 4 \left(x + 2\right) \left({x}^{2} - 2 x + 4\right)$


text:
• Drell–Yan process at Large Hadron Collider

• # Fulltext

https://www.ias.ac.in/article/fulltext/pram/076/03/0421-0430

• # Keywords

Drell–Yan; Linear Hadron Collider; compact moun solenoid.

• # Abstract

Drell–Yan process at LHC, $q\bar{q} \to Z/ \gamma^{\ast} \to \ell^+ \ell^-$, is one of the benchmarks for conﬁrmation of Standard Model at TeV energy scale. Since the theoretical prediction for the rate is precise and the ﬁnal state is clean as well as relatively easy to measure, the process can be studied at the LHC even at relatively low luminosity. Importantly, the Drell–Yan process is an irreducible background to several searches of beyond Standard Model physics and hence the rates at LHC energies need to be measured accurately. In the present study, the methods for measurement of the Drell–Yan mass spectrum and the estimation of the cross-section have been developed for LHC operation at the centre-of-mass energy of 10 TeV and an integrated luminosity of 100 pb-1 in the context of CMS experiment

• # Author Affiliations

1. Panjab University, Chandigarh 160 014, India
2. University of Florida, Gainesville, Florida, USA
3. Tata Institute of Fundamental Research, Homi Bhabha Road, Mumbai 400 005, India

• # Pramana – Journal of Physics

Volume 95, 2021
All articles
Continuous Article Publishing mode

• # Editorial Note on Continuous Article Publication

Posted on July 25, 2019


text:
Matrices can be added to scalars, vectors and other matrices. − x What does it mean to add two matrices together? − Then. Matrix Multiplication Calculator. However, matrix multiplication is not, in general, commutative (although it is commutative if and are diagonal and of the same dimension). Therefore, in order for matrix multiplication Matrix multiplication shares some properties with usual multiplication. ψ Soft. {\displaystyle \Leftrightarrow } For relations, a symmetric relation is analogous to a commutative operation, in that if a relation R is symmetric, then ... both matrices are 2×2 rotation matrices. More: Commutativity isn't just a property of an operation alone. 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" is a metalogical symbol representing "can be replaced in a proof with.". Records of the implicit use of the commutative property go back to ancient times. {\displaystyle -i\hbar } , If and are matrices Join the initiative for modernizing math education. Can you explain this answer? ) but That's it! 4 and If and are both matrices, then usually, . The following are truth-functional tautologies. ) ) 2 3 4 = 12 and 4 3 = 12). ℏ d The term "commutative" is used in several related senses. 0.0 0 … B= | 1 0 | | 1 0 | AB is not equal to BA therefor matrix multiplication is not commutative. Matrix multiplication Matrix multiplication is an operation between two matrices that creates a new matrix such that given two matrices A and B, each column of the product AB is formed by multiplying A by each column of B (Deﬁnition 1). 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When a commutative operator is written as a binary function then the resulting function is symmetric across the line y = x. For example, T for the matrix that makes it taller and L for the matrix that leans the N. Some students will have the question, “Do we lean the taller N or the orig-inal N?”Make sure this discussion point comes out. = 0 Since matrices form an Abelian ∂ under multiplication. ... one matrix is the Zero matrix. Either way, the result (having both socks on), is the same. 2X + 3X = 5X AX + BX = (A+B)X XA + XB = X(A+B) AX + 5X = (A+5I)X AX+XB does not factor Each of these operations has a precise definition. (also called products of operators) on a one-dimensional wave function w-R 6 There is no defined process for matrix division. b d Thus, this property was not named until the 19th century, when mathematics started to become formalized. It is a fundamental property of many binary operations, and many mathematical proofs depend on it. In general, matrix multiplication is not commutative (i.e., AB = BA). Putting on left and right socks is commutative. Explore thousands of free applications across science, mathematics, engineering, technology, business, art, finance, social sciences, and more. 4 f i Some forms of symmetry can be directly linked to commutativity. (video) | Khan Academy Due to associativity, matrices form a semigroup For square matrices, if it is not commutative for any pair of matrices, it is not commutative in general. b The word is a combination of the French word commuter meaning "to substitute or switch" and the suffix -ative meaning "tending to" so the word literally means "tending to substitute or switch." ( Shuffling a deck of cards is non-commutative. , d So to show that matrix multiplication is NOT commutative we simply need to give one example where this is not the … q-O 4 A 2X2 matrix cannot be added to a 2X1 matrix. a Since matrices form an Abelian group under addition, matrices form a ring. x The commutativity of addition is observed when paying for an item with cash. Commutative Operation. 2 ≠ The term then appeared in English in 1838[2] in Duncan Farquharson Gregory's article entitled "On the real nature of symbolical algebra" published in 1840 in the Transactions of the Royal Society of Edinburgh.[12]. This page was last edited on 4 December 2020, at 15:19. − Arfken, G. Mathematical Methods for Physicists, 3rd ed. These two operators do not commute as may be seen by considering the effect of their compositions − ⇔ You already know subtraction and division, which are neither associative nor commutative. In contrast, the commutative property states that the order of the terms does not affect the final result. = The next one most people come across is matrix multiplication, which is associative, but not commutative. Since the snowball stays spherical, we kno… l-B 3 A matrix multiplied by its inverse is one. x ( with rows and columns. d x Weisstein, Eric W. "Matrix Multiplication." The act of dressing is either commutative or non-commutative, depending on the items. C = mtimes (A,B) is an alternative way to execute A*B, but is rarely used. Some truth functions are noncommutative, since the truth tables for the functions are different when one changes the order of the operands. Subtraction, division, and composition of functions are not. . 2 = 1985. In higher branches of mathematics, such as analysis and linear algebra the commutativity of well-known operations (such as addition and multiplication on real and complex numbers) is often used (or implicitly assumed) in proofs.[16][17][18]. The product of two block matrices is given by multiplying This also applies more generally for linear and affine transformations from a vector space to itself (see below for the Matrix representation). x Division is noncommutative, since For example, 5 + 6 = 6 + 5 but 5 – 6 ≠ 6 – 5. ... both matrices are Diagonal matrices. − . 1 Show Instructions. . For example, let , so again the operators do not commute and the physical meaning is that the position and linear momentum in a given direction are complementary. 178-179, , respectively (where Today the commutative property is a well-known and basic property used in most branches of mathematics. Putting on underwear and normal clothing is noncommutative. The implied summation over repeated indices without the presence of an explicit sum , It canhave the same result (such as when one matrix is the Identity Matrix) but not usually. {\displaystyle 0-1\neq 1-0} Dec 04,2020 - Matrix multiplication isa)Associative but not commutativeb)Commutative but not associativec)Associative as well as commutatived)None of theseCorrect answer is option 'D'. Suppose (unrealistically) that it stays spherical as it melts at a constant rate of . The name is needed because there are operations, such as division and subtraction, that do not have it (for example, "3 − 5 ≠ 5 − 3"); such operations are not commutative, and so are referred to as noncommutative operations. g-A 2 Matrix multiplication is commutative. In particular, matrix multiplication is not "commutative"; you cannot switch the order of the factors and expect to end up with the same result. Now, since , , and are scalars, use The rules allow one to transpose propositional variables within logical expressions in logical proofs. ( ... one matrix is the Identity matrix. (basically case #2) 4. is the reduced Planck constant). False.. Matrix multiplication is not a commutative operation. Explore anything with the first computational knowledge engine. matrix multiplication for square matrices is not a commutative operation, but still satisfies the associative and distributive properties, Common Core High School: Number & Quantity, HSN-VM.C.9
2020 matrix multiplication is commutative