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# American Institute of Mathematical Sciences

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## Multiple spatiotemporal coexistence states and Turing-Hopf bifurcation in a Lotka-Volterra competition system with nonlocal delays

 School of Mathematics, Harbin Institute of Technology, Harbin 150001, China

* Corresponding author: Weihua Jiang

Received  October 2020 Revised  November 2020 Published  December 2020

Fund Project: The authors are supported by the National Natural Science Foundation of China (No. 11871176)

We consider a two-species Lotka-Volterra competition system with both local and nonlocal intraspecific and interspecific competitions under the homogeneous Neumann condition. Firstly, we obtain conditions for the existence of Hopf, Turing, Turing-Hopf bifurcations and the necessary and sufficient condition that Turing instability occurs in the weak competition case, and find that the strength of nonlocal intraspecific competitions is the key factor for the stability of coexistence equilibrium. Secondly, we derive explicit formulas of normal forms up to order 3 by applying center manifold theory and normal form method, in which we show the difference compared with system without nonlocal terms in calculating coefficients of normal forms. Thirdly, the existence of complex spatiotemporal phenomena, such as the spatial homogeneous periodic orbit, a pair of stable spatial inhomogeneous steady states and a pair of stable spatial inhomogeneous periodic orbits, is rigorously proved by analyzing the amplitude equations. It is shown that suitably strong nonlocal intraspecific competitions and nonlocal delays can result in various coexistence states for the competition system in the weak competition case. Lastly, these complex spatiotemporal patterns are presented in the numerical results.

Citation: Xianyong Chen, Weihua Jiang. Multiple spatiotemporal coexistence states and Turing-Hopf bifurcation in a Lotka-Volterra competition system with nonlocal delays. Discrete & Continuous Dynamical Systems - B, doi: 10.3934/dcdsb.2021013
##### References:

show all references

##### References:
$w_{*}>0$ is a positive root of $F(w) = 0$
System parameter values are taken as Table 1 shows. The left figure represents the enlarged figure inside the pink rectangular part of the right figure and the red hollow circles are intersection points of $\mathcal{L}_{n}$ and $\mathcal{S}_{n}$. The solid black circles means that the remaining curves $\mathcal{L}_{n}$ and $\mathcal{S}_{n}$ are omitted here
(a), (b) are bifurcation sets and phase portraits respectively
When the initial values $u_{1}(t,x) = 0.45-0.01$, $u_{2}(t,x) = 0.27-0.01$, $t\in[-\tau,0]$ and parameters $(\tau,d_{1}) = (\tau_{w_{*}},d_{1}^{1}(d_{2}))+(-0.01,0.01)\in \mathscr{D}_{1}$, the coexistence equilibrium is stable
When the initial values $u_{1}(t,x) = 0.45-0.01$, $u_{2}(t,x) = 0.27-0.01$, $t\in[-\tau,0]$ and parameters $(\tau,d_{1}) = (\tau_{w_{*}},d_{1}^{1}(d_{2}))+(0.01,0.01)\in \mathscr{D}_{2}$, a spatial homogeneous periodic orbit is stable
For $(\tau,d_{1}) = (\tau_{w_{*}},d_{1}^{1}(d_{2}))+(0.003,-0.01)\in \mathscr{D}_{4}$, (a), (b) with initial values $u_{1}(t,x) = 0.45-0.01\cos x$, $u_{2}(t,x) = 0.27-0.01\cos x$ and (c), (d) with initial values $u_{1}(t,x) = 0.45+0.01\cos x$, $u_{2}(t,x) = 0.27+0.01\cos x$, , $t\in[-\tau,0]$. A pair of spatial inhomogeneous periodic orbits is stable, which indicates $\mathscr{D}_{4}$ is a bistable region
For $(\tau,d_{1}) = (\tau_{w_{*}},d_{1}^{1}(d_{2}))+(-0.01,-0.01)\in \mathscr{D}_{6}$, (a), (b) with initial values $u_{1}(t,x) = 0.45-0.01\cos x$, $u_{2}(t,x) = 0.27-0.01\cos x$ and (c), (d) with initial values $u_{1}(t,x) = 0.45+0.01\cos x$, $u_{2}(t,x) = 0.27+0.01\cos x$, , $t\in[-\tau,0]$. a pair of spatial inhomogeneous steady states is stable, which indicates $\mathscr{D}_{6}$ is a bistable region
Values of system parameters
 Parameters $r_{1}$ $r_{2}$ $a_{11}$ $a_{12}$ $a_{21}$ $a_{22}$ $b_{11}$ $b_{12}$ $b_{21}$ $b_{22}$ Values $5$ $5$ $4$ $7$ $6$ $5$ $3$ $1$ $0.5$ $4$
 Parameters $r_{1}$ $r_{2}$ $a_{11}$ $a_{12}$ $a_{21}$ $a_{22}$ $b_{11}$ $b_{12}$ $b_{21}$ $b_{22}$ Values $5$ $5$ $4$ $7$ $6$ $5$ $3$ $1$ $0.5$ $4$
Values of system parameters
 Parameters $d_{2}$ $r_{1}$ $r_{2}$ $a_{11}$ $a_{12}$ $a_{21}$ $a_{22}$ $b_{11}$ $b_{12}$ $b_{21}$ $b_{22}$ Values $0.6$ $5$ $5$ $4$ $7$ $6$ $5$ $3$ $1$ $0.5$ $4$
 Parameters $d_{2}$ $r_{1}$ $r_{2}$ $a_{11}$ $a_{12}$ $a_{21}$ $a_{22}$ $b_{11}$ $b_{12}$ $b_{21}$ $b_{22}$ Values $0.6$ $5$ $5$ $4$ $7$ $6$ $5$ $3$ $1$ $0.5$ $4$
 [1] Guo-Bao Zhang, Ruyun Ma, Xue-Shi Li. Traveling waves of a Lotka-Volterra strong competition system with nonlocal dispersal. Discrete & Continuous Dynamical Systems - B, 2018, 23 (2) : 587-608. doi: 10.3934/dcdsb.2018035 [2] Dan Wei, Shangjiang Guo. Qualitative analysis of a Lotka-Volterra competition-diffusion-advection system. Discrete & Continuous Dynamical Systems - B, 2021, 26 (5) : 2599-2623. doi: 10.3934/dcdsb.2020197 [3] Qigang Yuan, Jingli Ren. Periodic forcing on degenerate hopf bifurcation. Discrete & Continuous Dynamical Systems - B, 2021, 26 (5) : 2857-2877. doi: 10.3934/dcdsb.2020208 [4] Chin-Chin Wu. Existence of traveling wavefront for discrete bistable competition model. Discrete & Continuous Dynamical Systems - B, 2011, 16 (3) : 973-984. doi: 10.3934/dcdsb.2011.16.973 [5] Yuncherl Choi, Taeyoung Ha, Jongmin Han, Sewoong Kim, Doo Seok Lee. Turing instability and dynamic phase transition for the Brusselator model with multiple critical eigenvalues. Discrete & Continuous Dynamical Systems - A, 2021  doi: 10.3934/dcds.2021035 [6] Rui Hu, Yuan Yuan. Stability, bifurcation analysis in a neural network model with delay and diffusion. Conference Publications, 2009, 2009 (Special) : 367-376. doi: 10.3934/proc.2009.2009.367 [7] Zhigang Pan, Chanh Kieu, Quan Wang. Hopf bifurcations and transitions of two-dimensional Quasi-Geostrophic flows. Communications on Pure & Applied Analysis, , () : -. doi: 10.3934/cpaa.2021025 [8] Shanshan Chen, Junping Shi, Guohong Zhang. Spatial pattern formation in activator-inhibitor models with nonlocal dispersal. Discrete & Continuous Dynamical Systems - B, 2021, 26 (4) : 1843-1866. doi: 10.3934/dcdsb.2020042 [9] Jianping Gao, Shangjiang Guo, Wenxian Shen. Persistence and time periodic positive solutions of doubly nonlocal Fisher-KPP equations in time periodic and space heterogeneous media. Discrete & Continuous Dynamical Systems - B, 2021, 26 (5) : 2645-2676. doi: 10.3934/dcdsb.2020199 [10] M. Grasselli, V. Pata. Asymptotic behavior of a parabolic-hyperbolic system. Communications on Pure & Applied Analysis, 2004, 3 (4) : 849-881. doi: 10.3934/cpaa.2004.3.849 [11] Elena Bonetti, Pierluigi Colli, Gianni Gilardi. Singular limit of an integrodifferential system related to the entropy balance. Discrete & Continuous Dynamical Systems - B, 2014, 19 (7) : 1935-1953. doi: 10.3934/dcdsb.2014.19.1935 [12] Dmitry Treschev. A locally integrable multi-dimensional billiard system. Discrete & Continuous Dynamical Systems - A, 2017, 37 (10) : 5271-5284. doi: 10.3934/dcds.2017228 [13] Nizami A. Gasilov. Solving a system of linear differential equations with interval coefficients. Discrete & Continuous Dynamical Systems - B, 2021, 26 (5) : 2739-2747. doi: 10.3934/dcdsb.2020203 [14] Eduardo Casas, Christian Clason, Arnd Rösch. Preface special issue on system modeling and optimization. Mathematical Control & Related Fields, 2021  doi: 10.3934/mcrf.2021008 [15] Dugan Nina, Ademir Fernando Pazoto, Lionel Rosier. Controllability of a 1-D tank containing a fluid modeled by a Boussinesq system. Evolution Equations & Control Theory, 2013, 2 (2) : 379-402. doi: 10.3934/eect.2013.2.379 [16] Yanqin Fang, Jihui Zhang. Multiplicity of solutions for the nonlinear Schrödinger-Maxwell system. Communications on Pure & Applied Analysis, 2011, 10 (4) : 1267-1279. doi: 10.3934/cpaa.2011.10.1267 [17] Xu Zhang, Xiang Li. Modeling and identification of dynamical system with Genetic Regulation in batch fermentation of glycerol. Numerical Algebra, Control & Optimization, 2015, 5 (4) : 393-403. doi: 10.3934/naco.2015.5.393 [18] Manoel J. Dos Santos, Baowei Feng, Dilberto S. Almeida Júnior, Mauro L. Santos. Global and exponential attractors for a nonlinear porous elastic system with delay term. Discrete & Continuous Dynamical Systems - B, 2021, 26 (5) : 2805-2828. doi: 10.3934/dcdsb.2020206 [19] Zaihong Wang, Jin Li, Tiantian Ma. An erratum note on the paper: Positive periodic solution for Brillouin electron beam focusing system. Discrete & Continuous Dynamical Systems - B, 2013, 18 (7) : 1995-1997. doi: 10.3934/dcdsb.2013.18.1995 [20] Misha Bialy, Andrey E. Mironov. Rich quasi-linear system for integrable geodesic flows on 2-torus. Discrete & Continuous Dynamical Systems - A, 2011, 29 (1) : 81-90. doi: 10.3934/dcds.2011.29.81

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# Homework Help: Prove a limit using epsilon-delta definition

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1. Sep 8, 2016

### Math_QED

1. The problem statement, all variables and given/known data

Proof that:

$\lim_{x \to 1} \frac{1}{1+x} = \frac{1}{2}$

using the epsilon-delta definition of a limit.

(Problem below)

2. Relevant equations

$\lim_{x \to 1} \frac{1}{1+x} = \frac{1}{2} \iff \forall ε>0, \existsδ>0, \forall x ∈ \mathbb{R}\backslash\{-1\}: 0 < |x-1| < δ \Rightarrow |\frac{1}{1+x} - \frac{1}{2}|<ε$

3. The attempt at a solution

Draft

So, I have to find δ(ε) such that:

$|x-1| < δ \Rightarrow |\frac{1}{1+x} - \frac{1}{2}|<ε$
and, after simplifying I got:
$|\frac{1}{1+x} - \frac{1}{2}|<ε \Rightarrow |x-1| < ε|2x+2|$

Now, I chose $δ = 1$, then $|x-1| < 1$, thus $2< 2x + 2 < 6$. So I decided to take $δ = min\{1,6ε\}$

Proof

We take ε arbitrary and we choose $δ = min\{1,6ε\}$

Case 1: δ = 1, then 6ε > 1 as $δ = min\{1,6ε\}$.
We then have, $|x-1|<δ \Rightarrow |x-1| < 1 \Rightarrow |x-1| < 6ε \Rightarrow \frac{|x-1|}{6} < ε$

Now, here comes the problem. I wanted to use that $6>|2x + 2|$ (this result was obtained in the draft). However, when I would do this, the total fraction will become larger (as the denominator becomes smaller), and it would be possible that the inequality does not hold anymore. I got a similar problem with case 2. I'm learning this on my own, so it is possible that I'm totally wrong. Thank you for helping me in advance. Also, posting in Latex for the first time. Any hints on this matter would also be appreciated.

Last edited: Sep 8, 2016
2. Sep 8, 2016

### Staff: Mentor

The 6 ε are too large, you'll need something smaller.
There is no need to include prefactors larger than one in general. It does not improve anything, it just makes the inequalities harder to satisfy.

3. Sep 8, 2016

### Math_QED

How can I find what delta is better to pick?

4. Sep 8, 2016

### PeroK

You started off well, but then you took a wrong turn. Try starting from:

$|\frac{1}{1+x} - \frac{1}{2}| = |\frac{1-x}{2(1+x)}|$

Also, in these proofs, I would leave out the epsilons and deltas until I'm ready. Bringing them in at the start is not always a good idea.

5. Sep 9, 2016

### epenguin

Yes I think you just calculated |x - 1| when correct calculation would have given you |1 - x| .

But I want to ask the professional mathematicians and teachers here, it seems to me this shows something.

Firstly this concept of limits is supposed to be a difficult one OK? And that is the reason in order to get it across, students are made to do tons and tons of exercises on it. But actually it looks to me that really their difficulty is usually not concepts of limits, but the ordinary algebraic manipulations in doing these calculations! As in this case. Whether then they 'get it' in a way that stays as a result of that drilling I don't know, do you?

And look at this – above you have an expression which is obviously ½ when x = 1 There is nothing peculiar about it. It is not looking like 0/0 or anything weird like that. Can you make students who are going to be at most ornery users of math like engineers plough through this? Which seems a bit too much of a hoo-ha to me.

Ah but, I expect you will say this is just to get them used to easy examples and then we will present them with some things that are weird and do look like 0/0. Except that then, as I said, you get the students going crazy on the supposedly easy examples! Maybe things like
(sin x)/x are a bit more fun. But that I can think of, even those who go on to be regular users of math are rarely going to meet these things. In fact that I can recall there is only one place a lot of people meet it usually, that is in the basic thing of calculus where you say (lim δx →0 of) [f(x + δx) - f(x)]/δx and top and bottom do look to be 0. But then for generations I think that one was explained in just minutes, part of one lesson, and people took it in their stride.

Maybe you could say that limits come a bit more into their own with sums of infinite series - okay it may not be strictly quite right to say they ever add up to something and you should say is their sum has that number as limit. But I mean to say even this is such a big deal?

When I was at school, mnya mnyah, I don't remember we had any of this except the above examples. At my one math year in university we did, and it wasn't any problem for me. And just because of the relief of a thing which wasn't a problem I never questioned it. But if people do have a big problem then perhaps we should?

6. Sep 9, 2016

### Staff: Mentor

|x - 1| and |1 - x| are equal for all values of x.
If you're proving the limit of a function whose graph is a straight line, the proof is pretty easy. As soon as there's a curve involved, things get more difficult.
The business with $\delta$ and $\epsilon$ brings rigor, rather than hand-waving, to the argument.

format: plain


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# Cloud Data Manager

#### Examples of using RECALL

The CLI can be used to recall data sets using the RECALL command. RECALL is allowed for cataloged data sets only, therefore the command accepts a specific data set name as input. For example, an SMS-managed data set is recalled with its original SMS attributes as input to the ACS routines. For a non-SMS managed data set, the recall is performed on the original volume that accommodated the data set.

##### Scenario A: Recalling a data set when there is not enough space on its original volume

A data set needs to be recalled, but the disk that originally accommodated the data set does not have enough free space. A manual recall operation is needed to recall the data set with a new volume parameter. The RECALL command will be as follows:

TSO M9CLI RECALL M9.APPL.SOURCE NEWV(M9AP01)
##### Scenario B: Ensuring valid execution of a job with archived data sets as input

A batch job requires several archived non-SMS data sets as input. Normally, automatic recall will be used during execution to transparently recall the data sets to their original volumes. But, since the data sets were archived a long time ago, their original volumes no longer exist. To prevent failures during the batch job execution, the system programmer uses the RECALL command in a preliminary job to recall the data sets to new volumes.

//M9CLISMF JOB ACCT#,TIME=NOLIMIT,REGION=0M
//*
//* RECALLING FILES PRIOR TO EXECUTION
//*
//M9CLI EXEC PGM=IKJEFT01
//


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# aliquote

## < a quantity that can be divided into another a whole number of time />

R for applied epidemiology and public health. #rstats


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# Exponents and Exponential Functions

## Objective

Add and subtract polynomial expressions using properties of operations.

## Common Core Standards

### Core Standards

?

• A.APR.A.1 — Understand that polynomials form a system analogous to the integers, namely, they are closed under the operations of addition, subtraction, and multiplication; add, subtract, and multiply polynomials.

?

• 7.EE.A.1

• 8.EE.C.7

## Criteria for Success

?

1. Understand a polynomial as a sum of terms in which each term includes only multiplication as an operator.
2. Identify the degree and leading coefficient for polynomials.
3. Find the sum or difference of polynomials using the properties of operations and write polynomials in standard form.

## Tips for Teachers

?

In Lessons 2 and 3, students briefly study polynomials in order to apply properties of operations and exponents to mathematical situations. They will study polynomials more in depth in Algebra 2.

## Anchor Problems

?

### Problem 1

A polynomial is given below.

${10-4x+x^5+3x^2-x-6x^2}$

a.   Write the polynomial using the fewest number of terms possible. Write the polynomial in standard form.

b.   Identify the degree and leading coefficient of the polynomial.

### Problem 2

a.   ${(4x^2+x+7)+(2x^2+3x+1)}$

b.   ${(3x^3-x^2+8)-(x^3+5x^2+4x-7)}$

c.   ${3(x^3+8x)-2(x^3+12)}$

d.   ${(3+1)+6(p-8)-(p+2)}$

#### References

EngageNY Mathematics Algebra I > Module 1 > Topic B > Lesson 8Exercise 4

Algebra I > Module 1 > Topic B > Lesson 8 of the New York State Common Core Mathematics Curriculum from EngageNY and Great Minds. © 2015 Great Minds. Licensed by EngageNY of the New York State Education Department under the CC BY-NC-SA 3.0 US license. Accessed Dec. 2, 2016, 5:15 p.m..

Modified by Fishtank Learning, Inc.

## Problem Set

?

The following resources include problems and activities aligned to the objective of the lesson that can be used to create your own problem set.

${x^2-4(x-8x^2)+(5+10x)}$
Write a polynomial addition problem that has a sum of ${m^8-m^6+m^4}$.


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Ethics

There are some things you want to do, and some things you have to do.
But is there anything you truly "ought" to do?  What if anything do we
owe to other people (or animals, or ourselves)?  You've probably heard
it said that we should treat other people like we want to be treated.
But why?  What if they don't want to be treated that way?  Are there
any genuine moral or ethical facts?  Are they just relative to
culture?  Do the ends always justify the means, or are there some
things we shouldn't do no matter what good may come of them?  What
does it mean to be a good person?  This discussion-based class will be
a comprehensive survey of normative ethics, meta-ethics, and applied
ethics.  Special topics include: killing, liberty, punishment,
equality, and justice.

1 hr, Fridays, 10:00am - 11:00am


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# When is a surjective homomorphism with a right-inverse an isomorphism?

Let $G$ be an abelian group (or more generally a module over a ring $R$) and $\phi:G\to G$ a surjective group homomorphism with a right-inverse $\psi: G\to G$, i.e. $\phi\circ \psi = id_G$. So, we have a $split$ short exact sequence $$0\to K:=\ker(\phi)\to G\overset{\phi}{\to} G\to 0$$ and hence $G\cong K\oplus G$.

Question: Under which conditions on $G$ can we conclude that $\phi$ is an isomorphism (i.e. that $K=0$)?

I suspect that there is a notion/name for such groups $G$ that I'm ignorant of. So, references are very welcome as well. Thanks in advance for any contribution.

• If $G$ is a finitely generated abelian group or module over a noetherian ring, every surjective endomorphism of $G$ is an isomorphism (the verification is easy). If the ring is not noetherian, the same is true, but the proof (though short) is not so obvious (it is a result by Vasconcelos, I think). On the other side, there are surjective endomorphisms of a free not finitely generated abelian group (and so split) that are not isomorphisms: take $F\oplus F\to F$ surjective not injective and sum to each side an infinite number of copies of $F$. – A.G Dec 16 '16 at 17:34
• math.stackexchange.com/questions/239364/… – A.G Dec 16 '16 at 17:41
• Have you heard of hopfian groups? Anyway if your abelian group $G$ is finitely generated (as a group) then your $K = 0$. – M.U. Dec 16 '16 at 20:06
• @A.G. Thanks for your comment. The groups I'm interested in (in particular $\mathbb{R}/\mathbb{Z}$) are typically not finitely generated. – Dave Dec 19 '16 at 8:46
• @M.U. Thanks for the reference to Hopfian Groups, I'll definitely look into that. – Dave Dec 19 '16 at 8:46


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# Posts for Tag: graph

### Speed and Kinetic Energy of Sports Pitches, Shots and Kicks

Posted In: Science | Sports

I’ve been playing (and watching) alot of soccer recently with the kids and it got me thinking about how hard the pros can kick the ball compared to us. This got me thinking about how much energy athletes can impart to a soccer ball and how that compares to balls and projectiles in other sports. This is not a scientific study, as I just googled the fastest pitch, shot, serve, kick, throw etc. from a variety of sports and the weight of the respective balls/projectiles to calculate their kinetic energy and momentum. I added in the stats for a (sort of) human projectile for comparison as well (Usain Bolt).

The graph is color coded so orange refers to projectiles that require no additional equipment, while the blue requires a bat or racket or club to aid in hitting the ball. You can toggle between log and linear scale on the x-axis to better see the differences between different projectiles.

The hammer throw is interesting because it far exceeds the kinetic energy and momentum of the other balls. If you watch a video of olympic hammer throws, you’ll see how much energy these very large, strong athletes are able to put into the throw. I think another aspect is that the top kinetic energy projectiles are all throws where there is significant acceleration of the projectile over a longer period of time rather than an instantaneous kick or hit.

Switching to the speed tab, all of the fastest projectiles are aided by equipment to achieve their very high speeds, but generally these projectiles have lower weights. This is also seen in the momentum tab, where the heavier projectiles are mostly unaided by equipment, probably because of the challenge of imparting enough momentum onto a heavy ball/projectile would require accelerating an even heavier racket/bat.

### Equations and stuff

The equation for kinetic energy is $E = {1\over2} mv^2$,
where E is kinetic energy (expressed in joules or kilojoules), m is mass and v is velocity (or speed).

The equation for momentum is $P = mv$, where P is momentum.

The difference between momentum and kinetic energy is slightly tricky. The momentum rankings seem to prioritize the mass of the projectile while kinetic energy is a balance between speed (velocity) and mass. In kicking, throwing or hitting a ball/projectile, the player needs to put impart the energy into the ball. In a collision, total momentum of the system (player and ball) is conserved but kinetic energy is not, although total energy is (some energy may be “lost” as heat, sound, etc). In terms of being “hit” by the projectile, I believe that kinetic energy is probably more important than momentum for gauging the overall effect of the impact, but the total energy is not the only concern.The area over which the impact would occur is also important. Honestly, the table tennis (ping-pong) ball is the only one I think I’d be okay getting hit by (at least at these world record speeds).

Data sources and tools:
Mostly google for ball weights and trying to find some mention of the “fastest” throw or kick or whatever. Calculations are made using the equations above and plotted using Plot.ly javascript library.

Posted In: College

### Real Country Sizes Shown on Mercator Projection

Posted In: Maps

I remember as a child thinking that Alaska was as large as 1/2 of the continental US. Later, however, I learned that while it is the largest state, it is actually only about 1/5 the size of the lower 48 states. My son has also remarked that Greenland is very big. And while it is very big, it’s nowhere near the size of the continent of Africa.

The map above shows the distortion in sizes of countries due to the mercator projection. Pressing on the button animates the country ‘shrinking’ to its actual size or ‘growing’ to the size shown on the mercator projection. It was inspired by a similar animation that I saw on reddit and decided I wanted to try to build the same thing.

The mercator projection is a commonly used projection on computer maps because it has perpendicular latitude and longitude lines (forming rectangles). It is formed by projecting the glob onto a cylinder A variant of the was adopted by Google maps, which helped establish it as the informal standard for web-based maps (although Google maps now uses a globe view, instead of a map projection when zooming out to a very wide view).

Areas far from the equator are distorted in terms of their distances and are shown much larger than they actually are. This is one of the major issues with a projection of a globe onto a cylinder area. This is why Greenland, Russia and Canada shrink so much in the animation, they are fairly high in latitude in the Northern Hemisphere.

This next graph shows each country plotted with their actual land area and apparent land area as shown on a Mercator projection. The further the countries are from the 1:1 line the greater the overestimate of their size from the Mercator (also color coded to be red). It is a logarithmic plot showing many different orders of magnitude in country size. The table also shows the top 10 countries whose size is overestimated (and the difference in land area in square kilometers or as a percentage reduction from the size in the Mercator projection).

As it shows, Greenland is the country that has the largest percent difference between its apparent size in a Mercator projection and it’s real size (it’s only about 1/4 of the apparent size). And Russia is the country with the largest absolute difference between these two sizes.

Data and tools: This visualization was made using the Leafletjs javascript mapping library and country shapefiles (converted to geojson). I calculated the area in two ways, one assuming latitude and longitude are rectangular coordinates (i.e. Mercator projection) and the other was the actual area. Then I calculated the latitude and longitude coordinates for the outline of the “real” size by modifying the original latitude and longitude by the ratio of these two areas to draw the new smaller, “real” country size.

### Bitcoin Energy Consumption – Does It Consume More Electricity Than Your State?

Posted In: Energy | Money

This visualization looks at the staggeringly high energy use of Bitcoin and puts it into context: comparing it to electricity usage of US states. Unfortunately for Bitcoin, high energy usage is an intended feature of the system, rather than an unintended consequence. This is because mining is an increasingly energy intensive process, based upon increasingly computationally intensive calculations that are performed on high powered computers and graphical processing units.

Currently, 28 out of 50 states plus the District of Columbia all have lower electricity consumption than estimated annual bitcoin electricity consumption (~73 TWh per year). These states are highlighted in variations of yellow. This is approximately equal to the average annual electricity usage across all US States. States with higher electricity consumption than bitcoin are highlighted in shades of red.

When dividing the total energy use (73 TWh) by the current number of transactions (93 million), we get an average energy consumption of 783 kWh per transaction. Click on the “Energy per Transaction” button to see this visualization. What’s crazy is that a transaction is simply a transfer of bitcoin between “wallets”, recording the transaction, and a validation of the process. There’s no good reason why verifying digital transactions should take this much energy, except that it was built into the fundamental process of validating and mining bitcoin. 783 kWh is larger than the monthly per capita electricity consumption in 10 US states. It could also drive you and your family over 2000 miles in an electric car (e.g. Tesla Model S).

I’m not expert enough in this area to know how much more energy consumption will rise into the future, but if crypto advocates’ predictions come true and bitcoin is used extensively, millions of transactions will occur per hour instead of per year and the price of bitcoin may rise much higher than it currently is. If the price rises, then miners will be willing to expend more energy to “mine” the more valuable bitcoin. Needless to say, this sounds like a very bad idea from an energy consumption and sustainability standpoint.

Data and Tools:
State energy data comes from the US Department of Energy. Estimates of Bitcoin energy use come from Digiconomist’s Bitcoin Energy Consumption Index. The choropleth map is visualized using javascript and the Leaflet.js library with Open Street Map tiles.

### Solar (Sun) Intensity By Location and Time

Posted In: Energy | Science

This visualization shows the amount of solar intensity (also called solar insolation and measured in watts per square meter) all across the globe as a function of time of day and day of year. This is an idealized calculation as it does not take into account reductions in solar intensity due to cloud cover or other things that might block the sun from reaching the earth (e.g dust and pollution).

As would be expected, the highest amount of solar intensity occurs on the globe right where the sun is overhead and as the angle of the sun lowers, the solar intensity declines. This is why the area around the equator and up through the tropics is so sunny, the sun is overhead here the most. If you click on the map you should see a popup of the intensity of sunlight at that location.

As the earth rotates over the course of a day, the angle of the sun changes and eventually the angle is so low, the sun is blocked by the horizon (this is sunset).

##### Instructions
• The default is to show the sunlight intensity for the current date and time but you can change it by moving the sliders for hour or day.
• You can also toggle between the orientation of the surface that you measure the sunlight on. The default shows the intensity of sunlight on a horizontal surface. The other option shows the intensity on a surface that is oriented to face the sun (i.e. perpendicular)

Again, the intensity will depend on the angle it makes with the sun and so it depends on your location on earth (i.e. latitude). Latitudes around the equator will receive more sunlight because their angle is closer to perpendicular.

Shifting through the days of the year, you can start to see the cause of the seasons as the amount of sunlight changes and more or less sunlight goes to each of the northern and southern hemispheres.

Calculations and Tools:
The calculations for solar intensity are based on equations from “Renewable and Efficient Electric Power Systems” by Gilbert Masters Chapter 7. Calculations were made using javascript and visualized using the Leaflet.js library with Open Street Map tiles.

This was a fun project for me to learn online mapping tools and programming.

### When Can I Retire? Early Retirement Calculator / FIRE Calculator

Posted In: Financial Independence | Money

#### How long do I need to save before I can retire?

This early retirement calculator / visualizer is designed to project the number of years until you can retire, based upon a few key inputs such as annual income and spending, income growth rate, expected annual spending in retirement and asset allocation. It is a pre-retirement calculator that is useful before you retire to get a sense of how many years it is likely to take to accumulate enough money to retire. The three primary modes that are available in the early retirement calculator are: (1) constant, single fixed-percentage real return rates, (2) historical series of real returns are applied to account for likely variability in future returns and (3) monte carlo simulation of the variable returns based upon user-specified input parameters.

This interactive calculator was built to let you play with the inputs and help you understand how savings rate and retirement spending strongly determine how long it will take you to save up for retirement. Note: it does not simulate the post-retirement period when you start to draw down your savings. That can be done on this post-retirement calculator (Rich, Broke or Dead) which compares the frequency of various outcomes in retirement (running out of money, ending up with way too much money, and life-expectancy).
(more…)


text:
# Two arrays lookup

The task is to return array with values which satisfy the condition: for each max the less or equal numbers count.

Example Input: numbers = [1, 2, 1, 4] maxes = [2, 4]. Output: [3, 4]

Explanation => 1, 2 and 1 are 3 numbers less than or equal to 2, and all 4 numbers satisfy the condition for 4.

Here is my naive approach code in C#, I have a feeling there can be an optimization, and would be thankful to see one:

    static int[] counts(int[] nums, int[] maxes)
{
int queries = maxes.Length;
int numsLength = nums.Length;
int[] counts = new int[queries];

for (int i = 0; i < queries; i++)
{
for (int j = 0; j < numsLength; j++)
{
if (nums[j] <= maxes[i])
{
++counts[i];
}
}
}

return counts;
}

• The code gives no idea what it is supposed to accomplish. You didn't state what you are trying to achieve, other than meeting the requirements stated. Jun 4, 2018 at 6:10
• Entire method can be replaced with return maxes.Select(max => nums.Count(n => n <= max)); Jun 4, 2018 at 16:03
• Are there any boundaries? Like whats the maximum number of items that nums and maxes can contain? Jul 5, 2018 at 19:36

In this case performance depends on the number of maxes. The longer the array the more time it takes to check it and the slower it gets.

As an alternative you can group all numbers first and get their counts. Then just sum the counts of the numbers that meet the criteria.

public static int[] counts2(this int[] nums, int[] maxes)
{
var counts =
nums
.GroupBy(
n => n,
(k, items) => (num: k, count: items.Count()))
.ToArray();

return
maxes
.Select(m =>
counts
.Where(x => x.num <= m)
.Sum(x => x.count))
.ToArray();
}

On my machine the results are:

Number of runs: 10000

Test-1:

nums:  1000
maxes: 10

OP vs t3chb0t

00:00:00.2005513 <  00:00:00.5321118

Test-2:

nums:  1000
maxes: 100

OP vs t3chb0t

00:00:02.3541534 >  00:00:01.7438213

Test-3:

nums:  1000
maxes: 1000

OP vs t3chb0t

00:00:22.9002408 >  00:00:12.8527922

Test-4: with AsParallel()

nums:  1000
maxes: 1000

OP vs t3chb0t

00:00:22.9081026 >  00:00:03.8906315

Chaining AsParallel() to maxes makes it run even faster.

As always with performance: test different solutions before you judge which version is the fastest one because it often depends on many other factors. Using plain loops unwisely does not have to be better the using clever linq.

I would rather see different names

int maxesLength = maxes.Length;

static int[] MaxesCounts

It is efficient enough just a little hard to follow. You could sort and do a binary search but not sure that would be any more efficient.

static int[] MaxesCounts(int[] nums, int[] maxes)
{
int[] counts = new int[maxes.Length];
foreach (int num in nums)
{
for (int m = 0; m < maxes.Length; m++)
{
if (num <= maxes[m])
{
counts[m]++;
}
}
}
return counts;
}

I'm not sure about performance but using a LINQ query will certainly simplify your algorithm:

static int[] counts(int[] nums, int[] maxes)
{
return (from int max in maxes
select nums.Count(x => x <= max)).ToArray();
}

• About performance, LINQ is pretty much always slower than the equivalent code using loops. It's also more memory intensive than iterating an array. Jun 12, 2018 at 14:53


text:
## January 15th, 2021 2:30-3:30 pm CET

Title : A density of ramified primes

Abstract : Let $K$ be a cyclic number field of odd degree over ℚ with odd narrow class number, such that $2$ is inert in $K$/ℚ. We extend the definition of spin (a special quadratic residue symbol) to all odd ideals in $K$, not necessarily principal. We discuss some of the ideas involved in obtaining an explicit formula, depending only on [$K$:ℚ], for the density of rational prime ideals satisfying a certain property of spins, conditional on a standard conjecture on short character sums. This talk is based on joint work with Christine McMeekin and Djordjo Milovic.


text:
# What is the role of an intercept in an I-Spline basis?

I have the following data to which I want to fit a monotone non-decreasing spline.

x <- c(300, 377, 455, 533, 611, 688, 766, 844, 922, 1000)
y <- c(0.5, 0.8, 1, 1, 1, 1, 1, 1, 1, 1)

To achieve this, I use an I-Spline basis (i.e., splines2::iSpline) paired with non-negative coefficients (i.e., nnls::nnls).

# Inner knots.
k <- quantile(x, probs = c(.25, .5, .75))

h <- splines2::iSpline(x, knots = k, Boundary.knots = range(x), intercept = TRUE)
a <- nnls::nnls(h, y)\$x

And this gives me the following:

# Fitted.
f <- h %*% a

print(f)
#             [,1]
#   [1,] 0.0000000
#   [2,] 0.8106201
#   [3,] 0.9699174
#   [4,] 0.9992354
#   [5,] 1.0035000
#   [6,] 1.0035515
#   [7,] 1.0035515
#   [8,] 1.0035515
#   [9,] 1.0035515
#  [10,] 1.0035515

However, with intercept = TRUE, I would expect the spline to start somewhere above the first data point. For instance, this seems to be the case when I set the first basis function to 1.

h[, 1] <- 1

If I set intercept = FALSE, then things go haywire.

Can you please help me understand what is going on, i.e., why is the spline not starting at (300, 0.5)?

### Further Details

I want to clarify my question further. My confusion stems from what is written in this paper on p. 15:

To get a basis for the increasing splines we need to add the constant function to the I-splines and allow it to enter the linear combination with either sign.

In the splines2 package documentation, where the intercept argument for iSpline() is described, it says:

Notice that when using I-Spline for monotonic regression, intercept = TRUE should be set even when an intercept term is considered additional to the spline bases in the model.

What I get from this is that if I want to fit a monotone spline with, say, nnls::nnls(), then I need to add a constant function to the I-Spline basis, i.e., h <- cbind(1, h), assuming that h is the basis matrix output of iSpline().

However, in the paper linked above, that's not always the case. Sometimes the author replaces the first column of the basis matrix (i.e., h <- cbind(1, h[ -1]) with the constant function, and, other times, he appends the constant function to the basis matrix.

For instance, if I create the I-Spline basis matrix as follows:

h <- splines2::iSpline(x, knots = c(3, 5, 8), degree = 0, intercept = TRUE)

Then, depending on how I "add" the constant function (or not) I get different splines:

What's not clear to me is how the basis matrix provided by iSpline() is intended to be used in this case.

From the definition of k in your code we know the first segment is fitting more than 2 data points.

Only if it is fitting first 2 data points and without the constraint that the function to be continuous, the spline will pass the first point. Note, if that is the case, the model is not a spline.

library(ggplot2)
x <- c(300, 377, 455, 533, 611, 688, 766, 844, 922, 1000)
y <- c(0.5, 0.8, 1, 1, 1, 1, 1, 1, 1, 1)
group <- c(0,0,1, 1, 1, 1, 1, 1, 1, 1)
d = data.frame(x=x,y=y, group=group)
ggplot(d,aes(x,y, group=group))+geom_point()+geom_smooth(method='lm')+theme_gray(20)

• Haitao Du, what you wrote makes sense. However, I am still confused because in this paper on p. 15 (last sentence), it says that a constant function needs to be added to the I-Spline basis matrix. However, in the paper, the first column of the basis matrix is sometimes replaced with ones (i.e., h <- cbind(1, h[ -1]) and other times simply another column is appended (i.e., h <- cbind(1, h)). Can you please shed some light on why this is the case? – Mihai Apr 20 at 8:59
• I will append this to the question above. – Mihai Apr 20 at 9:15


text:
## Self, non-self and segregation: a very basic look at agent-based lattice models

In our college days, a part time physics teacher from an old and respected V$_1$ clan used to chat with us about issues of mutual interest that were beyond that of the rest of the class (or for that matter the rest of the teachers) and well out of the scope of the syllabus. He was the only one among the physics staff with an interest in science for science’s sake. We always felt he had it in him to be a scientist and he was indeed was pursuing a doctoral program at his own pace on the side. However, he clarified to us that he was the big fish in the small pond and that every man’s ambition is like a rocket set off on a Dīpāvalī night — drawing out a parabola on the board he declared with his characteristic smirk: “It will come down; hence, why trouble yourself with a dizzying fall”. In course of one involved conversation spanning thermodynamics, dynamical systems and biological ensembles, he declared to us: “I agree with you that there are several problems where the actual entities are fungible. It doesn’t matter if we are dealing with atoms, cells or animals, they could as well just be numbers. You should explore the Ising models — maybe you will find something there to answer your questions.” We are not really into “proving a few theorems”; however, playing with things on paper or a computer has always excited us. Hence, the next time we could access a computer we began looking into those models and soon realized that it could be used to understand some basic aspects of biological systems.

Here we shall describe some experiments with such models that go no further than the most basic exploration of these systems. In physics, such models were first proposed by Lenz and his student Ising. In sociology, they were introduced by Schelling (of whose work we learnt much later) who carried out the experiments with a graph paper and coins. Today we can do them easily by writing some code on a little computer. The basic rules for the games we shall look at go thus:

1) The games are played on a lattice on the surface of a torus but for visualization we shall cut open the torus and render it as a square board. Thus it would look like this:

2) Each cell of the lattice can be occupied by an “agent” of one of two colors (as above) or be empty.
3) The system has strict conservation laws: the agents can neither be created nor destroyed and the number of agents of each color will be conserved.
4) The neighborhood of an agent is determined by the “span” $l$ which defines a square grid of a centered on it with $(2l+1)^2-1$ neighbors. $l=1$ means a $3 \times 3$ grid with the agent at the center and 8 lattice positions available for neighbors around it. Thus, the agent marked with a black dot (above) has 5 neighbors at $l=1$: 2 blue and 3 yellow. $l=2$ means a $5 \times 5$ grid with the agent in the center and 24 available lattice positions for neighbors in two concentric shells around it.
5) The agency of the agent manifests as its ability to read the number and color of its neighbors and either stay put where it is or move over to a random empty cell in the lattice.
6) Beyond this is there is occupancy, $o$, i.e. the fraction of the total available cells in the lattice that are occupied by agents.

All the experiments described below are played on a $50 \times 50$ lattice, i.e. there are 2500 cells available to the agents. In all experiments, the agents move to an available empty cell if the number of its neighbors of any color are $\le s$, the sociality factor. Thus, if $s=0$ then the agent will move from their current location if they have no neighbors at all. In the first set of experiments, they additionally sense the the absolute number of non-self neighbors, i.e. those of a different color and move if it is $\ge c_n$, the non-self count. The movements of the agents are repeated over and over until stability is reached or 30 successive rounds of movement have elapsed. The games are illustrated thus: the plot to the left is the initial configuration where the agents are randomly introduced into the lattice and the plot to the right is the final configuration that is reached as mentioned above. We measure segregation by looking at the $\tfrac{n}{s}$, i.e. the mean non-self: self ratio in the neighborhood of the agents. As the agents are randomly introduced, the game would start with $\tfrac{n}{s} \approx 1$. The degree of segregation can be statistically assessed at the beginning and end of the run by means of the t-test to see if the mean number of self and non-self agents in the neighborhood of any given agent are significantly different. At the start of the run the difference would be insignificant.

Game 1 is a run with low occupancy $o=0.4$ and an equal number of agents of the two colors (blue and orange); $l=1$, i.e. a neighborhood with 8 available cells; $s=0$; $c_n=5$, i.e. the agents tolerate up to half of the 8 available lattice points in the neighborhoods being occupied by non-self agents.

Game 1

We see that at the end of the run $\tfrac{n}{s}$ remains close to 1 and the mean number of self and non-self neighbors of an agent is not significantly different suggesting that tolerating non-self agents in up to half of the available neighborhoods does not result in segregation at low occupancy and low sociality.

Game 2 is run with the same parameters as above, except that we increase sociality $s=3$.

Game 2

Notably, the increased sociality results in highly significant segregation. It also results in greater clumping of the agents, resulting in clustered but clearly segregated domains.

Game 3 is run with the same parameters as Game 1 but we increase the occupancy $o=0.6$.

Game 3

Here, we see a small but significant reduction of $\tfrac{n}{s}$. Thus, increasing the population with same level of tolerance for non-self by itself results in some segregation that is not seen at low occupancy.

Game 4 is run with $l=2$; thus, the agent responds to the status of the 24 available lattice points of the neighborhood around it. The occupancy is low $o=.4$; $c_n= 11$ i.e. less than half the available neighborhood positions tolerated as non-self; $s=6$.

Game 4

In these runs we often seen no significant segregation of the agents despite the relatively low tolerance to non-self; however, we see greater clumping of the agents resulting in a more anisotropic distribution of the agents at the conclusion of the run. Thus, when larger neighborhoods are sensed by the agents, even relatively low tolerance for non-self is overridden under low occupancy leading to paradoxical clumping of self and non-self into spatially restricted domains.

Game 5 is similar to 4 but the sociality of the agents is increased to $s=8$.

Game 5

This change has the dramatic effect of moving the agents towards strong segregation along with formation closely packed monotypic domains of the two agents, with clear boundaries. Thus, the sociality parameter drives a phase transition from packing with little segregation despite relatively strong non-self tolerance to nearly complete separation into domains with shared borders.

Game 6 is different from the previous ones in that it senses the relative non-self fraction rather than the absolute count of non-self in the neighborhood, $f_n=\tfrac{c_n}{c_t}$, where $c_t$ is the total number of occupied lattice points in the neighborhood. The sociality factor $s$ is applied just as in the above cases. In this run, we set $s=0$, i.e. the agent moves only if it has no neighbors. $f_n=.7$, i.e., the agent move only if the fraction of its non-self neighbors is greater or equal to $f_n$. Thus, if an agent has 5 neighbors and 4 of them are non-self then $f_n=0.8$ and it would move. We set it to a low occupancy of $o=0.4$.

Game 6

We observe that even with a low occupancy, high tolerance for non-self and low sociality, relative sensing drives significant segregation. Thus, with relative sensing the $f_n$ is the primary determinant of segregation.

What might be some real-life scenarios where these games matter? We can easily imagine the system of agents being a set of living cells showing their agency in response to chemotactic signals. At a macroscopic level, we can imagine these as individual animals (e.g. humans). Indeed, Game 6 was the one played by Schelling as a model for the sociology of segregation. In terms of cells, we can conceive of the following mechanisms: 1) cells are primitively motile; hence, they can move. 2) they can exhibit sociality by means of adhesion molecules (usually proteins) that are expressed on the cell surface. Unless these adhesion interactions are satisfied to a given degree, they would keep moving. 3) They can sense self from non-self. In the simplest case this can again happen via adhesion molecules. Indeed, such mechanisms are used by fungi and ciliates, among others, for discriminating self from non-self for mating or hyphal anastomosis. This kind of adhesion-based self-non-self discrimination can be a mechanism of sensing the absolute number of non-self neighbors. Alternatively, it can happen through sensing of diffusible signals. This mechanism is best suited for relative sensing of the ratio of non-self to total neighbors. Either way the cellular agent could respond to nonself agents in the environment.

Thus, these simple models show how the interplay of sociality and non-self tolerance can result in segregation or paradoxical grouping. In the case of relative sensing (game 6), even with high tolerance and low sociality we see segregation. This is the reality of human societies that many modern occidentally oriented social observers find very hard to swallow. But they are simply tilting against a mathematical reality, much like their medieval representative, old Don Quixote, charging a windmill. Similarly, absolute-count-sensing shows the role of sociality and crowding in segregation (e.g. Games 2, 3 and 5). Greater crowding and sociality can lead to greater segregation when compared to the same, relatively high, tolerance for non-self under low crowding and sociality. Similarly, certain middling level of sociality with sensing of larger neighborhoods can trigger long-lasting clustering without strong segregation despite lower than half tolerance for absolute number non-self agents in the available neighborhood spots (Game 4). This scenario could explain the grouping of bacteria in mixed species biofilms or relatively long-lived clustering of distinct groups in a social setting, e.g. the jāti-s in an Indian village. In conclusion, some social phenomena can be accounted for by simple lattice models of agents which are entirely agnostic to the actual mechanism of agency and sensing.

This entry was posted in Life, Scientific ramblings and tagged , , , , , , , , , . Bookmark the permalink.


text:
# ProfessorFrancescaToni

Faculty of EngineeringDepartment of Computing

Professor in Computational Logic

//

### Contact

+44 (0)20 7594 8228f.toni

//

### Location

430Huxley BuildingSouth Kensington Campus

//

## Publications

Publication Type
Year
to

351 results found

Rago A, Cocarascu O, Bechlivanidis C, Toni Fet al., 2021, Argumentation as a framework for interactive explanations for recommendations, KR 2020, 17th International Conference on Principles of Knowledge Representation and Reasoning, Publisher: IJCAI, Pages: 805-815, ISSN: 2334-1033

As AI systems become ever more intertwined in our personallives, the way in which they explain themselves to and inter-act with humans is an increasingly critical research area. Theexplanation of recommendations is, thus a pivotal function-ality in a user’s experience of a recommender system (RS),providing the possibility of enhancing many of its desirablefeatures in addition to itseffectiveness(accuracy wrt users’preferences). For an RS that we prove empirically is effective,we show how argumentative abstractions underpinning rec-ommendations can provide the structural scaffolding for (dif-ferent types of) interactive explanations (IEs), i.e. explana-tions empowering interactions with users. We prove formallythat these IEs empower feedback mechanisms that guaranteethat recommendations will improve with time, hence render-ing the RSscrutable. Finally, we prove experimentally thatthe various forms of IE (tabular, textual and conversational)inducetrustin the recommendations and provide a high de-gree oftransparencyin the RS’s functionality.

Conference paper

Rago A, Cocarascu O, Bechlivanidis C, Lagnado D, Toni Fet al., 2021, Argumentative explanations for interactive recommendations, Artificial Intelligence, Vol: 296, Pages: 1-22, ISSN: 0004-3702

A significant challenge for recommender systems (RSs), and in fact for AI systems in general, is the systematic definition of explanations for outputs in such a way that both the explanations and the systems themselves are able to adapt to their human users' needs. In this paper we propose an RS hosting a vast repertoire of explanations, which are customisable to users in their content and format, and thus able to adapt to users' explanatory requirements, while being reasonably effective (proven empirically). Our RS is built on a graphical chassis, allowing the extraction of argumentation scaffolding, from which diverse and varied argumentative explanations for recommendations can be obtained. These recommendations are interactive because they can be questioned by users and they support adaptive feedback mechanisms designed to allow the RS to self-improve (proven theoretically). Finally, we undertake user studies in which we vary the characteristics of the argumentative explanations, showing users' general preferences for more information, but also that their tastes are diverse, thus highlighting the need for our adaptable RS.

Journal article

Cyras K, Oliveira T, Karamlou M, Toni Fet al., 2021, Assumption-based argumentation with preferences and goals for patient-centric reasoning with interacting clinical guidelines, Argument and Computation, Vol: 12, Pages: 149-189, ISSN: 1946-2166

A paramount, yet unresolved issue in personalised medicine is that of automated reasoning with clinical guidelines in multimorbidity settings. This entails enabling machines to use computerised generic clinical guideline recommendations and patient-specific information to yield patient-tailored recommendations where interactions arising due to multimorbidities are resolved. This problem is further complicated by patient management desiderata, in particular the need to account for patient-centric goals as well as preferences of various parties involved. We propose to solve this problem of automated reasoning with interacting guideline recommendations in the context of a given patient by means of computational argumentation. In particular, we advance a structured argumentation formalism ABA+G (short for Assumption-Based Argumentation with Preferences (ABA+) and Goals) for integrating and reasoning with information about recommendations, interactions, patient’s state, preferences and prioritised goals. ABA+G combines assumption-based reasoning with preferences and goal-driven selection among reasoning outcomes. Specifically, we assume defeasible applicability of guideline recommendations with the general goal of patient well-being, resolve interactions (conflicts and otherwise undesirable situations) among recommendations based on the state and preferences of the patient, and employ patient-centered goals to suggest interaction-resolving, goal-importance maximising and preference-adhering recommendations. We use a well-established Transition-based Medical Recommendation model for representing guideline recommendations and identifying interactions thereof, and map the components in question, together with the given patient’s state, prioritised goals, and preferences over actions, to ABA+G for automated reasoning. In this, we follow principles of patient management and establish corresponding theoretical properties as well as illustrate our approach in realis

Journal article

Santos M, Toni F, 2021, Artificial intelligence-empowered technology in the home, The Home in the Digital Age, Pages: 38-55, ISBN: 9780367530174

Book chapter

Cyras K, Rago A, Emanuele A, Baroni P, Toni Fet al., 2021, Argumentative XAI: A Survey, The 30th International Joint Conference on Artificial Intelligence (IJCAI-21)

Conference paper

Cyras K, Heinrich Q, Toni F, 2021, Computational complexity of flat and generic assumption-based argumentation, with and without probabilities, Artificial Intelligence, Vol: 293, Pages: 1-36, ISSN: 0004-3702

Reasoning with probabilistic information has recently attracted considerable attention in argumentation, and formalisms of Probabilistic Abstract Argumentation (PAA), Probabilistic Bipolar Argumentation (PBA) and Probabilistic Structured Argumentation (PSA) have been proposed. These foundational advances have been complemented with investigations on the complexity of some approaches to PAA and PBA, but not to PSA. We study the complexity of an existing form of PSA, namely Probabilistic Assumption-Based Argumentation (PABA), a powerful, implemented formalism which subsumes several forms of PAA and other forms of PSA. Specifically, we establish membership (general upper bounds) and completeness (instantiated lower bounds) of reasoning in PABA for the class FP#P (of functions with a #P-oracle for counting the solutions of an NP problem) with respect to newly introduced probabilistic verification, credulous and sceptical acceptance function problems under several ABA semantics. As a by-product necessary to establish PABA complexity results, we provide a comprehensive picture of the ABA complexity landscape (for both flat and generic, possibly non-flat ABA) for the classical decision problems of verification, existence, credulous and sceptical acceptance under those ABA semantics.

Journal article

Albini E, Lertvittayakumjorn P, Rago A, Toni Fet al., 2021, DAX: deep argumentative eXplanation for neural networks, Publisher: arXiv

Despite the rapid growth in attention on eXplainable AI (XAI) of late,explanations in the literature provide little insight into the actualfunctioning of Neural Networks (NNs), significantly limiting theirtransparency. We propose a methodology for explaining NNs, providingtransparency about their inner workings, by utilising computationalargumentation (a form of symbolic AI offering reasoning abstractions for avariety of settings where opinions matter) as the scaffolding underpinning DeepArgumentative eXplanations (DAXs). We define three DAX instantiations (forvarious neural architectures and tasks) and evaluate them empirically in termsof stability, computational cost, and importance of depth. We also conducthuman experiments with DAXs for text classification models, indicating thatthey are comprehensible to humans and align with their judgement, while alsobeing competitive, in terms of user acceptance, with existing approaches to XAIthat also have an argumentative spirit.

Working paper

Rago A, Albini E, Baroni P, Toni Fet al., 2021, Influence-driven explanations for bayesian network classifiers, Publisher: arXiv

One of the most pressing issues in AI in recent years has been the need toaddress the lack of explainability of many of its models. We focus onexplanations for discrete Bayesian network classifiers (BCs), targeting greatertransparency of their inner workings by including intermediate variables inexplanations, rather than just the input and output variables as is standardpractice. The proposed influence-driven explanations (IDXs) for BCs aresystematically generated using the causal relationships between variableswithin the BC, called influences, which are then categorised by logicalrequirements, called relation properties, according to their behaviour. Theserelation properties both provide guarantees beyond heuristic explanationmethods and allow the information underpinning an explanation to be tailored toa particular context's and user's requirements, e.g., IDXs may be dialecticalor counterfactual. We demonstrate IDXs' capability to explain various forms ofBCs, e.g., naive or multi-label, binary or categorical, and also integraterecent approaches to explanations for BCs from the literature. We evaluate IDXswith theoretical and empirical analyses, demonstrating their considerableadvantages when compared with existing explanation methods.

Working paper

Cocarascu O, McLean A, French P, Toni Fet al., 2021, An Explanatory Query-Based Framework for Exploring Academic Expertise

The success of research institutions heavily relies upon identifying theright researchers "for the job": researchers may need to identify appropriatecollaborators, often from across disciplines; students may need to identifysuitable supervisors for projects of their interest; administrators may need tomatch funding opportunities with relevant researchers, and so on. Usually,finding potential collaborators in institutions is a time-consuming manualsearch task prone to bias. In this paper, we propose a novel query-basedframework for searching, scoring, and exploring research expertiseautomatically, based upon processing abstracts of academic publications. Givenuser queries in natural language, our framework finds researchers with relevantexpertise, making use of domain-specific knowledge bases and word embeddings.It also generates explanations for its recommendations. We evaluate ourframework with an institutional repository of papers from a leading university,using, as baselines, artificial neural networks and transformer-based modelsfor a multilabel classification task to identify authors of publicationabstracts. We also assess the cross-domain effectiveness of our framework witha (separate) research funding repository for the same institution. We show thatour simple method is effective in identifying matches, while satisfyingdesirable properties and being efficient.

Journal article

Oksanen J, Cocarascu O, Toni F, 2021, Automatic Product Ontology Extraction from Textual Reviews

Ontologies have proven beneficial in different settings that make use oftextual reviews. However, manually constructing ontologies is a laborious andtime-consuming process in need of automation. We propose a novel methodologyfor automatically extracting ontologies, in the form of meronomies, fromproduct reviews, using a very limited amount of hand-annotated training data.We show that the ontologies generated by our method outperform hand-craftedontologies (WordNet) and ontologies extracted by existing methods (Text2Ontoand COMET) in several, diverse settings. Specifically, our generated ontologiesoutperform the others when evaluated by human annotators as well as on anexisting Q&A dataset from Amazon. Moreover, our method is better able togeneralise, in capturing knowledge about unseen products. Finally, we considera real-world setting, showing that our method is better able to determinerecommended products based on their reviews, in alternative to using Amazon'sstandard score aggregations.

Journal article

Lertvittayakumjorn P, Choshen L, Shnarch E, Toni Fet al., 2021, GrASP: A Library for Extracting and Exploring Human-Interpretable Textual Patterns

Data exploration is an important step of every data science and machinelearning project, including those involving textual data. We provide a Pythonlibrary for GrASP, an existing algorithm for drawing patterns from textualdata. The library is equipped with a web-based interface empowering human usersto conveniently explore the data and the extracted patterns. We alsodemonstrate the use of the library in two settings (spam detection and argumentmining) and discuss future deployments of the library, e.g., beyond textualdata exploration.

Journal article

Lauren S, Belardinelli F, Toni F, 2021, Aggregating Bipolar Opinions (With Appendix)

We introduce a novel method to aggregate Bipolar Argumentation (BA)Frameworks expressing opinions by different parties in debates. We use BipolarAssumption-based Argumentation (ABA) as an all-encompassing formalism for BAunder different semantics. By leveraging on recent results on judgementaggregation in Social Choice Theory, we prove several preservation results,both positive and negative, for relevant properties of Bipolar ABA.

Journal article

Lertvittayakumjorn P, Toni F, 2021, Explanation-Based Human Debugging of NLP Models: A Survey

To fix a bug in a program, we need to locate where the bug is, understand whyit causes the problem, and patch the code accordingly. This process becomesharder when the program is a trained machine learning model and even harder foropaque deep learning models. In this survey, we review papers that exploitexplanations to enable humans to debug NLP models. We call this problemexplanation-based human debugging (EBHD). In particular, we categorize anddiscuss existing works along three main dimensions of EBHD (the bug context,the workflow, and the experimental setting), compile findings on how EBHDcomponents affect human debuggers, and highlight open problems that could befuture research directions.

Journal article

Paulino-Passos G, Toni F, 2021, Monotonicity and Noise-Tolerance in Case-Based Reasoning with Abstract Argumentation (with Appendix)

Recently, abstract argumentation-based models of case-based reasoning($AA{\text -} CBR$ in short) have been proposed, originally inspired by thelegal domain, but also applicable as classifiers in different scenarios.However, the formal properties of $AA{\text -} CBR$ as a reasoning systemremain largely unexplored. In this paper, we focus on analysing thenon-monotonicity properties of a regular version of $AA{\text -} CBR$ (that wecall $AA{\text -} CBR_{\succeq}$). Specifically, we prove that $AA{\text -}CBR_{\succeq}$ is not cautiously monotonic, a property frequently considereddesirable in the literature. We then define a variation of $AA{\text -}CBR_{\succeq}$ which is cautiously monotonic. Further, we prove that suchvariation is equivalent to using $AA{\text -} CBR_{\succeq}$ with a restrictedcasebase consisting of all "surprising" and "sufficient" cases in the originalcasebase. As a by-product, we prove that this variation of $AA{\text -}CBR_{\succeq}$ is cumulative, rationally monotonic, and empowers a principledtreatment of noise in "incoherent" casebases. Finally, we illustrate $AA{\text-} CBR$ and cautious monotonicity questions on a case study on the U.S. TradeSecrets domain, a legal casebase.

Journal article

Dejl A, He P, Mangal P, Mohsin H, Surdu B, Voinea E, Albini E, Lertvittayakumjorn P, Rago A, Toni Fet al., 2021, Argflow: A Toolkit for Deep Argumentative Explanations for Neural Networks., Publisher: ACM, Pages: 1761-1763

Conference paper

Lauren S, Belardinelli F, Toni F, 2020, Aggregating Bipolar Opinions, 20th International Conference on Autonomous Agents and Multiagent Systems

Conference paper

Kotonya N, Toni F, 2020, Explainable Automated Fact-Checking: A Survey, Barcelona. Spain, 28th International Conference on Computational Linguistics (COLING 2020), Publisher: International Committee on Computational Linguistics, Pages: 5430-5443

A number of exciting advances have been made in automated fact-checkingthanks to increasingly larger datasets and more powerful systems, leading toimprovements in the complexity of claims which can be accurately fact-checked.However, despite these advances, there are still desirable functionalitiesmissing from the fact-checking pipeline. In this survey, we focus on theexplanation functionality -- that is fact-checking systems providing reasonsfor their predictions. We summarize existing methods for explaining thepredictions of fact-checking systems and we explore trends in this topic.Further, we consider what makes for good explanations in this specific domainthrough a comparative analysis of existing fact-checking explanations againstsome desirable properties. Finally, we propose further research directions forgenerating fact-checking explanations, and describe how these may lead toimprovements in the research area.v

Conference paper

Lertvittayakumjorn P, Specia L, Toni F, 2020, FIND: human-in-the-loop debugging deep text classifiers

Since obtaining a perfect training dataset (i.e., a dataset which isconsiderably large, unbiased, and well-representative of unseen cases) ishardly possible, many real-world text classifiers are trained on the available,yet imperfect, datasets. These classifiers are thus likely to have undesirableproperties. For instance, they may have biases against some sub-populations ormay not work effectively in the wild due to overfitting. In this paper, wepropose FIND -- a framework which enables humans to debug deep learning textclassifiers by disabling irrelevant hidden features. Experiments show that byusing FIND, humans can improve CNN text classifiers which were trained underdifferent types of imperfect datasets (including datasets with biases anddatasets with dissimilar train-test distributions).

Working paper

Kotonya N, Toni F, 2020, Explainable Automated Fact-Checking for Public Health Claims, 2020 Conference on Empirical Methods in Natural Language Processing (EMNLP(1) 2020), Publisher: ACL

Fact-checking is the task of verifying the veracity of claims by assessing their assertions against credible evidence. The vast major-ity of fact-checking studies focus exclusively on political claims. Very little research explores fact-checking for other topics, specifically subject matters for which expertise is required. We present the first study of explainable fact-checking for claims which require specific expertise. For our case study we choose the setting of public health. To support this case study we construct a new datasetPUBHEALTHof 11.8K claims accompanied by journalist crafted, gold standard explanations(i.e., judgments) to support the fact-check la-bels for claims1. We explore two tasks: veracity prediction and explanation generation. We also define and evaluate, with humans and computationally, three coherence properties of explanation quality. Our results indicate that,by training on in-domain data, gains can be made in explainable, automated fact-checking for claims which require specific expertise.

Conference paper

Lertvittayakumjorn P, Specia L, Toni F, 2020, FIND: Human-in-the-Loop Debugging Deep Text Classifiers, 2020 Conference on Empirical Methods in Natural Language Processing, Publisher: ACL

Since obtaining a perfect training dataset (i.e., a dataset which is considerably large, unbiased, and well-representative of unseen cases)is hardly possible, many real-world text classifiers are trained on the available, yet imperfect, datasets. These classifiers are thus likely to have undesirable properties. For instance, they may have biases against some sub-populations or may not work effectively in the wild due to overfitting. In this paper, we propose FIND–a framework which enables humans to debug deep learning text classifiers by disabling irrelevant hidden features. Experiments show that by using FIND, humans can improve CNN text classifiers which were trained under different types of imperfect datasets (including datasets with biases and datasets with dissimilar train-test distributions).

Conference paper

Paulino-Passos G, Toni F, 2020, Cautious Monotonicity in Case-Based Reasoning with Abstract Argumentation, arxiv

Recently, abstract argumentation-based models of case-based reasoning($AA{\text -}CBR$ in short) have been proposed, originally inspired by thelegal domain, but also applicable as classifiers in different scenarios,including image classification, sentiment analysis of text, and in predictingthe passage of bills in the UK Parliament. However, the formal properties of$AA{\text -}CBR$ as a reasoning system remain largely unexplored. In thispaper, we focus on analysing the non-monotonicity properties of a regularversion of $AA{\text -}CBR$ (that we call $AA{\text -}CBR_{\succeq}$).Specifically, we prove that $AA{\text -}CBR_{\succeq}$ is not cautiouslymonotonic, a property frequently considered desirable in the literature ofnon-monotonic reasoning. We then define a variation of $AA{\text-}CBR_{\succeq}$ which is cautiously monotonic, and provide an algorithm forobtaining it. Further, we prove that such variation is equivalent to using$AA{\text -}CBR_{\succeq}$ with a restricted casebase consisting of all"surprising" cases in the original casebase.

Journal article

Baroni P, Toni F, Verheij B, 2020, On the acceptability of arguments and its fundamental role in nonmonotonic reasoning, logic programming and n-person games: 25 years later Foreword, ARGUMENT & COMPUTATION, Vol: 11, Pages: 1-14, ISSN: 1946-2166

Journal article

Albini E, Rago A, Baroni P, Toni Fet al., 2020, Relation-Based Counterfactual Explanations for Bayesian Network Classifiers, The 29th International Joint Conference on Artificial Intelligence (IJCAI 2020)

Conference paper

Cocarascu O, Cabrio E, Villata S, Toni Fet al., 2020, A dataset independent set of baselines for relation prediction in argument mining., Publisher: arXiv

Argument Mining is the research area which aims at extracting argument components and predicting argumentative relations (i.e.,support and attack) from text. In particular, numerous approaches have been proposed in the literature to predict the relations holding between the arguments, and application-specific annotated resources were built for this purpose. Despite the fact that these resources have been created to experiment on the same task, the definition of a single relation prediction method to be successfully applied to a significant portion of these datasets is an open research problem in Argument Mining. This means that none of the methods proposed in the literature can be easily ported from one resource to another. In this paper, we address this problem by proposing a set of dataset independent strong neural baselines which obtain homogeneous results on all the datasets proposed in the literature for the argumentative relation prediction task. Thus, our baselines can be employed by the Argument Mining community to compare more effectively how well a method performs on the argumentative relation prediction task.

Working paper

Cocarascu O, Stylianou A, Cyras K, Toni Fet al., 2020, Data-empowered argumentation for dialectically explainable predictions, 24th European Conference on Artificial Intelligence (ECAI 2020), Publisher: IOS Press

Today’s AI landscape is permeated by plentiful data anddominated by powerful data-centric methods with the potential toimpact a wide range of human sectors. Yet, in some settings this po-tential is hindered by these data-centric AI methods being mostlyopaque. Considerable efforts are currently being devoted to defin-ing methods for explaining black-box techniques in some settings,while the use of transparent methods is being advocated in others,especially when high-stake decisions are involved, as in healthcareand the practice of law. In this paper we advocate a novel transpar-ent paradigm of Data-Empowered Argumentation (DEAr in short)for dialectically explainable predictions. DEAr relies upon the ex-traction of argumentation debates from data, so that the dialecticaloutcomes of these debates amount to predictions (e.g. classifications)that can be explained dialectically. The argumentation debates con-sist of (data) arguments which may not be linguistic in general butmay nonetheless be deemed to be ‘arguments’ in that they are dialec-tically related, for instance by disagreeing on data labels. We illus-trate and experiment with the DEAr paradigm in three settings, mak-ing use, respectively, of categorical data, (annotated) images and text.We show empirically that DEAr is competitive with another transpar-ent model, namely decision trees (DTs), while also providing natu-rally dialectical explanations.

Conference paper

Albini E, Baroni P, Rago A, Toni Fet al., 2020, PageRank as an Argumentation Semantics, Biennial International Conference on Computational Models of Argument (COMMA), Publisher: IOS PRESS, Pages: 55-66, ISSN: 0922-6389

Conference paper

Čyras K, Karamlou A, Lee M, Letsios D, Misener R, Toni Fet al., 2020, AI-assisted schedule explainer for nurse rostering, Pages: 2101-2103, ISSN: 1548-8403

We present an argumentation-supported explanation generating system, called Schedule Explainer, that assists with makespan scheduling. Our stand-alone generic tool explains to a lay user why a resource allocation schedule is good or not, and offers actions to improve the schedule given the user's constraints. Schedule Explainer provides actionable textual explanations via an interactive graphical interface. We illustrate our system with a proof-of-concept application tool in a nurse rostering scenario whereby a shift-lead nurse aims to account for unexpected events by rescheduling some patient procedures to nurses and is aided by the system to do so.

Conference paper

Cocarascu O, Toni F, 2020, Deploying Machine Learning Classifiers for Argumentative Relations “in the Wild”, Argumentation Library, Pages: 269-285

Argument Mining (AM) aims at automatically identifying arguments and components of arguments in text, as well as at determining the relations between these arguments, on various annotated corpora using machine learning techniques (Lippi & Torroni, 2016).

Book chapter

Cocarascu O, Rago A, Toni F, 2020, Explanation via Machine Arguing, Pages: 53-84, ISBN: 9783030600662

As AI becomes ever more ubiquitous in our everyday lives, its ability to explain to and interact with humans is evolving into a critical research area. Explainable AI (XAI) has therefore emerged as a popular topic but its research landscape is currently very fragmented. Explanations in the literature have generally been aimed at addressing individual challenges and are often ad-hoc, tailored to specific AIs and/or narrow settings. Further, the extraction of explanations is no simple task; the design of the explanations must be fit for purpose, with considerations including, but not limited to: Is the model or a result being explained? Is the explanation suited to skilled or unskilled explainees? By which means is the information best exhibited? How may users interact with the explanation? As these considerations rise in number, it quickly becomes clear that a systematic way to obtain a variety of explanations for a variety of users and interactions is much needed. In this tutorial we will overview recent approaches showing how these challenges can be addressed by utilising forms of machine arguing as the scaffolding underpinning explanations that are delivered to users. Machine arguing amounts to the deployment of methods from computational argumentation in AI with suitably mined argumentation frameworks, which provide abstractions of “debates”. Computational argumentation has been widely used to support applications requiring information exchange between AI systems and users, facilitated by the fact that the capability of arguing is pervasive in human affairs and arguing is core to a multitude of human activities: humans argue to explain, interact and exchange information. Our lecture will focus on how machine arguing can serve as the driving force of explanations in AI in different ways, namely: by building explainable systems with argumentative foundations from linguistic data focusing on reviews), or by extracting argumentative reasoning from existin

Book chapter

Jha R, Belardinelli F, Toni F, 2020, Formal Verification of Debates in Argumentation Theory., CoRR, Vol: abs/1912.05828

Journal article

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# How to calculate the equilibrium pressure when ammonia and hydrogen gasses react in a closed system?

There is a container that has a volume of $5~\mathrm{L}$ on one side and $3~\mathrm{L}$ on the other and is separated in the middle. There are $2~\mathrm{mol}$ of $\ce{N2}$ on one side at $0~^\circ\mathrm{C}$ and $4~\mathrm{mol}$ of $\ce{H2}$ on the other at $0~^\circ\mathrm{C}$. When the valve is turned and the gases react, what will the total pressure be?

The balanced reaction is $$\ce{N2 + 3H2 <=> 2NH3}$$ My thinking was that the reaction occurs at STP so the pressure would stay at 1 atm?
Or that you could use \begin{align} pV &= nRT\\ p(8~\mathrm{L}) &=6(.082)(273.15)\\ p &= 0.06 \end{align} The $8~\mathrm{L}$ comes from adding the two volumes, and the 6 from adding the number moles together.

• Welcome to Chemistry.SE. Take the tour to get familiar with this site. This appears to be a homework question, please share your thoughts and attempts towards the solution. It'll make us certain that we aren't doing your homework for you.
– user15489
Jun 9 '15 at 2:08
• You are neglecting the fact, that these gasses react with each other. Also the temperature is given as $0~^\circ\mathrm{C}$, so STP it is not. I feel like there is an equilibrium constant missing in the question. Jun 9 '15 at 3:56
– user15489
Jun 9 '15 at 4:13
• That is all the information that was given for the problem. And I thought that STP was 1 atm and 0 degrees C, thats why I said that.
– user16813
Jun 9 '15 at 10:13

Note: Not all the reactants will react to form the products, but rather exist in equilibrium. However since the question doesn't mention anything about the equilibrium constant, I will assume that the reaction goes to completion as otherwise it will be impossible to do this question.

If you are aware of the various gas laws, doing this question is actually really simple and doesn't involved many calculations.

Step 1: Finding Initial Pressure

Firstly, the question doesn't specify what the initial pressure of the container is. Therefore you can't just assume that it is $1\ \mathrm{atm}$. So the first step would be finding the initial pressure: $$p = \frac{nRT}{V} = \frac{6\ \mathrm{mol}\times 8.3125\ \mathrm{L\ kPa\ mol^{-1}\ K^{-1}}\times273\ \mathrm{K}}{8\ \mathrm{L}} = 1702.4\ \mathrm{kPa} = 16.8\ \mathrm{atm}$$

Step 2: Find Moles of Products

Looking at the stoichiometry of the chemical equation, $4$ moles of gas react to form $2$ moles of gas. Therefore the ratio is $2:1$. In this case, we originally had $6$ moles of gas. Therefore at the end of the reaction, there should be $3$ moles of gas.

Step 3: Find Final Pressure

Now we know that the total number of moles of gas has halved (from $6$ moles to $3$ moles). One of the gas laws state that the number of moles of gas is directly proportional to the pressure of the system. This is clearly obviously by looking at the ideal gas equation. Therefore, by halving the original amount of moles of gas should result in halving of the original pressure of the system.

Therefore the final pressure will be: $8.4\ \mathrm{atm}$


text:
# `int (x^3+3x-4)/(x^3-4x^2+4x) dx` Use partial fractions to find the indefinite integral

For the given integral problem: `int (x^3+3x-4)/(x^3-4x^2+4x)dx` , we may simplify  by applying long division since the highest degree of x is the same from numerator and denominator side.

`(x^3+3x-4)/(x^3-4x^2+4x) = 1+(4x^2-x-4)/(x^3-4x^2+4x)` .

Apply partial fraction decomposition on the expression `(4x^2-x-4)/(x^3-4x^2+4x)` .

The pattern on setting up partial fractions will depend on the factors of the denominator. For the given problem, the factored form of the denominator will be:

`(x^3-4x^2+4x) =(x)(x^2-4x+4)`

`=(x) (x-2)(x-2)`  or `x(x-2)^2`

For the linear factor `(x)` , we  will have partial fraction: `A/x`

For the repeated linear factor `(x-2)^2` , we will have partial fractions: `B/(x-2) + C/(x-2)^2` .

The rational expression becomes:

`(4x^2-x-4)/(x^3-4x^2+4x) =A/x +B/(x-2) + C/(x-2)^2`

Multiply both side by the `LCD =x(x-2)^2` :

`((4x^2-x-4)/(x^3-4x^2+4x)) (x(x-2)^2)=(A/x +B/(x-2) + C/(x-2)^2)(x(x-2)^2)`

`4x^2-x-4=A*(x-2)^2+B*(x(x-2)) + C*x`

We apply zero-factor property on x(x-2)^2 to solve for value we can assign on x.

`x=0`

`x-2 = 0` then `x=2` .

To solve for `A` , we plug-in `x=0` :

`4*0^2-0-4=A*(0-2)^2+B*(0(0-2)) + C*0`

`0-0-4 = A*(-2)^2 +0 +0`

`-4 =4A`

`-4/4 =(4A)/4`

`A =-1`

To solve for `C` , we plug-in `x=2` :

`4*2^2-2-4=A*(2-2)^2+B*(2(2-2)) + C*2`

`16-2-4 = A*0 +B*0 +2C`

`10= 0 + 0 +2C`

`10 =2C`

`(10)/2= (2C)/2`

`C=5`

To solve for B, plug-in `x=1` ,`A=-1` , and `C=5` :

`4*1^2-1-4=(-1)*(1-2)^2+B*(1(1-2)) + 5*1 `

`4-1-4= (-1)*(-1)^2+B(1*(-1)) +5`

`-1= -1-B +5`

`-1= -B+4`

`-1-4= -B`

`-5=-B`

`(-5)/(-1) = (-B)/(-1)`

`B =5`

Plug-in `A = -1` , `B =5,` and `C=5` , we get the partial decomposition:

`(4x^2-x-4)/(x^3-4x^2+4x) =-1/x +5/(x-2) + 5/(x-2)^2`

Then the integrand becomes:

`(x^3+3x-4)/(x^3-4x^2+4x) = 1+(4x^2-x-4)/(x^3-4x^2+4x)` .

` =1-1/x +5/(x-2) + 5/(x-2)^2`

Apply the basic integration property:`int (u+-v) dx = int (u) dx +- int (v) dx` .

`int (x^3+3x-4)/(x^3-4x^2+4x) dx = int [1-1/x +5/(x-2) + 5/(x-2)^2] dx`

`=int1 dx - int 1/x dx +int 5/(x-2)dx + int 5/(x-2)^2dx`

Apply basic integration property: ` int(a) dx = ax+C`

`int1 dx = 1x` or `x`

Apply integration formula for logarithm: `int 1/u du = ln|u|+C` .

`int 1/x dx=ln|x|`

`int 5/(x-2)dx= int 5/udu`

`= 5ln|u|`

`=5 ln|x-2|`

Note: Let `u =x-2` then `du = dx` .

Apply the Power Rule for integration: `int (u^n) dx =u^(n+1)/ (n+1) +C` .

`int 5/(x-2)^2dx=int 5/u^2du`

`=int 5u^(-2)du`

`= 5 * u^(-2+1)/(-2+1)`

`= 5* u^-1/(-1)`

`= -5/u`

`= -5/(x-2)`

Note: Let `u =x-2` then `du = dx`

Combining the results, we get the indefinite integral as:

`int (x^3+3x-4)/(x^3-4x^2+4x)dx =x-ln|x| +5 ln|x-2|-5/(x-2)+C`


text:
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The number of distinct simple graphs with up to three nodes is $15$ $10$ $7$ $9$
Consider an undirected graph $G$ where self-loops are not allowed. The vertex set of $G$ is $\{(i,j) \mid1 \leq i \leq 12, 1 \leq j \leq 12\}$. There is an edge between $(a,b)$ and $(c,d)$ if $|a-c| \leq 1$ and $|b-d| \leq 1$. The number of edges in this graph is______.
Let $G$ be a simple undirected graph. Let $T_D$ be a depth first search tree of $G$. Let $T_B$ be a breadth first search tree of $G$. Consider the following statements. No edge of $G$ is a cross edge with respect to $T_D$. (A cross edge in $G$ is between two nodes ... $\mid i-j \mid =1$. Which of the statements above must necessarily be true? I only II only Both I and II Neither I nor II
Which of the following graphs has an Eulerian circuit? Any $k$-regular graph where $k$ is an even number. A complete graph on $90$ vertices. The complement of a cycle on $25$ vertices. None of the above
The maximum number of edges in a n-node undirected graph without self loops is $n^2$ $\frac{n(n-1)}{2}$ $n-1$ $\frac{(n+1)(n)}{2}$
How many perfect matching are there in a complete graph of $6$ vertices? $15$ $24$ $30$ $60$
The $2^n$ vertices of a graph $G$ corresponds to all subsets of a set of size $n$, for $n \geq 6$. Two vertices of $G$ are adjacent if and only if the corresponding sets intersect in exactly two elements. The number of vertices of degree zero in $G$ is: $1$ $n$ $n + 1$ $2^n$
What is the largest integer $m$ such that every simple connected graph with $n$ vertices and $n$ edges contains at least $m$ different spanning trees ? $1$ $2$ $3$ $n$
The minimum number of edges in a connected cyclic graph on $n$ vertices is: $n-1$ $n$ $n+1$ None of the above


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Interactive mirror (Ctrl + M) doesn't mirror object

I'm trying to simply mirror an object (maybe for the first time)

So I want to use the mirror command CtrlM for it.

When use it, blender asks to select the axis I want to use for the mirror operation, then I type X, Y or Z, the axis is selected, I press enter or click anywhere, and nothing happens, the object just sits there the same way.

What could it be the problem? I can't find info about this problem. Thanks for your help!

• Could you upload some screenshots or a .blend?
– gandalf3
May 21 '14 at 22:01
• OK, i edited the question, now it has screenshots. should i mention i'm using mac? May 21 '14 at 22:17
• I think it's working, but as your object is symmetrical on all three axes it will look the same. Try it on a Suzanne.
– gandalf3
May 21 '14 at 22:18
• But how does it need to be done to face in a different way?...and what's a Suzanne? May 21 '14 at 22:23
• @invicente It actually mirrors in global by default. May 21 '14 at 22:44

When using Interactive mirror as with the mirror modifier, you need to mirror about a point else pertaining to the former, the object will just mirror/flip in place and if the object is symmetrical you won't see any changes. This point can be the object origin, another object or the 3d cursor depending on which mode you are using.

You can either change the object's origin or simply switch the pivot point to use the 3d cursor and then mirror.

It is also worth noting that mirroring basically just scales the object on the desired axis by -1 (so without applied rotation you might not get expected results). So as a cheap and quick alternative to CtrlM, you could just select the object and use SX -1.

Mirroring about a point

What I did in the last example was to first translate the object over a bit on the X axis and then change the pivot mode to the 3d cursor as the mirror tool needs a point to work around. I then invoked interactive mirror and selected the axis I wanted to mirror on which in the example is the X axis. In the first one, I simply mirrored a single object and it flipped it to the opposite side, in the second, I duplicated the object and then mirrored so I had identical copies.

• Also as a side note, maybe add that mirroring actually just scales by -1 (so without applied rotation you might not get expected results)?
– gandalf3
May 21 '14 at 22:54
• @gandalf3 another side note: Better use it in Object mode. Scale by negative value will flip normals when in Edit mode. May 22 '14 at 0:18

It only works in edit mode: As per the Blender documentation

• While this link may answer the question, it is better to include the essential parts of the answer here and provide the link for reference. Link-only answers can become invalid if the linked page changes. - From Review Sep 29 '17 at 18:06
• @batFINGER: Thank you very much for guide line, I will do in future posts Sep 29 '17 at 18:36
• Nothing in the page linked corresponds to that. The first sentense states otherwise (and it's actually how it works), pointing out Object or Mesh selection (to have object selection one needs to be in Object mode, mesh - Edit mode). Sep 29 '17 at 18:48
• @batFINGER what do you mean? I was mostly answering the statement in the answer, not sure what should be undone as the answer was reviewed correctly as I understand Sep 29 '17 at 19:28


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# Math Help - [SOLVED] /chi hangs low

1. ## [SOLVED] /chi hangs low

I've had a need for the \chi symbol in a version of tex with a subscript. For some reason the symbol hangs low relative to other symbols, so the subscript does not line up visually in the document. I've been using a "double subscript" to make it work (e.g., \chi_ { _E }; a cumbersome hack. Is there a more elegant approach?

2. Originally Posted by huram2215
I've had a need for the \chi symbol in a version of tex with a subscript. For some reason the symbol hangs low relative to other symbols, so the subscript does not line up visually in the document. I've been using a "double subscript" to make it work (e.g., \chi_ { _E }; a cumbersome hack. Is there a more elegant approach?
I once had a similar problem, which I dealt with by giving the symbol an invisible superscript, which has the effect of pushing down the subscript. Compare these three, which are respectively \chi_E, \chi_{_E} and \chi^{\mathstrut}_E:

$\chi_E\ \chi_{_E}\ \chi^{\mathstrut}_E$

But maybe my cumbersome hack is just as inelegant as yours.


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# Creating Functions - Lesson 7

## Questions

• How can I define new functions?
• What’s the difference between defining and calling a function?
• What happens when I call a function?

## Objectives

• Define a function that takes parameters.
• Return a value from a function.
• Set default values for function parameters.
• Document our functions.
• Explain why we should divide programs into small, single-purpose functions.

## Jupyter Notebook File

Please use the Jupyter notebook named "07CreatingFunctions.ipynb" for this lesson. If you haven't downloaded them already, you may find them here.

# Key Points

• Define a function using def function_name(parameter).
• The body of a function must be indented.
• Call a function using function_name(value).
• Numbers are stored as integers or floating-point numbers.
• Variables defined within a function can only be seen and used within the body of the function.
• If a variable is not defined within the function it is used, Python looks for a definition before the function call
• Use help(thing) to view help for something.
• Put docstrings in functions to provide help for that function.
• Specify default values for parameters when defining a function using name=value in the parameter list.
• Parameters can be passed by matching based on name, by position, or by omitting them (in which case the default value is used).