OpenLLM-France/wikimedia huggingface dataset:
# Tight closure/Not plus closure/Dimension two/Standard-graded/Fact/Proof

Consider
    $R={\mathbb {F} }_{2}(t)[x,y,z]/(g)\,.$
In this ring $y^{3}z^{3}\in I^{*}$, but it can not belong to the plus closure. Else there would be a curve morphism
    $Y\longrightarrow C_{{\mathbb {F} }_{2}(t)}$
which annihilates the cohomology class $c$ and this would extend to a morphism of relative curves over ${\mathbb {A} }_{{\mathbb {F} }_{2}}^{1}$ almost everywhere.
# Tight closure/Plus closure/Two-dimensional/Standard-graded/Fact

Let $R$ be a standard-graded, two-dimensional normal domain over (the algebraic closure of) a finite field. Let $I$ be an $R_{+}$-primary graded ideal.
    $I^{*}=I^{+}\,.$
# Tight closure/Regular ring/Solid closure/Motivation/Section

An important property of tight closure is that it is trivial for regular rings, i.e.
    $I^{*}=I\,$
for every ideal $I$. This rests upon Kunz's theorem saying that the Frobenius homomorphism for regular rings is flat. This property implies the following cohomological property of torsors.

## Corollary

Let $(R,{\mathfrak {m}})$ denote a regular local ring of dimension $d$ and of positive characteristic, let $I={\left(f_{1},\ldots ,f_{n}\right)}$ be an ${\mathfrak {m}}$-primary ideal and $f\in R$ be an element with $f\notin I$. Let $B=R[T_{1},\ldots ,T_{n}]/{\left(f_{1}T_{1}+\cdots +f_{n}T_{n}+f\right)}$ be the corresponding forcing algebra. Then the extended ideal ${\mathfrak {m}}B$ satisfies
    $H_{{\mathfrak {m}}B}^{d}(B)=H^{d-1}(D({\mathfrak {m}}B),{\mathcal {O}}_{B})=0\,.$

### Proof

This follows from fact and $f\notin I^{*}$.
$\Box$
In dimension two this is true in every (even mixed) characteristic.

## Theorem

Let $(R,{\mathfrak {m}})$ denote a two-dimensional regular local ring, let $I={\left(f_{1},\ldots ,f_{n}\right)}$ be an ${\mathfrak {m}}$-primary ideal and $f\in R$ an element with $f\notin I$. Let
    $B=R[T_{1},\ldots ,T_{n}]/{\left(f_{1}T_{1}+\cdots +f_{n}T_{n}+f\right)}\,$
be the corresponding forcing algebra. Then for the extended ideal ${\mathfrak {m}}B$ we have
    $H_{{\mathfrak {m}}B}^{2}(B)=H^{1}(D({\mathfrak {m}}B),{\mathcal {O}}_{B})=0\,.$
$D({\mathfrak {m}}B)$
$f\notin I$
The main point for the proof of this result is that for $f\notin I$, the natural mapping
    $H^{1}(U,{\mathcal {O}}_{X})\longrightarrow H^{1}(T,{\mathcal {O}}_{T})$
is not injective by a Matlis duality argument. Since the local cohomology of a regular ring is explicitly known, this map annihilates some cohomology class of the form ${\frac {1}{fg}}$ where $f,g$ are parameters. But then it annihilates the complete local cohomology module and then $T$ is an affine scheme.
For non-regular two-dimensional rings it is a difficult question in general to decide whether a torsor is affine or not. A satisfactory answer is only known in the normal two-dimensional graded case over a field, which we will deal with in the final lectures.
In higher dimension in characteristic zero it is not true that a regular ring is solidly closed (meaning that every ideal equals its solid closure), as was shown by the following example of Paul Roberts.

## Example

Let $K$ be a field of characteristic $0$ and let
    $B=K[X,Y,Z][U,V,W]/{\left(X^{3}U+Y^{3}V+Z^{3}W-X^{2}Y^{2}Z^{2}\right)}\,.$
Then the ideal ${\mathfrak {a}}=(X,Y,Z)B$ has the property that $H_{\mathfrak {a}}^{3}(B)\neq 0$. This means that in $R=K[X,Y,Z]$, the element $X^{2}Y^{2}Z^{2}$ belongs to the solid closure of the ideal ${\left(X^{3},Y^{3},Z^{3}\right)}$, and hence the three-dimensional polynomial ring is not solidly closed.
This example was the motivation for the introduction of parasolid closure, which has all the good properties of solid closure but which is also trivial for regular rings.
# Tight closure/Localization problem/Remark

Let $S\subseteq R$ be a multiplicative system and $I$ an ideal in $R$. Then the localization problem of tight closure is the question whether the identity
    $(I^{*})_{S}=(IR_{S})^{*}\,$
holds.
Here the inclusion $\subseteq$ is always true and $\supseteq$ is the problem. The problem means explicitly:
If $f\in (IR_{S})^{*}$, can we find an $h\in S$ such that $hf\in I^{*}$ holds in $R$?
# Tight closure/Monsky-quartic/Explicit non-inclusion/Fact

Let ${\mathbb {F} }_{q}={\mathbb {F} }_{p}(\alpha )$, $t\mapsto \alpha$, $\deg(\alpha )=d$. Set $Q=2^{d-1}$.
    $xyf^{Q}\notin I^{[Q]}\,.$
# Tight closure/Plus closure/Tantalizing question/Remark

For an ideal $I\subseteq R$ in a domain $R$ define its plus closure by
    $I^{+}={\left\{f\in R\mid {\text{there exists a finite domain extension }}R\subseteq T{\text{ such that }}f\in IT\right\}}\,.$
Equivalent: Let $R^{+}$ be the absolute integral closure of $R$. This is the integral closure of $R$ in an algebraic closure of the quotient field $Q(R)$ (first considered by Artin). Then
    $f\in I^{+}{\text{ if and only if }}f\in IR^{+}.$
The plus closure commutes with localization.
We also have the inclusion $I^{+}\subseteq I^{*}$. Here the question arises:
Question: Is $I^{+}=I^{*}$?
This question is known as the tantalizing question in tight closure theory.
# Tight closure/Situation/Domain/Remark

Let $R$ be a noetherian domain of positive characteristic, let
    $F\colon R\longrightarrow R,f\longmapsto f^{p},$
be the Frobenius homomorphism and
    $F^{e}\colon R\longrightarrow R,f\longmapsto f^{q},$
(mit $q=p^{e}$) its $e$th iteration. Let $I$ be an ideal and set
    $I^{[q]}={\text{ extended ideal of }}I{\text{ under }}F^{e}\,.$
Then define the tight closure of $I$ to be the ideal
    $I^{*}={\left\{f\in R\mid {\text{ there exists }}z\neq 0{\text{ such that }}zf^{q}\in I^{[q]}{\text{ for all }}q=p^{e}\right\}}\,.$
# Tight closure/Relation to Hilbert-Kunz multiplicity/Fact

Let $(R,{\mathfrak {m}})$ be an analytically unramified and formally equidimensional local noetherian ring of positive characteristic, let $I\subseteq R$ be an ${\mathfrak {m}}$-primary ideal. Let $f\in R$.
    $f\in I^{*}{\text{ if and only if }}e_{HK}((I,f))=e_{HK}(I).$
# Tight closure/Monsky-quartic/Explicit non-inclusion/Fact/Proof

This is an elementary but tedious computation.
# Russian Revolution/Questions
## Questions

Note: these questions are not well related to the weeks of the course. --Abd 14:07, 29 August 2010 (UTC)

### Week 1

* Why would I (you the student) study the Russian Revolution?
* What impacts would you say the Revolution had on Russia?
* What do you think you'll learn from this course?
* How do you think you'll grow as a person after this course?
* What are your goals regarding this course?

### Week 2

* Who were the Romanovs?
* How did they come to power?
* Why did they rule the way they ruled?

### Week 3

* What affected the 1905 revolution?
* Who were some of the major players in the 1905 revolution?
* How was this revolution different from past Russian revolutions?
* How was this revolution different from other Russian revolutions to come?

### Weeks 4 and 5

* What affected the February 1917 revolution?
* Who were some of the major players in the February 1917 revolution?
* How was this revolution different from other past Russian revolutions?
* How was this revolution different from other Russian revolutions to come?
* How did Lvov react towards this revolutions?
* How did Kerensky react towards this revolutions?

### Week 6

What were the causes of the 1917 revolution?
# Russian Revolution/Related Books
## Related Books
### Books on Specific Revolutions

* 10 Days that Shook the World, John Reed
* Bolshevik Revolution, Richard Pipes

### Books on Russian History

* 20th Century Russia, Robert Service
* Lenin, Robert Service
# Russian Revolution/Related Wikipedia pages
## People
### Individuals

* Vladimir Lenin
* Joseph Stalin

### Groups

* Bolshevik
* Menshevik
* White movement
* Black Guards
* Green Army

## Events
### Revolutions

* Russian Revolution of 1917
* February Revolution
* October Revolution
* Russian Civil War
# Russian Revolution/Week 1
## Week 1
### Why

The purpose of this course is to introduce you, the students, to the events that shocked the world: Russian workers and peasants turned on the ruling elite of the Russian Empire in the middle of World War I. According to Robert Service in his book, Twentieth Century Russia, “No imperial power before the First World War was more reviled in Europe than the Russian Empire” (Page Number --). Russia was a nation that until the creation of St. Petersburg (named Petrograd after Russia went to war against Germany in 1914) by Peter the Great in -. Why did Russia become such a great power? Why did the Russian Empire evolve into the Union of Socialistic Soviet Republics which rivaled the United States from 1945-1991? Russia was behind European nations in most everything from its military, businesses, and people. Russia was a modern day pre-industrial revolution nation found in the mid 17th century.

### What were the influences of the Revolutions

Russia, rose and fell because of the revolutions that occurred in it's nation history. How did these revolutions effect the people it was trying to liberate or help? What substantial difference can be physically evaluated?

#### Karl Marx

It all started with Karl Marx. Karl Marx believed that the proletariat (working class, who made up over 88% of the population) class could only gain freedom by rising up against the bourgeoisie (Capitalist/Ruling class, who made up 2%)

### Questions

Week 1 Questions

### Next week's materials

Week 2
# Software applications/Overview/Part-up/Overview Part-ups/HeldCare

* Part-up: https://part-up.com/partups/heldcare-urw7iPZ8Xi7v9tz9f
* Website: http://www.heldcare.com
HeldCare is an initiative of Social Enterprise NL and Achmea Foundation.
You can join the program until 01 January 2016.

## Questions

* How long will the program run?
# Software applications/Overview/Part-up/Overview Part-ups/Innovisie

The goal of this part-up is to share information about innovations in healthcare.
This part-up is not about starting commercial activities. It's about 'intervisie' (see https://nl.wikipedia.org/wiki/Intervisie for more info).

## Agenda

* On 17 November a second meeting is planned. This meeting will be about why it's hard to implement technology. Is it because employees of healthcare institutions are not motivated or because IT does not facilitate innovation?

## Recent development

* http://www.amerpoort.nl/blogitem/berichten/2015/innovisie.html (Dutch)
# Software applications/Overview/Part-up/Overview Part-ups/P21

This part-up is about exploring the 21st century skills.
# Software applications/Overview/Part-up/Overview Part-ups/Zetel 151

The idea in this Dutch part-up is to place screens in parlement where you can see the opinion of the people.
# Software applications/Overview/Part-up/Overview Part-ups/The Next Bank

In this part-up the future of banking is discussed.
# Software applications/Overview/Part-up/Overview Part-ups/Hospitality Unlimited

This part-up is about developing a new hotel concept.
# Software applications/Overview/Part-up/Product backlog/Tasklist
## Status

On 18.11.15 Timboliu shared this idea in part-up https://part-up.com/partups/part-up-product-ownership-FKGKXh7icbaDnFq4N.

## General

When your active in many part-ups it would be nice if the system could generate a personal tasklist with all the activities you joined. It would be great if you could change the sequence of the activities.
# Software applications/Overview/Part-up/Overview Part-ups/Finno

This Dutch part-up is about creating a budget app.
# Software applications/Overview/Part-up/Projects

* At the moment there are no, or just af few, part-ups that are making money.
* In this learning project you can find information about how we are going to change this.

## Ideas

* Call prospects

## To-do

* Create part-up: Create a part-up for this initiative called 'Projects' or can we think of a better name?
* Part-up wiki: to share information about the case we want to set up a wiki. On 12.11.15 Timboliu sent a mail to Laurens about this idea.
# Universal Bibliography/Law/New Zealand

This page is part of a pan-jurisdictional bibliography of law. This part of the Universal Bibliography is a bibliography of New Zealand law.
See Laws of Australia and New Zealand

## Encyclopedias

* The Laws of New Zealand. Butterworths. Wellington. Robin Cooke was editor. Review: [1994] NZLJ 278
* Adams, Ernest Claude (ed). New Zealand Encyclopaedia of Forms and Precedents. (New Zealand Encyclopaedia of Forms and Precedents other than Court Forms). 1963 to 1968. See [1964] NZLJ 243 to 244. As to Adams, see

## Dictionaries

* New Zealand Law Dictionary. 9th Ed: 2019: . 8th Ed: 2014: .
** Butterworths New Zealand Law Dictionary. 7th Ed: 2011: . 6th Ed: 2005: . 5th Ed: 2001. 4th Ed: 1995.
** Hinde & Hinde. Hinde and Hinde. George W Hinde and Marian S Hinde. New Zealand Law Dictionary. 3rd Ed: 1979: . Revised 3rd Ed: 1986. Bibliography:
** Hinde's Law Dictionary. New Zealand Law Dictionary: being the Second New Zealand edition of Mozley and Whiteley's Law dictionary. 2nd Ed: 1972.
** George W Hinde. Mozley and Whiteley's Law Dictionary. New Zealand Edition. 1964. Review:
* New Zealand Legal Words and Phrases. 1995.
Maori
* Stephens and Boyce. Dictionary of Māori Legal Terms. 2013.

## Computer-assisted legal research

* NZLII (nzlii.org)
* Westlaw NZ

## Introductions

* John Aeneas Byron O'Keefe and Walter Leslie Farrands. Introduction to New Zealand Law. Third Edition. Butterworths. 1976.
* Mulholland, Raymond Douglas. Introduction to the New Zealand Legal System. Butterworths. 1985.
* Whitta, Horton and Mullholland. Introduction to the New Zealand Legal System. 1968.
* Lawrennce. Legal Studies: A First Book on New Zealand Law. 2nd Ed: 1990: .
* Greville and Polaschek. Access to New Zealand Law. 2019. GlobaLex. NYU Law.
# Survey research and design in psychology/Readings/Textbooks/Francis

Francis, G. (2007). Introduction to SPSS for Windows: v. 15.0 and 14.0 with notes for studentware (5th ed.). Sydney: Pearson Education. ISBN 9780733989384 | Publisher site | Co-op
# Survey research and design in psychology/Readings/Textbooks/Howell/2010

Howell, D. C. (2010). Statistical methods for psychology (7th ed.). Belmont, CA: Wadsworth. ISBN 0495012874 | ISBN 9780495012870. ~AU$130 | Hardbound | Companion site | Author's site | UC library
# Survey research and design in psychology/Readings/Textbooks/Howell/2013

Howell, D. C. (2013). Statistical methods for psychology (8th ed.). Belmont, CA: Wadsworth. ISBN 1111835489 | ISBN 9781111835484 ~AU$155 | Hardbound | Companion site | Author's site | UC library
# Survey research and design in psychology/Readings/Textbooks/AllenBennett/2010

Allen, P. & Bennett, K. (2010). PASW Statistics by SPSS: A Practical Guide: Version 18.0 (1st ed.). South Melbourne, Victoria, Australia: Cengage Learning Australia. ISBN 9780170222907 | RR: AU$69.95 Co-op member: AU$65.05 | Product details | Student companion site | Instructor companion site | Data files
# Survey research and design in psychology/Readings/Textbooks/AllenBennett/2014

Allen, P. & Bennett, K. (2014). SPSS Statistics Version 22: A practical guide (3rd ed.). South Melbourne, Victoria, Australia: Cengage Learning Australia. Product details | Co-op.
# Survey research and design in psychology/Readings/Textbooks/Howell/2007

Howell, D. C. (2007). Statistical methods for psychology (6th ed.). Belmont, CA: Wadsworth. ISBN 0495012874 | ISBN 9780495012870. ~AU$120 | Hardbound | Companion site | Instructor site | (or you can use earlier editions)
# Survey research and design in psychology/Readings/Textbooks/HowittCramer
## 2017

* Howitt, D. & Cramer, D. (2017a). Understanding Statistics in Psychology with SPSS (7th ed.). Harlow, UK: Pearson. ISBN 1292134216 | eBook | Educator resources
* Howitt, D. & Cramer, D. (2017b). Introduction to SPSS in psychology (7th ed.). Harlow, UK: Pearson. ISBN 1292186666 | Publisher site

## 2014

* Howitt, D. & Cramer, D. (2014a). Introduction to statistics in psychology (6th ed.). Harlow, UK: Pearson. ISBN 1292000740 | RR: AU$87.95 | Co-op member: AU$70.36 | Publisher site | Companion site | Educator resources | UC library
* Howitt, D. & Cramer, D. (2014b). Introduction to SPSS in psychology (6th ed.). Harlow, UK: Pearson. ISBN 1292000694 | RR AU$59.40 | Co-op: AU$47.52 | Publisher site | Companion site | Educator resources
# Survey research and design in psychology/Readings/Textbooks/AllenBennett/2008

Allen, P. & Bennett, K. (2008). SPSS for the health and behavioural sciences. South Melbourne, Victoria, Australia: Thomson. | ~AU$53 Companion site
# Survey research and design in psychology/Readings/Textbooks/Francis/2007

Francis, G. (2007). Introduction to SPSS for Windows: v. 15.0 and 14.0 with notes for studentware (5th ed.). Sydney: Pearson Education. ISBN 9780733989384 | Publisher site | Co-op
# Survey research and design in psychology/Readings/Textbooks/Howell
## Subpages

Special:Prefixindex/Survey research and design in psychology/Readings/Textbooks/Howell/
# Power series/R/Third power/Fourth coefficient/Exercise

    $\sum _{n=0}^{\infty }a_{n}x^{n}$
$x^{0},x^{1},x^{2},x^{3},x^{4}$
    $\sum _{n=0}^{\infty }c_{n}x^{n}={\left(\sum _{n=0}^{\infty }a_{n}x^{n}\right)}^{3}\,.$
# Power series/R/Introduction/Section
## Definition

Let ${\left(c_{n}\right)}_{n\in \mathbb {N} }$ be a sequence of real numbers and $x$ another real number. Then the series
    $\sum _{n=0}^{\infty }c_{n}x^{n}$
$x$
${\left(c_{n}\right)}_{n\in \mathbb {N} }$
For a power series, it is important that $x$ varies and that the power series represents in some convergence interval a function in $x$. Every polynomial is a power series, but one for which all coefficients starting with a certain member are $0$. In this case, the convergence is everywhere.
We have encountered an important power series earlier, the geometric series $\sum _{n=0}^{\infty }x^{n}$ (here all coefficients equal $1$), which converges for $\vert {x}\vert <1$ and represents the function $1/(1-x)$, see fact. Another important power series is the exponential series, which for every real number converges and represents the real exponential function. Its inverse function is the natural logarithm.
The behavior of convergence of a power series is given by the following theorem.

## Theorem

Let
    $f(x):=\sum _{n=0}^{\infty }c_{n}x^{n}\,$
be a power series and suppose that there exists some $x_{0}\neq 0$ such that $\sum _{n=0}^{\infty }c_{n}x_{0}^{n}$ converges. Then there exists a positive $R$ (where $R=\infty$ is allowed) such that for all $x\in \mathbb {R}$ fulfilling $\vert {x}\vert <R$ the series converges absolutely. On such an (open) interval of convergence, the power series $f(x)$ represents a

### Proof

$\Box$
If two functions are given by power series, then their sum is simply given by the (componentwise defined) sum of the power series. It is not clear at all by which power series the product of two power series is described. The answer is given by the Cauchy-product of series.

## Definition

For two series $\sum _{i=0}^{\infty }a_{i}$ and $\sum _{j=0}^{\infty }b_{j}$ of real numbers, the series
    $\sum _{k=0}^{\infty }c_{k}{\text{ with }}c_{k}:=\sum _{i=0}^{k}a_{i}b_{k-i}$
Also, for the following statement we do not provide a proof.

## Lemma

Let
    $\sum _{k=0}^{\infty }a_{k}{\text{ and }}\sum _{k=0}^{\infty }b_{k}$
be absolutely convergent series of real numbers. Then also the Cauchy product $\sum _{k=0}^{\infty }c_{k}$ is absolutely convergent and for its sum the equation
    $\sum _{k=0}^{\infty }c_{k}={\left(\sum _{k=0}^{\infty }a_{k}\right)}\cdot {\left(\sum _{k=0}^{\infty }b_{k}\right)}\,$
From this we can infer that the product of power series is given by the power series whose coefficients are those which arise by the multiplication of polynomials, see exercise.
# Power series/R/n-th derivative in a/Direct/Exercise

Let $\sum _{n=0}^{\infty }c_{n}(x-a)^{n}$ be a convergent power series. Determine the derivatives $f^{(k)}(a)$.
# Power series approach/1 over polynomial/0/Degree 3/1/Exercise

Determine the Taylor polynomial of the third order of the function ${\frac {1}{x^{2}+1}}$ in the zero point, using the power series approach described in remark.
# Power series/R/Same variable/Cauchy product/Exercise

Let $\sum _{n=0}^{\infty }a_{n}x^{n}$ and $\sum _{n=0}^{\infty }b_{n}x^{n}$ be two power series absolutely convergent in $x\in \mathbb {R}$. Prove that the Cauchy product of these series is exactly
    $\sum _{n=0}^{\infty }c_{n}x^{n}{\text{ where }}c_{n}=\sum _{i=0}^{n}a_{i}b_{n-i}.$
# Power systems
## Introduction

* Introduction
* History
* Sources

## Components in Power Systems

* Synchronous Generator
* Automatic Voltage Regulator
* Overhead Lines
* Representation of Lines
* Underground Cables
* Transformers
* Voltage Characteristic of Loads

## Modeling Power Systems

* Ybus

## Learning readings
### Wikipedia

* Electric power system
* Power system protection
* Energy system
* Power-system automation
* Smart grid
* Smart meter
* Net metering
* Distributed generation
* Net metering in the United States
* Category:Electric power systems components
* Continual power system
* Power system simulation
* Power system operations and control
* Power systems CAD
* Electric Power Systems Research
* Emergency power system
* Electric power transmission
* Electrical grid
* Power engineering software
* History of electric power transmission
* Three-phase electric power
* Grid-connected photovoltaic power system
* List of countries by renewable electricity production
* Variable renewable energy
* Building-integrated photovoltaics
* Grid energy storage
# Power sums/Induction/Exercise

Prove the following formulas by induction.
    $\sum _{i=1}^{n}i={\frac {n(n+1)}{2}}\,,$
    $\sum _{i=1}^{n}i^{2}={\frac {n(n+1)(2n+1)}{6}}\,.$
    $\sum _{i=1}^{n}i^{3}={\left({\frac {n(n+1)}{2}}\right)}^{2}\,.$
# PlanetPhysics/A Cats Meow Application of Gauss Law

If you know the amount of charge contained within a Gaussian surface, then the total flux of the Electric Field generated by the enclosed charge is calculated from Gauss' Law.
As a demonstration, imagine a pair of cats that have charges placed on them by their loyal masters. Although the contours of the cats' elegant frames represent a complicated geometry, calculating the flux is a simple task if the charge on the cats is known. The flux through the Gaussian surface in Figure 1 is given by Gauss' law
   $\Phi ={\frac {q_{1}+q_{2}}{\epsilon _{0}}}$
\vspace{10 pt}
   \includegraphics[scale=.6]{Cats.eps}
   {\mathbf Figure 1:} Gaussian Surface Encompassing Two Cats
Note that we add the charges in equation (1) because it is the net enclosed charge. For example if the charge on cat 1 is $10.5\,\,[\mu C]$ and the charge on the cat 2 is $12.2[\mu C]$, then the total flux through $G$ is
$\Phi ={\frac {10.5\times 10^{-8}\,[C]+12.2\times 10^{-8}\,[C]}{8.85\times 10^{-12}\,[C^{2}/Nm^{2}]}}$ $\Phi =3073.4\,\,[Nm^{2}/C]$
The reverse of this problem is another important result. If we measure the flux through a given Gaussian surface, then we can calculate the amount of enclosed charge.
# PlanetPhysics/1D Example of the Relation Between Force and Potential Energy

For a simple one dimensional example of the relationship between force and potential energy, assume that the potential energy of a particle is given by the equation
   $U(x)=-{\frac {A}{x}}\left[1+Be^{-x/c}\right]$
The relationship between force and potential energy is
   $\mathbf {F} =-\nabla U$
For our 1D example, where the potential energy is dependent only on the position, $x$
   $F=-{\frac {dU}{dx}}$
Taking the derivative yields
   ${\frac {dU}{dx}}={\frac {A}{x^{2}}}\left[1+Be^{-x/c}\right]-{\frac {A}{x}}\left[-{\frac {B}{C}}e^{-x/c}\right]$
Therefore, the force on the particle is governed by the equation
   $F=-{\frac {A}{x^{2}}}\left[1+Be^{-x/c}\right]+{\frac {A}{x}}\left[-{\frac {B}{C}}e^{-x/c}\right]$
If we further assume that the constant C is so much larger than x, the force will simplify. Rearranging to get
   $F=-{\frac {A}{x^{2}}}\left[1+Be^{-x/c}-{\frac {Bx}{C}}e^{-x/c}\right]$
Because $x\ll C$, $e^{-x/c}\Rightarrow e^{0}=1$, we get
   $F=-{\frac {A}{x^{2}}}\left[1+B-{\frac {Bx}{C}}\right]$
Further simplifying ${\frac {Bx}{C}}=0$ gives the force under this assumption
   $F=-{\frac {A}{x^{2}}}\left[1+B\right]$