OpenLLM-France/wikimedia huggingface dataset:
# The Integral

What we introduced last time:
$L(f)=\sup _{P}L(P,f)$
$U(f)=\inf _{P}U(P,f)$
Are actually known as the lower and upper integrals respectively. Both are simultaneously the integral provided that the function that they are built from satisfy the following condition, the definition of integrability (but not the integrability criterion).
A bounded function $f$ on $[a,b]$ is integrable if $L(f)=U(f)$.
And that's all the integral really is!

## Yet ANOTHER return of that summing exercise

This is just a quick little question, probably trivial. What does this mean for $f(x)=x$ ?

# Integrability

As previously defined we can prove the integrability of a function by noting that $U(f)=L(f)$
However, there is a much more useful way to prove that a function, or an entire class of functions, is integrable. This is the theorem called the Integrability Criterion:
A bounded function $f:[a,b]\to \mathbb {R}$ is integrable on $[a,b]$ iff for all $\varepsilon >0$ there exists a partition $P$ of $[a,b]$ such that $U(P,f)-L(P,f)<\varepsilon$
You'll notice that it has properties similar to the definition of the limit.
Since this is a proof of necessity assume that $f$ is integrable on $[a,b]$ . Let $\varepsilon >0$ be given and say $I=\int \limits _{a}^{b}f(x)dx$ . Also $I=\inf _{Q}U(Q,f)=\sup _{R}L(R,f)$ . Now by the approximation property there exists a partition $Q$ such that $U(Q,f)<I+{\tfrac {\varepsilon }{2}}$ and a partition $R$ such that $L(R,f)>I-{\tfrac {\varepsilon }{2}}$ . If we take the refinement $P=Q\sup R$ and apply the properties of refinement that we proved before:
$U(P,f)<I+{\frac {\varepsilon }{2}}$
and
$L(P,f)>I-{\frac {\varepsilon }{2}}$
We can manipulate this to:
$U(P,f)-L(P,f)<\varepsilon$
Now for the proof to finish up the biconditional...

## Integrability criterion proof

You do it. If for all $\varepsilon >0$ there exists a partition $P$ of $[a,b]$ such that $U(P,f)-L(P,f)<\varepsilon$ then $f$ is integrable on $[a,b]$ .
Some hints: show that it implies $U(f)-L(f)<\varepsilon$ for all $\varepsilon >0$ then show this is so only if $U(f)=L(f)$ .
# Calculus/Surface area

This page or section of Calculus is a stub. You can help Wikibooks by expanding it.
Suppose we are given a function $f$ and we want to calculate the surface area of the function $f$ rotated around a given line. The calculation of surface area of revolution is related to the arc length calculation.
If the function $f$ is a straight line, other methods such as surface area formulae for cylinders and conical frusta can be used. However, if $f$ is not linear, an integration technique must be used.
Recall the formula for the lateral surface area of a conical frustum:
    $A=2\pi rl$
where $r$ is the average radius and $l$ is the slant height of the frustum.
For $y=f(x)$ and $a\leq x\leq b$ , we divide $[a,b]$ into subintervals with equal width $\delta x$ and endpoints $x_{0},x_{1},\ldots ,x_{n}$ . We map each point $y_{i}=f(x_{i})$ to a conical frustum of width Δx and lateral surface area $A_{i}$ .
We can estimate the surface area of revolution with the sum
    $A=\sum _{i=0}^{n}A_{i}$
As we divide $[a,b]$ into smaller and smaller pieces, the estimate gives a better value for the surface area.

## Definition (Surface of Revolution)

The surface area of revolution of the curve $y=f(x)$ about a line for $a\leq x\leq b$ is defined to be
$A=\lim _{n\to \infty }\sum _{i=0}^{n}A_{i}$

## The Surface Area Formula

Suppose $f$ is a continuous function on the interval $[a,b]$ and $r(x)$ represents the distance from $f(x)$ to the axis of rotation. Then the lateral surface area of revolution about a line is given by
    $A=2\pi \int _{a}^{b}r(x){\sqrt {1+f'(x)^{2}}}\,dx$
And in Leibniz notation
    $A=2\pi \int _{a}^{b}r(x){\sqrt {1+\left({\tfrac {dy}{dx}}\right)^{2}}}\,dx$
Proof:
    $A$ $=\lim _{n\to \infty }\sum _{i=1}^{n}A_{i}$ $=\lim _{n\to \infty }\sum _{i=1}^{n}2\pi r_{i}l_{i}$ $=2\pi \cdot \lim _{n\to \infty }\sum _{i=1}^{n}r_{i}l_{i}$
As $n\to \infty$ and $\Delta x\to 0$, we know two things:
* the average radius of each conical frustum $r_{i}$ approaches a single value
* the slant height of each conical frustum $l_{i}$ equals an infitesmal segment of arc length
From the arc length formula discussed in the previous section, we know that
    $l_{i}={\sqrt {1+f'(x_{i})^{2}}}$
Therefore
    $A$ $=2\pi \cdot \lim _{n\to \infty }\sum _{i=1}^{n}r_{i}l_{i}$ $=2\pi \cdot \lim _{n\to \infty }\sum _{i=1}^{n}r_{i}{\sqrt {1+f'(x_{i})^{2}}}\Delta x$
Because of the definition of an integral $\int _{a}^{b}f(x)dx=\lim _{n\to \infty }\sum _{i=1}^{n}f(c_{i})\Delta x_{i}$ , we can simplify the sigma operation to an integral.
    $A=2\pi \int _{a}^{b}r(x){\sqrt {1+f'(x)^{2}}}dx$
Or if $f$ is in terms of $y$ on the interval $[c,d]$
    $A=2\pi \int _{c}^{d}r(y){\sqrt {1+f'(y)^{2}}}dy$
# Calculus/Systems of ordinary differential equations

We have already examined cases where we have a single differential equation and found several methods to aid us in finding solutions to these equations. But what happens if we have two or more differential equations that depend on each other? For example, consider the case where
    $D_{t}x(t)=3y(t)^{2}+x(t)t$
and
    $D_{t}y(t)=x(t)+y(t)$
Such a set of differential equations is said to be coupled. Systems of ordinary differential equations such as these are what we will look into in this section.

## First order systems

A general system of differential equations can be written in the form
    $D_{t}\mathbf {x} =\mathbf {F} (\mathbf {x} ,t)$
Instead of writing the set of equations in a vector, we can write out each equation explicitly, in the form:
    $D_{t}x_{1}=F_{1}(x_{1},\ldots ,x_{n},t)$
          $\vdots \!\;$
    $D_{t}x_{i}=F_{i}(x_{1},\ldots ,x_{n},t)$
If we have the system at the very beginning, we can write it as:
    $D_{t}\mathbf {x} =\mathbf {G} (\mathbf {x} ,t)$
where
    $\mathbf {x} =(x(t),y(t))=(x,y)$
and
    $\mathbf {G} (\mathbf {x} ,t)=(3y^{2}+xt,x+y)$
or write each equation out as shown above.
Why are these forms important? Often, this arises as a single, higher order differential equation that is changed into a simpler form in a system. For example, with the same example,
    $D_{t}x(t)=3y(t)^{2}+x(t)t$
    $D_{t}y(t)=x(t)+y(t)$
we can write this as a higher order differential equation by simple substitution.
    $D_{t}y(t)-y(t)=x(t)$
then
    $D_{t}x(t)=3y(t)^{2}+(D_{t}y(t)-y(t))t$
    $D_{t}x(t)=3y(t)^{2}+tD_{t}y(t)-ty(t)$
Notice now that the vector form of the system is dependent on t since
    $\mathbf {G} (\mathbf {x} ,t)=(3y^{2}+xt,x+y)$
the first component is dependent on t. However, if instead we had
    $\mathbf {H} (\mathbf {x} )=(3y^{2}+x,x+y)$
notice the vector field is no longer dependent on t. We call such systems autonomous. They appear in the form
    $D_{t}\mathbf {x} =\mathbf {H} (\mathbf {x} )$
We can convert between an autonomous system and a non-autonomous one by simply making a substitution that involves t, such as y=(x, t), to get a system:
    $D_{t}\mathbf {y} =(\mathbf {F} (\mathbf {y} ),1)=(\mathbf {F} (\mathbf {x} ,t),1)$
In vector form, we may be able to separate F in a linear fashion to get something that looks like:
    $\mathbf {F} (\mathbf {x} ,t)=A(t)\mathbf {x} +\mathbf {b} (t)$
where A(t) is a matrix and b is a vector. The matrix could contain functions or constants, clearly, depending on whether the matrix depends on t or not.
# Canadian Criminal Law/Offences/Use of Firearm in Commission of an Offence
## Legislation

   Using firearms in commission of offence 85. (1) Every person commits an offence who uses a firearm, whether or not the person causes or means to cause bodily harm to any person as a result of using the firearm,
       (a) while committing an indictable offence, other than an offence under section 220 (criminal negligence causing death), 236 (manslaughter), 239 (attempted murder), 244 (discharging firearm with intent), 244.2 (discharging firearm — recklessness), 272 (sexual assault with a weapon) or 273 (aggravated sexual assault), subsection 279(1) (kidnapping) or section 279.1 (hostage taking), 344 (robbery) or 346 (extortion);
       (b) while attempting to commit an indictable offence; or
       (c) during flight after committing or attempting to commit an indictable offence.
   Using imitation firearm in commission of offence (2) Every person commits an offence who uses an imitation firearm
       (a) while committing an indictable offence,
       (b) while attempting to commit an indictable offence, or
       (c) during flight after committing or attempting to commit an indictable offence,
   whether or not the person causes or means to cause bodily harm to any person as a result of using the imitation firearm.
   R.S., 1985, c. C-46, s. 85; 1995, c. 39, s. 139; 2003, c. 8, s. 3; 2008, c. 6, s. 3; 2009, c. 22, s. 3.
   – CCC

## Proving the Offence
### Elements of Proof

Using a Firearm in Commission of an Offence - s. 85(1)
* ♰ identity of accused
* ♰ date and time of incident
* ♰ jurisdiction (incl. region and province)
* ♰ the accused used a firearm
** ♰ committing or attempting to commit an indictable offence; or,
** ♰ during flight after committing or attempting to commit an indictable offence.
Using an Imitation Firearm in Commission of an Offence - s. 85(2)
* ♰ identity of accused
* ♰ date and time of incident
* ♰ jurisdiction (incl. region and province)
* ♰ the accused was using an imitation firearm while
** ♰ committing or attempting to commit an indictable offence; or,
** ♰ during flight after committing or attempting to commit an indictable offence.
Items in bold are essential elements that make out the offence.

### Tendering Exhibits

* ♰ the firearm or imitation firearm

## Interpretation

A firearm must not merely "proximate for future use”, but rather must be ready for present use during the offence.
* ↑ R. v. Steele, 2007 SCC 36, [2007] 3 SCR 3

## Motions and Orders

* s.486.2(2) or 486.2(4) order for private testimony
* s. 486.3(2) Order prohibiting cross examination
* s. 486.5(1) or (2) publication ban
* Video-recorded evidence by disabled person (s.715.2) or person under age 18 (s.715.1)
# Canadian Criminal Law/Offences/Unlawfully in a Dwelling
## Legislation

   Being unlawfully in dwelling-house 349. (1) Every person who, without lawful excuse, the proof of which lies on that person, enters or is in a dwelling-house with intent to commit an indictable offence in it is guilty of an indictable offence and liable to imprisonment for a term not exceeding ten years or of an offence punishable on summary conviction.
   Presumption (2) For the purposes of proceedings under this section, evidence that an accused, without lawful excuse, entered or was in a dwelling-house is, in the absence of any evidence to the contrary, proof that he entered or was in the dwelling-house with intent to commit an indictable offence therein.
   – CCC

## Proof of the Offence

The following should be proven:
* identity of accused
* date and time of incident
* jurisdiction (incl. region and province)
* the accused entered into the premises
* the premises was a dwelling house
* the accused had no justification for entering the premises or permission to enter
* the accused intended to commit an indictable offence (presumed under s. 349(2))
Essential elements are in bold.
* ↑ R. v. Payne, [2007] N.J. No. 303 (P.C.)

## Application

The offence can be committed in two ways, either by "entering" or "being in" a dwelling-house without a lawful excuses with the intent to commit an indictable offence.
The mens rea is made out where 1) there is a general intent to enter the dwelling-house without lawful excuse, and 2) the specific intent to commit an indictable offence.
The crown must prove an entry and and intent to commit an indictable offence therein.
The presumption under s.349(2) is a mandatory inference. Once the actus reus is proven that the accused was in the dwelling the mes rea is presumed.
* ↑ R v Beyo 2000 CanLII 5683 (ONCA)
* ↑ R. v. Ellis, [2012] N.J. No. 355 (P.C.) (“the first element of the offence [of unlawfully being in a dwelling-house] that must be proven by the Crown is that the defendant actually entered the [complainant’s] home.”)
* ↑ see R. v. Nagy, (1988), 45 C.C.C. (3d) 350 (Ont. C.A.) R. v. Beyo 2000 CanLII 5683 (ON CA), (2000), 144 C.C.C. (3d) 15 (Ont. C.A.) R v E(S) 1993 CanLII 3410 (NWTCA)
* ↑ R v Austin [1969] 1 CCC 97 SCC
* ↑ R v Nagy 1988 45 CCC 3d 350 (ONCA)
* ↑ R. v. Atkinson, [2012] O.J. No. 2520 (C.A.), at para 97, (“from evidence of the actus reus, the trier of fact presumes the mens rea, absent evidence to the contrary.”)

## See Also

* Canadian Criminal Law/Offences/Break and Enter
* Canadian Criminal Law/Offences/Forceable Entry
# Canadian Criminal Law/Offences/Unauthorized Possession of a Firearm
## Legislation

   Unauthorized possession of firearm 91. (1) Subject to subsection (4), every person commits an offence who possesses a firearm without being the holder of
       (a) a licence under which the person may possess it; and
       (b) in the case of a prohibited firearm or a restricted firearm, a registration certificate for it.
   Unauthorized possession of prohibited weapon or restricted weapon (2) Subject to subsection (4), every person commits an offence who possesses a prohibited weapon, a restricted weapon, a prohibited device, other than a replica firearm, or any prohibited ammunition, without being the holder of a licence under which the person may possess it. ...
   Exceptions (4) Subsections (1) and (2) do not apply to
       (a) a person who possesses a firearm, a prohibited weapon, a restricted weapon, a prohibited device or any prohibited ammunition while the person is under the direct and immediate supervision of a person who may lawfully possess it, for the purpose of using it in a manner in which the supervising person may lawfully use it; or
       (b) a person who comes into possession of a firearm, a prohibited weapon, a restricted weapon, a prohibited device or any prohibited ammunition by the operation of law and who, within a reasonable period after acquiring possession of it,
          (i) lawfully disposes of it, or
          (ii) obtains a licence under which the person may possess it and, in the case of a prohibited firearm or a restricted firearm, a registration certificate for it.
   R.S., 1985, c. C-46, s. 91; 1991, c. 28, s. 7, c. 40, ss. 5, 36; 1995, c. 22, s. 10, c. 39, s. 139; 2008, c. 6, s. 4; 2012, c. 6, s. 2.
   – CCC

## Proof of Offence
### Unauthorized possession of firearm

* time and date of the offence
* jurisdiction of the offence
* identity of the accused
* the accused possessed an object
* the object is a firearm
* the accused does not hold a licence to possess it
* if firearm is restricted or prohibited, the accused does not hold a registration certificate.

### Unauthorized possession of prohibited weapon or restricted weapon

* time and date of the offence
* jurisdiction of the offence
* identity of the accused
* the accused possessed an object
* the object is:
** prohibited weapon,
** a restricted weapon,
** a prohibited device, other than a replica firearm, or
** prohibited ammunition
* the accused does not hold a licence to possess it
# Canadian Criminal Law/Offences/Trespassing at Night
## Legislation

   Trespassing at night 177. Every one who, without lawful excuse, the proof of which lies on him, loiters or prowls at night on the property of another person near a dwelling-house situated on that property is guilty of an offence punishable on summary conviction.
   R.S., c. C-34, s. 173.
   – CCC

## Proof of the Offence

In addition to the essential elements of time, jurisdiction and identity, the crown should prove:
* the accused loitered or prowled
* it was night at the time
* the accused was on private property at the time
* a private dwelling-house is situated on that property
* there was no lawful excuse for the conduct

## Application

"loitering" is purposeless wandering about the area. It includes standing idly around, hanging around, lingering, sauntering, dawdling.
"prowling" is conduct with a specific unlawful goal.
A "lawful excuse" must include (1) a reason that is permissible under the law of the land to be on the property and (2) he must carry out his purpose in a lawful manner.
The burden is on the defence to prove a lawful excuse.
See also: R. v. Butt, 2010 NLTD 58
* ↑ R. v. Willis, (1987), 37 C.C.C. (3d) 184, at 186 (B.C. Co. Ct.) R. c. Cloutier, 1991 CanLII 2831 (QC CA) at para. 154 R. v. Lozowchuk 1984 CarswellSask 268
* ↑ R. v. Dillon, [1964] 3 C.C.C. 205 (Alta. S.C. App. Div.) R. c. Cloutier, 1991 CanLII 2831 (QC CA) R. v. Walczak, 2008 ONCJ 70 R. v. Priestap, 2006 CanLII 12288 (ON CA)
* ↑ R. v. Andsten 1960 128 C.C.C. 311 (BCCA)
* ↑ R. v. Johnson, 1985 CarswellBC 680 (BCPC)
# Canadian Criminal Law/Offences/Theft and Forgery of a Credit Card
## Legislation

   Theft, forgery, etc., of credit card 342. (1) Every person who
       (a) steals a credit card,
       (b) forges or falsifies a credit card,
       (c) possesses, uses or traffics in a credit card or a forged or falsified credit card, knowing that it was obtained, made or altered
          (i) by the commission in Canada of an offence, or
          (ii) by an act or omission anywhere that, if it had occurred in Canada, would have constituted an offence, or
       (d) uses a credit card knowing that it has been revoked or cancelled,
   is guilty of
       (e) an indictable offence and is liable to imprisonment for a term not exceeding ten years, or
       (f) an offence punishable on summary conviction.
   Jurisdiction (2) An accused who is charged with an offence under subsection (1) may be tried and punished by any court having jurisdiction to try that offence in the place where the offence is alleged to have been committed or in the place where the accused is found, is arrested or is in custody, but where the place where the accused is found, is arrested or is in custody is outside the province in which the offence is alleged to have been committed, no proceedings in respect of that offence shall be commenced in that place without the consent of the Attorney General of that province.
   Unauthorized use of credit card data (3) Every person who, fraudulently and without colour of right, possesses, uses, traffics in or permits another person to use credit card data, including personal authentication information, whether or not the data is authentic, that would enable a person to use a credit card or to obtain the services that are provided by the issuer of a credit card to credit card holders is guilty of
       (a) an indictable offence and is liable to imprisonment for a term not exceeding ten years; or
       (b) an offence punishable on summary conviction.
   ... R.S., 1985, c. C-46, s. 342; R.S., 1985, c. 27 (1st Supp.), ss. 44, 185(F); 1997, c. 18, s. 16; 2009, c. 28, s. 4.
   – CCC
# ETD Guide/Universities/Resources List

General Information:
Association of Research Libraries. ARL Statistics and Measurement Program. http://www.arl.org/stats
Association of Research Libraries. Supplementary Statistics 1998-99. Washington, DC: ARL, 2000.
Astin, Alexander W. Assessment for Excellence The Philosophy and Practice of Assessment and Evaluation In Higher Education. American Council on Education/Oryx Press. 1991.
Developing National Library Network Statistics & Performance Measures.
EQUINOX: Library Performance Measurement and Quality Management System. http://equinox.dcu.ie/
International Coalition of Library Consortia (ICOLC). Guidelines for Statistical Measures of Usage of Web-Based Indexed, Abstracted, and Full Text Resources. November 1998. http://www.library.yale.edu/consortia/webstats.html
McClure, Charles R. and Cynthia L. Lopata. Assessing the Academic Networked Environment. Washington, DC: Coalition for Networked Information, 1996.
NDLTD: Networked Digital Library of Theses and Dissertations. http://www.ndltd.org/
Ryan, Joe. Library Statistics & Measures. http://web.syr.edu/~jryan/infopro/stats.html
Technology in Higher Education and Web Studies:
The Campus Computing Project. http://www.campuscomputing.net/
Coalition for Networked Information. Assessing the Academic Networked Environment. http://www.cni.org/projects/assessing/
Katz, Richard N. and Julia A. Rudy. Information Technology in Higher Education: Assessing Its Impact and Planning for the Future. New Directions for Institutional Research No. 102. San Francisco: Jossey-Bass, 1999.
Lyman, Peter and Hal R. Varian. How Much Information? http://www2.sims.berkeley.edu/research/projects/how-much-info-2003/
Online Computer Library Center, Inc. Web Characterization Project. http://wcp.oclc.org/
TLT Group. Flashlight Program. http://www.tltgroup.org/flashlightP.htm
University of Washington. UW Libraries Assessment. http://www.lib.washington.edu
Virginia Polytechnic Institute and State University. Electronic Thesis and Dissertation Initiative http://etd.vt.edu/
Next Section: Policy Initiatives: National, Regional, and Local; Discipline specific; Language specific
# ETD Guide/Universities/Scenarios illustrating approaches, schedules and workflow

A common workflow at a campus involved in NDLTD is as follows. Each student will use a home computer, or computer in a lab, to submit his or her ETD. His or her work will have been created on the computer he or she is working with, or will have been moved there using some media format, such as diskette or CD-RW. The student will go to a well-known URL (e.g., at Virginia Tech, will start at http://etd.vt.edu and then follow links).
First, students will provide an ID and password that authenticates them. Next, they will enter in metadata about their ETD. When that is complete and checked, they will use their browser to locate each of the files that make up the ETD, so uploading can proceed. Eventually, the entire ETD will be uploaded.
Later, a person in the graduate school will find that a new ETD has been uploaded. He or she will check the submission. If there are problems, he or she will use email to contact the student so that changes/corrections can be made, and the process repeated up through this step. Once a proper ETD is submitted, a cataloguer in the Library will be notified. He or she will catalog the work, adding in additional categories, and checking the metadata. Eventually he or she will release the work to allow proper access, according to the student/faculty preferences.
More refined workflow models can be applied. Let us suppose that at the Polytechnic University of Valencia, Spain the process starts from a catalog of ETD proposals, published by faculty members filling in the appropriate form; each proposal may include the title, keywords, abstract, level of expertise required, and other useful information. A student can apply for a number of proposals (specifying an order of preference) filling a form including his/her personal data. A faculty committee makes the final assignment of proposals to candidates. At this point, most of the ETD metadata have been collected, and there is no need to introduce them during the submission process.
Next Section: Role of the Graduate School and Graduate Program