text:
## Tuesday, October 11, 2011

### Blogs I Follow: Henry Rzepa

Just in case you have not run into Henry's blog yet, check it out. His blog makes me so jealous I did not follow up on my basic quantum chemistry education. Implementing Hartree-Fock in Fortran is not nearly as interesting or useful as the stuff he has been blogging about. A second reason you should, is his brilliant use of Jmol (Henry is one of those using Jmol for more than 10 years). This is his blog in action:

His double gravatar is not vanity, but a bug in one of blog extensions he is using creating those blue z icons. I think the French in the blog title is, while unlikely for British I always understood, deliberate.

Henry, if you're reading this... what about adding permalinks to those Jmol visualizations (or DOIs, like data cites)? Second, I'd love to be able to download to be able to download a visualization, and open it in Jmol or Bioclipse. Would such be possible in CML or JVXL (think datument)?

1. "Denouement" is the technical term for the conclusion in classic dramatic structure. We learned that term in my high school English class in the US, and its use here seems more a literary than French allusion.

2. Yes, the double Avatar is a bug! It seems to apply only to the latest post and seems to be due to the Chemicalize extension.

As for permalinks, a small number of the posts are in fact permanently archived using WebCite; http://www.webcitation.org/ It would be nice to have a WordPress extension that could do this routinely, but no such exists. Unfortunately, WebCite does not of itself allocate a DOI handle to this.

I do try to quote a DOI handle for the digital repository entry for a calculation. This would in principle allow for the entry to be scrapped by a script. See this slide via something like:

wget --quiet --no-check-certificate https://spectradspace.lib.imperial.ac.uk:8443/dspace/handle/10042/to-$T/logfile.out -O to-$T.log;

I also wrote this post which describes how data can be extracted from a Jmol instance. But that method depends on a human; to my knowledge there is no way of scripting this to happen automatically. However, Bob Hanson has achieved miracles with scripting Jmol, and it might be possible.

Clearly however, a mechanism for exposing better metadata declarations of the data contained in a blog post such that bots can acquire it without human intervention are needed. If they exist, do let me know!

3. Henry, I was more thinking about citations for the Jmol visualizations... but when you mention a repos for the calculations, you must be using Quixote, I guess... that would be a great combo and boost for Quixote!

Maybe we should talk to Sam and ask him how a Quixote page can easily be embedded in blog posts :)

Regarding the wget command... how would I learn what URL to use?

4. The first point; citations for the Jmol visualisations. Unfortunately, when Webcite archives a blog post, it does not replicate the Jmol functionality. That aspect of it is not active in the archive.

Second point. We use Dspace, and have done so for 6 years (see here). Quixote is quite new. To my knowledge it does not have Handle functionality, and in effect cannot be resolved using the same mechanisms as a DOI. Dspace has many faults, but it has been stable for us over this period, and we now have some 10,000 deposits using it. In our case, the Dspace repository is actually linked to the batch queues for our high performance computing cluster, which means the two talk to each other very effectively.

Third point. Dspace actually archives a whole collection of files. Thus see here for the files. These include an RDF declaration of appropriate metadata in the form of a METS manifest. In the wget command I listed, replace the string "logfile.out" by any of the other files you might want to acquire. These include a CML file, and possibly other useful information.

5. Ah, I did not know you had been using DSpace that active! Did you ever write up how you've integrated it into your workflow? Particularly the HPC platform?

6. An article got half written once! But I have now been asked by a forward looking publisher to write something called a data descriptor to help both prospective authors of research articles, and readers of journals, to understand the chemistry data life cycle better. Its almost finished, and it will be distributed under a CC 2.5 license. This contains something on the topic you allude to.

7. I'm looking forward to reading that!

8. You can read my take on a data-descriptor at www.ch.ic.ac.uk/rzepa/data-descriptors/.

It is not meant as a data schema, but more of a human readable data life cycle, linked to a real example. It could of course expand (possibly uncontrollably), but I would hope it could induce other chemists who create and share data to write something similar, as a way of improving the profile of molecular data amongst researchers, and not just amongst cheminformaticians.


text:
# nLab differentiation

Contents

### Context

#### Differential geometry

synthetic differential geometry

Introductions

from point-set topology to differentiable manifolds

Differentials

V-manifolds

smooth space

Tangency

The magic algebraic facts

Theorems

Axiomatics

cohesion

• (shape modality $\dashv$ flat modality $\dashv$ sharp modality)

$(\esh \dashv \flat \dashv \sharp )$

• dR-shape modality$\dashv$ dR-flat modality

$\esh_{dR} \dashv \flat_{dR}$

tangent cohesion

differential cohesion

singular cohesion

$\array{ && id &\dashv& id \\ && \vee && \vee \\ &\stackrel{fermionic}{}& \rightrightarrows &\dashv& \rightsquigarrow & \stackrel{bosonic}{} \\ && \bot && \bot \\ &\stackrel{bosonic}{} & \rightsquigarrow &\dashv& \mathrm{R}\!\!\mathrm{h} & \stackrel{rheonomic}{} \\ && \vee && \vee \\ &\stackrel{reduced}{} & \Re &\dashv& \Im & \stackrel{infinitesimal}{} \\ && \bot && \bot \\ &\stackrel{infinitesimal}{}& \Im &\dashv& \& & \stackrel{\text{étale}}{} \\ && \vee && \vee \\ &\stackrel{cohesive}{}& \esh &\dashv& \flat & \stackrel{discrete}{} \\ && \bot && \bot \\ &\stackrel{discrete}{}& \flat &\dashv& \sharp & \stackrel{continuous}{} \\ && \vee && \vee \\ && \emptyset &\dashv& \ast }$

Models

Lie theory, ∞-Lie theory

differential equations, variational calculus

Chern-Weil theory, ∞-Chern-Weil theory

Cartan geometry (super, higher)

# Contents

## Idea

Differentiation is the process that assigns to a function $f : X \to Y$ its derivative, sometimes denoted $d f$. The derivative is a function that, roughly speaking, assigns to each point $x \in X$ the linear transformation $d f_x$ that maps infinitesimal differences $y - x$ (for points $y$ infinitesimally close to $x$) to infinitesimal differences $f(y) - f(x)$.

## Definitions

### Classical (analytic) definitions

Classically, if $f: \mathbb{R}^n \to \mathbb{R}$ is a function, the derivative $d f_x$ at a point $x \in \mathbb{R}^n$ is defined by a limiting process: it is the unique (when it exists) linear transformation $L: \mathbb{R}^n \to \mathbb{R}$ such that

$\lim_{h \to 0} \frac{f(x + h) - f(x) - L h}{{|h|}} = 0$

or in other words such that

$f(y) = f(x) + L(y - x) + g(y)$

where $g(y)$ is $o({|y - x|})$ (see o-notation?). We say in that case that $f$ is differentiable at $x$, with derivative $d f_x = L$. When $n = 1$, such a linear transformation $L: \mathbb{R} \to \mathbb{R}$ is multiplication by a scalar $\lambda \in \mathbb{R}$. This is often denoted $f'(x)$, so that the derivative may be regarded as another function $f': \mathbb{R} \to \mathbb{R}$.

We say $f: \mathbb{R}^n \to \mathbb{R}$ is differentiable if it is differentiable at every point $x$ of its domain. In that case, we get a global derivative function $\mathbb{R}^n \times \mathbb{R}^n \to \mathbb{R}$ which takes a pair $(x, h)$ to $d f_x(h)$. We can then ask whether this function is itself differentiable, i.e., whether the differentiation process can be iterated. The $n$-th derivative is the result of differentiating $n$ times (the $0$-th derivative of $f$ is just $f$). We say that $f: \mathbb{R}^n \to \mathbb{R}$ is infinitely differentiable (or more briefly, that $f$ is smooth) if it has an $n$-th derivative for each natural number $n$.

Notice that a differentiable function is necessarily continuous (indeed, it is locally a Lipschitz function). One denotes the class of functions having a continuous1 $n^{th}$ derivative by $C^n(\mathbb{R}^n)$. Thus continuous functions $f: \mathbb{R}^n \to \mathbb{R}$ form a filtration

$\ldots \subset C^n(\mathbb{R}^k) \subset \ldots \subset C^1(\mathbb{R}^k) \subset C^0(\mathbb{R}^k)$

and the intersection $C^\infty(\mathbb{R}^k) \coloneqq \bigcap_{n \geq 0} C^n(\mathbb{R}^k)$ coincides with the class of smooth functions.

Evidently differentiation gives a function $D: C^{n+1}(\mathbb{R}^k) \to C^n(\mathbb{R}^k \times \mathbb{R}^k)$. The convenient setting for developing the differential calculus is to go directly to the smooth case, where we have $D: C^\infty(\mathbb{R}^k) \to C^\infty(\mathbb{R}^k \times \mathbb{R}^k)$ without having to keep track of the degree of differentiability.

This development carries over more generally to smooth manifolds $X$, which are locally (smoothly) isomorphic to Euclidean spaces $\mathbb{R}^k$.

### Differentiation as a functor

Differentiation in more conceptual terms is a functor $d : Diff \to Diff$ on the category Diff of smooth manifolds that sends

• a manifold $X$ to its tangent bundle $T X$; recall that the points of $T X$ are ordered pairs $(x, v)$ where $x \in X$ and $v$ is a tangent vector at $x$, i.e., an (augmented) derivation $v: C^\infty(X) \to \mathbb{R}$ on the algebra of smooth functions, augmented by evaluation $ev_x: C^\infty(X) \to \mathbb{R}$ at $x$;

• a smooth function $f : X \to Y$ to its differential $d f : T X \to T Y$:

if $\gamma : [-1,1] \to X$ is a path in $X$ that represents a vector $v \in T_x X$, then $(d f)(v) \in T_{f(x)} Y$ is the vector represented by the path $[-1,1] \stackrel{\gamma}{\to} X \stackrel{f}{\to} Y$.

Equivalently, a smooth map $f: X \to Y$ induces an algebra map $f^\ast = - \circ f: C^\infty(Y) \to C^\infty(X)$, and hence by composition sends a derivation $v: C^\infty(X) \to \mathbb{R}$ to a derivation $v \circ f^\ast: C^\infty(Y) \to \mathbb{R}$ augmented by $ev_{f(x)}: C^\infty(Y) \to \mathbb{R}$). Concretely: $v \circ f^\ast(\phi) = v(\phi \circ f)$ for $\phi \in C^\infty(Y)$. Then the differential $d f: T X \to T Y$ is defined by $d f(x, v) = (f(x), v \circ f^\ast)$.

According to either of these descriptions, differentiation is manifestly functorial. In the English literature, this fact is known as the “chain rule”. In more traditional form:

$d(f \circ g)_x = d f_{g(x)} \circ d g_x.$

In a context $H$ of synthetic differential geometry (SDG) the functor $d$ is simply the internal hom out of the infinitesimal interval $D$

$d = [D, -] : H \to H \,.$

where $D \hookrightarrow \mathbb{R}$ is the smooth locus defined by $x^2 = 0$. (Classically, i.e., in the topos $Set$, this locus consists of just a single point2, but in toposes that model SDG it should be regarded as an infinitesimal neighborhood about a point.) This description as internally representable functor underlies the following classical fact:

###### Theorem

Differentiation is a product-preserving functor.

For example, the product of two smooth functions $f, g: X \to \mathbb{R}$ is a composition

$X \stackrel{\Delta}{\to} X \times X \stackrel{f \times g}{\to} \mathbb{R} \times \mathbb{R} \stackrel{mult}{\to} \mathbb{R}$

and the product rule for differentiation derives by applying the functor $d$ which results in the composite

$T X \stackrel{\Delta}{\to} T X \times T X \stackrel{d f \times d g}{\to} T \mathbb{R} \times T \mathbb{R} \stackrel{d mult}{\to} T \mathbb{R}$

which reduces to computing $d mult$.

## Exposition of differentiation via infinitesimals

We attempt to give an exposition of differentiation not via taking limits in the sense of analysis, but rather, by working with infinitesimal spaces as in synthetic differential geometry.

### Setup

For that purpose we regard objects such as the real line $\mathbb{R}$ and more generally any smooth manifold as being objects of a well adapted smooth topos of smooth spaces. But in fact much less is required for most of the discussion. For the basic theory, it is sufficient to work in the canonical site of such a topos, namely, the category $SmoothLoc$ of smooth loci. This site is defined via a categorial equivalence with the opposite category of the category of smooth algebras:

$C^\infty: SmoothLoc \stackrel{\sim}{\longrightarrow} SmoothAlg^{op}$

Apart from the real line $\mathbb{R}$, the other object involved in our discussion of the basics of differentiation is the first order infinitesimal line segment $D$ (an infinitesimally thickened point), a subobject of $\mathbb{R}$. This subobject collects those points of $\mathbb{R}$ that square to zero under the canonical multiplicative structure of $\mathbb{R}$

$D = \{\epsilon \in \mathbb{R} | \epsilon^2 = 0\} \,.$

Although this is just $\{0\}$ when interpreted in the usual context of Set, here we are to interpret this in the ambient smooth topos, or equivalently just in our site $SmoothLoc$ for it. That is to say, $D$ satisfies a pullback square in $SmoothLoc$:

$\array{ D &\to& * \\ \downarrow && \downarrow^{\mathrlap{0}} \\ \mathbb{R} &\stackrel{(-)^2}{\to}& \mathbb{R} }$

Dually, the categorial equivalence $C^\infty$ from $SmoothLoc$ to the opposite category of smooth algebras carries the above pullback square to a pushout square of smooth algebras:

$\array{ C^\infty(D) &\leftarrow& C^\infty(*) \simeq \mathbb{R} \\ \uparrow && \uparrow^{\mathrlap{f \mapsto f(0)}} \\ C^\infty(\mathbb{R}) &\stackrel{x \mapsto x^2}{\leftarrow}& C^\infty(\mathbb{R}) }$

Here along the bottom and on the right are the ordinary smooth algebras of smooth functions on the smooth manifolds $\mathbb{R}^1 = \mathbb{R}$ and $\mathbb{R}^0 = *$, whereas $C^\infty(D)$ is new notation, suggestive and justified, for whatever the algebra of functions on the object $D$ is, where $D$ is defined by the fact that it has this algebra of functions. The bottom morphism sends a smooth function $f : \mathbb{R} \to \mathbb{R}$ to the function $f((-)^2) : \mathbb{R} \to \mathbb{R}$. If we write $x = Id : \mathbb{R} \to \mathbb{R}$ for the canonical coordinate function, then this sends $x$ to the function $x^2$.

Notice that the commutativity of this pushout square means that the image in $C^\infty(D)$ of the squared coordinate function $x^2 \in C^\infty(\mathbb{R})$ is zero. This is because the former is the image under the bottom morphism of the coordinate function $x$ and that maps under the right vertical morphism to $0 \in C^\infty(*) \simeq \mathbb{R}$.

In fact one can show that the pushout $C^\infty(D)$ is generated by the coordinate function $x \in C^\infty(\mathbb{R}))$ subject to the relation $x^2 = 0$: it is the ring of dual numbers

\begin{aligned} C^\infty(D) & \simeq C^\infty(\mathbb{R})/(x^2) \\ & \simeq \mathbb{R}[x]/(x^2) \end{aligned} \,.

(Here in the second step we have used Hadamard's lemma.)

This computation of the smooth algebra $C^\infty(D)$ of smooth functions on $D$ expresses the intuitive idea that $D$ is such a small neighbourhood of $0$ in $\mathbb{R}$ that while the canonical coordinate function of $\mathbb{R}$ restricted to $D$ is non-vanishing (it would vanish only when restricted entirely to the point 0) it is “so small that its square is always 0”. Readers who like this intuitive statement should keep it in mind, as it is accurate and useful, whereas readers who do not like it could stick with the above precise definition, which unwinds this slogan in the internal logic of the ambient smooth topos that we keep alluding to.

In order to remind us about the infinitesimal nature of the generator $x$ in $C^\infty(D)$ we shall usually call this “$dx$” and write

$C^\infty(D) = \mathbb{R}[dx]/((dx)^2) \simeq \mathbb{R} \oplus dx \mathbb{R} \,.$

### Functions and differential 1-forms

We discussion of differential 1-forms are functions on the space of infinitesimal paths. For more comprehensive such discussion see (Stel 13).

Define the smooth locus $\mathbb{R}^{(\Delta^1_{inf})}$ as the product $\mathbb{R} \times D$ of the real line with the first order infinitesimal line segment. We think of this as the space of infinitesimal paths in $\mathbb{R}$ (see Spaces of infinitesimal k-simplicies). We do so by thinking of a generalized element of $\mathbb{R}^{(\Delta^1_{inf})}$ in the category $SmoothLoc$, namely a pair $(x, \epsilon) \in \mathbb{R} \times D$, as the linear path in $\mathbb{R}$ stretching from $x$ to $x + \epsilon$ (both regarded as generalized elements of $\mathbb{R}$).

We say a smooth differential 1-form on $\mathbb{R}$ is a function (meaning: a morphism in $SmoothLoc$)

$\omega : X^{(\Delta^1_{inf})} \to \mathbb{R}$

with the property that its restriction to $\mathbb{R}$ vanishes, hence so that the composite morphism

$0 : \mathbb{R} \hookrightarrow X^{(\Delta^1_{inf})} \stackrel{\omega}{\to} \mathbb{R}$

vanishes, where the first morphism is the inclusion of $\mathbb{R}$ as the constant infinitesimal paths in $\mathbb{R}$, hence $(Id, 0) : \mathbb{R} \to \mathbb{R} \times D$.

We call the coproduct of smooth algebras their tensor product (classically it is the smoothly completed algebraic tensor product). For each smooth loci $X$ and $Y$, we have

$C^\infty(X \times Y) \simeq C^\infty(X) \otimes C^\infty(Y) \,.$

Using this property, we can dualize the above definition of a diferential 1-form. Namely, in the opposite category of smooth algebras, a 1-form on the line is a morphism

$C^\infty(\mathbb{R}) \otimes \mathbb{R}[dx]/((dx)^2) \leftarrow C^\infty(\mathbb{R}) : \omega^* \,,$

hence an assignment $x \mapsto g(x) + h(x) dx$ such that $g(x) = 0$. In other words, it is any element in $C^\infty(\mathrm{R}) \otimes \mathbb{R}[dx]/((dx)^2)$ that is proportional to the generator $dx$. We write $\omega = h dx$ with $h \in C^\infty(\mathbb{R})$.

### Derivatives as infinitesimal differences

We can now “synthetically” describe differentiation as the process of taking infinitesimal differences of functions.

Consider the two morphisms of smooth loci

$+ : \mathbb{R}^{(\Delta^1_{inf})} = \mathbb{R} \times D \to \mathbb{R}$

and

$p_1 : \mathbb{R}^{(\Delta^1_{inf})} = \mathbb{R} \times D \to \mathbb{R} \,,$

where the first is the restriction of the canonical addition operation $+ : \mathbb{R} \times \mathbb{R} \to \mathbb{R}$ of real numbers, and where the second is the projection on the first factor out of the product.

Given an element $f \in C^\infty(\mathbb{R})$, hence a smooth function $f : \mathbb{R} \to \mathbb{R}$, we can then form two functions on infinitesimal paths by precomposing with these two morphisms:

first we get

$f : X^{(\Delta^1_{inf})} \stackrel{p_1}{\to} \mathbb{R} \stackrel{f}{\to} \mathbb{R} \,.$

This is the function on infinitesimal paths in the line that only depends on the starting point of a path, not on the length of the path, and which sends that starting point to the value of $f$ on that point. Therefore by slight abuse of notation we just keep writing “$f$” for this function.

The second combination is

$\tilde f : X^{(\Delta^1_{inf})} \stackrel{+}{\to} \mathbb{R} \stackrel{f}{\to} \mathbb{R} \,.$

This is the function on infinitesimal paths that sends any path $(x) \to (x + \epsilon)$ to the value of the function on the endpoint $(x + \epsilon)$ of the paths: in terms of these generalized elements $\tilde f$ is the assignment

$\tilde f : (x , \epsilon) \mapsto f(x + \epsilon) \,.$

For instance consider the function $f : x \mapsto x^2$. Then

\begin{aligned} \tilde f : (x, \epsilon) &\mapsto (x + \epsilon)^2 \\ & = x^2 + 2 x \epsilon + \epsilon^2 \\ & = x^2 + 2 x \epsilon \end{aligned} \,.

This is evident, if maybe still somewhat mysterious, in the internal language of the ambient smooth topos that we keep alluding to, but to make it entirely explicit and concrete notice that the functions here are dually morphisms of smooth algebras

$C^\infty(\mathbb{R}) \otimes (\mathbb{R} \oplus dx \mathbb{R}) \leftarrow C^\infty(\mathbb{R}) : (\tilde f)^*$

and the equation

$(x + dx)^2 = x^2 + 2 x dx$

holds in $C^\infty(\mathbb{R}) \otimes (\mathbb{R} \oplus dx \mathbb{R})$ by the above discussion, where now $x \in C^\infty(\mathbb{R})$ and $dx \in C^\infty(D) = \mathbb{R} \oplus dx \mathbb{R}$ are the two canonical generators.

But notice that the expression $x^2 + 2 x dx$ is not a dfifferential 1-form by the above definition, equivalently the function $\tilde f$ does not vanish when restricted to constant infinitesimal paths.

To get such, we form the derivative or infinitesimal difference, the function

$\tilde f - f : X^{(\Delta^1_{inf})} \to \mathbb{R}$

that acts on generalized elements of the space of infinitesimal paths in the line as

$(\tilde f - f) : (x, \epsilon) \mapsto f(x + \epsilon) - f(x) \,.$

By construction on the right this now is something linear in the infinitesimal length $\epsilon$ of the path. Write $f' \in C^\infty(\mathbb{R})$ for the coefficient:

$f(x + \epsilon) - f(x) = \epsilon f'(x) \,.$

This function $f'$ is the differential of $f$. It is the function that gives the change of the function $f$ along a “unit infinitesimal path”.

Dually, we have a morphism

$C^\infty(\mathbb{R})\otimes (\mathbb{R} \oplus dx \mathbb{R}) \leftarrow C^\infty(\mathbb{R}) : (\tilde f - f)^*$

that sends

$x \mapsto f(x + dx) - f(x) = f'(x) dx \,.$

The 1-form on the right is the de Rham differential of $f$, usually written

$(d f)(x) = f'(x) dx \in C^\infty(\mathbb{R} \times D) \,.$

### As a map of tangent bundles

Equivalently we may rephrase this differential $d f$ as a morphism of tangent bundles $d f \colon T \mathbb{R} \longrightarrow T \mathbb{R}$ as follows.

First, by the Kock-Lawvere axiom of SDG we have the standard fact that $T \mathbb{R} := \mathbb{R}^D \simeq \mathbb{R} \times \mathbb{R}$. Therefore given a function

$\mathbb{R} \times D \longrightarrow \mathbb{R}$

this is by the internal hom-adjunction equivalently a function of the form

$\mathbb{R} \longrightarrow \mathbb{R}^D = T \mathbb{R} \,.$

The condition on the original function makes this adjunct be a section of the tangent bundle of $\mathbb{R}$. This section is $x \mapsto f'(x)$, hence is the derivative of $f$ regarded as a tangent vector on $\mathbb{R}$. Moreover, since $\mathbb{R}$ is a microlinear space, this induces by rescaling a function

$T \mathbb{R} = \mathbb{R} \times \mathbb{R} \longrightarrow T \mathbb{R} \,.$

This is the differential of $f$ regarded as a map of tangent bundles.

Examples of sequences of local structures

geometrypointfirst order infinitesimal$\subset$formal = arbitrary order infinitesimal$\subset$local = stalkwise$\subset$finite
$\leftarrow$ differentiationintegration $\to$
smooth functionsderivativeTaylor seriesgermsmooth function
curve (path)tangent vectorjetgerm of curvecurve
smooth spaceinfinitesimal neighbourhoodformal neighbourhoodgerm of a spaceopen neighbourhood
function algebrasquare-0 ring extensionnilpotent ring extension/formal completionring extension
arithmetic geometry$\mathbb{F}_p$ finite field$\mathbb{Z}_p$ p-adic integers$\mathbb{Z}_{(p)}$ localization at (p)$\mathbb{Z}$ integers
Lie theoryLie algebraformal grouplocal Lie groupLie group
symplectic geometryPoisson manifoldformal deformation quantizationlocal strict deformation quantizationstrict deformation quantization

## References

### General

The Kock-Lawvere axiom for the axiomatization of differentiation in synthetic differential geometry was introduced in

• Anders Kock, A simple axiomatics for differentiation, Mathematica Scandinavica Vol. 40, No. 2 (October 24, 1977), pp. 183-193 (JSTOR)

Another synthetic formalization of the chain rule in differential cohesive homotopy type theory is given in

Discussion of differential forms as functions on infinitesimal simplices is in

• Herman Stel, Cosimplicial $C^\infty$ rings and the de Rham complex of Euclidean space (arXiv:1310.7407)

### History

Discussion of the history of the concept of differentiation, with emphasis on its roots all the way back in Zeno's paradoxes of motion is in

• Carl Benjamin Boyer, The history of the Calculus and its conceptual development, Dover 1949

1. Without the continuity assumption on derivatives, the class of (once-)differentiable functions exhibits non-trivial descriptive set-theoretic complexity, as may be surmised by this paper by Kechris and Woodin. Thus the continuity assumption eliminates troublesome pathologies.

2. According to John Bell, the fact that classically this is merely a point bears comparison to Bishop Berkeley’s objections to the differential calculus he put forth in The Analyst.

Last revised on September 14, 2018 at 13:11:34. See the history of this page for a list of all contributions to it.

format: plain


text:
# Two identical tumblers are fully filled with water at a certain temperature . One of the tumbler is warmed up and the other cooled . But it is formed that water overflows from both the tumblers . The temperature at which they are fully filled is

$(a)\;4\;K\qquad(b)\;277\;K\qquad(c)\;277^{0}C\qquad(d)\;277^{0}F$

## 1 Answer

Answer : $\;277\;K$
Explanation :
$277\;K=4^{0}C$
Water expands when heated above $\;4^{0}C\;$ and also when cooled below $\;4^{0}C\;$ . water has maximum density at $\;4^{0}C$
answered Mar 24, 2014 by

1 answer

1 answer

1 answer

1 answer

1 answer

1 answer

1 answer


text:
# Wrong optimal solution

If we have a linear programming problem that is of the form as the following:

The initial tableau is the following:

Then we get this:

$\begin{matrix} B & b & P_1 & P_2 & P_3 & P_4 & P_5 & P_6 & \theta & \\ P_1 & 11 & 1 & 0 & \frac{3}{4} & 1 & \frac{1}{4} & 0 & &L_1'=L_1+ \frac{1}{2}L_2' \\ P_2 & 0 & 0 & 1 & -\frac{1}{2} & 0 & \frac{1}{2} & 0 & & L_2'=\frac{L_2}{2}\\ P_6 & 8 & 0 & 0 & 0 & 2 & 0 & 1 & &L_3'=L_3 \\ & z & 1 & 0 & 2 & 0 & 1 &0 & & L_4'=L_4+2L_2' \end{matrix}$

Any of $P_3, P_4, P_5$ we choose the solution that we will get will be non-degenerate, right?

I chose $P_3$ and I got the following tableau:

$\begin{matrix} B & b & P_1 & P_2 & P_3 & P_4 & P_5 & P_6 & & \theta\\ P_3 & \frac{44}{3} & \frac{4}{3} & 0 & 1 & \frac{4}{3} & \frac{1}{3} & 0 & 11 &L_1''=\frac{L_1' 4}{3} \\ P_2 & \frac{22}{3} & \frac{2}{3} & 1 & 0 & \frac{2}{3} & \frac{2}{3} & 0 & 11 & L_2''=L_2'+\frac{1}{2}L_1''\\ P_6 & 8 & 0 & 0 & 0 & 2 & 0 & 1 & 4 &L_3''=L_3' \\ &z & -\frac{5}{3} & 0 & 0 & -\frac{8}{3} & \frac{1}{3} & 0 & & L_4''=L_4'-2L_1'' \end{matrix}$

$-\frac{5}{3}>-\frac{8}{3}$, thus $P_4$ gets in the basis and $P_6$ gets out of the basis.

Then I got the following tableau:

$\begin{matrix} B & b & P_1 & P_2 & P_3 & P_4 & P_5 & P_6 & & \theta\\ P_3 & \frac{28}{3} & \frac{4}{3} & 0 & 1 & 0 & \frac{1}{3} & -\frac{2}{4} & \frac{28}{4} &L_1'''=L_1''-\frac{4}{3}L_3'' \\ P_2 & \frac{14}{3} & 0 & 1 & 0 & 0 & \frac{2}{3} & -\frac{1}{3} & - & L_2'''=L_2''-\frac{2}{3}L_3'''\\ P_4 & 4 & 0 & 0 & 0 & 1 & 0 & \frac{1}{2} & - &L_3'''=\frac{L_3''}{2} \\ &z & -\frac{5}{3} & 0 & 0 & 0 & \frac{1}{3} & \frac{4}{3} & & L_4'''=L_4''+\frac{8}{3}L_3''' \end{matrix}$

Then $P_1$ gets in the basis and $P_3$ gets out of the basis:

$\begin{matrix} B & b & P_1 & P_2 & P_3 & P_4 & P_5 & P_6 & & \theta\\ P_1 & \frac{28}{4} & 1 & 0 & \frac{3}{4} & 0 & \frac{1}{4} & -\frac{1}{2} & &L_1''''=\frac{L_1''' 3}{4} \\ P_2 & \frac{14}{3} & 0 & 1 & 0 & 0 & \frac{2}{3} & -\frac{1}{3} & & L_2''''=L_2'''\\ P_4 & 4 & 0 & 0 & 0 & 1 & 0 & \frac{1}{2} & - &L_3''''=L_3''' \\ &z & 0 & 0 & \frac{5}{4} & 0 & \frac{3}{4} & \frac{3}{4} & & L_4''''=L_4'''+\frac{5}{3}L_1'''' \end{matrix}$

Is everything right?

If so, do we deduce now from the fact that $z_k''''-c_k \geq 0 \forall k$ and that we have a non-degenerate basic feasible solution, that it is the optimal one?

But the optimal solution should be equal to $-7$. Are the values of one of the tableaus wrong?

EDIT: I found this now in my textbook:

Degenerate solutions:

• Obviously if $\theta_0=\min_i \left\{ \frac{x_{i0}}{x_{ij}}: x_{ij}>0\right\}$ is achieved in more than one rows ( for example $\theta_0=\frac{x_{10}}{x_{1j}}=\frac{x_{20}}{x_{2j}}$), then the corresponding solution $x_1$ is degenerate.

• If the initial solution is degenerate , then we might have had $x_{10}=0, x_{1j}>0$, thus $\theta_0=0$ and therefore $x_1=x_0$ and $z_1=z_0$, i.e. that the solution couldn't be improved.

At both of the above cases, we turn the degenerate solution to non-degenerate, replacing $0$ of the basic variable by $\epsilon>0$, arbitrarily small and we continue normally till we find the optimal solution, and then we set again $\epsilon=0$.

If we do this, will we get a different tableau?

• The solution $x_1=7$ and $x_4=4$ looks right. Therefore $z^*=-7$. You are finish with the simplex algorithm if all $c_j\geq 0$. But at the simplex algorithm you usually introduce an artificial variable if you have an equality as a constraint. Thus $a_1,a_2$ and $a_3$ are missing. – callculus Jan 4 '16 at 15:32
• @callculus Could you explain it further to me? Why do we have to introduce an artificial variable in this case where we just have equalities? – Evinda Jan 4 '16 at 15:49
• If you apply the simplex algorithm you always introduce a artificial variable for every $=$- and $\geq$-constraint. This is the algorithm. In the case of a $\geq$-constraint you additional have a negative slack variable. – callculus Jan 4 '16 at 16:20
• If we have equalities we don't have to introduce artificial variables. So why do we have to indruce one in this case? @callculus – Evinda Jan 4 '16 at 16:56
• That is not true. See here: www3.nd.edu/~dgalvin1/30210/30210_F07/presentations/… – callculus Jan 4 '16 at 17:28

Staring form the tableau with basis $P_3,P_2,P_4$, some entries are incorrect. In the tableau with basis $P_3,P_2,P_4$, the entry at column $P_1$ row $P_2$ should be $\frac23$ instead of $0$.

You may verify the GNU Octave code online.

format rat;
c = [-1 2 -3 0 0 0]'; b = [11 0 8]';
A = [
1 -.5 1 1 0 0;
0 2 -1 0 1 0;
0 0 0 2 0 1];
basis = [3 2 6]; B = A(:,basis); cB = c(basis);
T = [B\b B\A; 0 cB'*(B\A)-c']
T =

44/3        4/3          0          1        4/3        1/3          0
22/3        2/3          1          0        2/3        2/3          0
8          0          0          0          2          0          1
0       -5/3          0          0       -8/3        1/3          0
basis = [3 2 4]; B = A(:,basis); cB = c(basis);
T = [B\b B\A; 0 cB'*(B\A)-c']
T =

28/3        4/3          0          1          0        1/3       -2/3
14/3        2/3          1          0          0        2/3       -1/3
4          0          0          0          1          0        1/2
0       -5/3          0          0          0        1/3        4/3
basis = [1 2 4]; B = A(:,basis); cB = c(basis);
T = [B\b B\A; 0 cB'*(B\A)-c']
T =

7          1          0        3/4          0        1/4       -1/2
0          0          1       -1/2          0        1/2          0
4          0          0          0          1          0        1/2
0          0          0        5/4          0        3/4        1/2

Thererfore, the optimal solution is $(7,0,0,4,0,0)^T$, with an optimal value of $-7+2(0)-3(0) = -7$.

The degeneracy of the BFS and its optimality are two different things. Let me use an adapted example in another question to illustrate this.

$\max y$ subject to

\begin{align} \color{blue}{x+y}&\le\color{blue}{1}\\ \color{red}x\phantom{+y}&\le\color{red}1\\ x,y&\ge0 \end{align}

Obviously, exactly one of the blue and red constraints is redundant, so this LPP has degenerate solution.

We transform it to the standard form by adding slack variable $\color{blue}{s_1}, \color{red}{s_2}$.

$\max x$ subject to

\begin{align} \color{blue}{x+y+s_1\phantom{+s_2}}&=\color{blue}{1}\\ \color{red}{x\phantom{+y+s_1}+s_2}&=\color{red}1\\ x,y,\color{blue}{s_1},\color{red}{s_2}&\ge0 \end{align}

Thus, each of $x=0,y=0,\color{blue}{s_1=0},\color{red}{s_2=0}$ corresponds to a line which bounds the feasible region. If we choose $x,y$ as basic variables, then the basic solution is $x_B=(x,y)=(1,0)$, and we calculate the simplex tableau at this degenerate point.

format rat;
c = [0 1 0 0]'; b = [1 1]'; A = [1 1 1 0; 1 0 0 1];
basis = [1 2]; B = A(:,basis); cB = c(basis);
T = [B\b B\A; cB'*(B\b) cB'*(B\A) - c']
T =

1          1          0          0          1
-0         -0          1          1         -1
0          0          0          1         -1

Therefore, the simplex tableau is

\begin{array}{c|r|rrrr|r} B & b & x & y & s_1 & s_2 & \theta \\ \hline x & 1 & 1 & 0 & 0 & 1^* & 1 \\ y & 0 & 0 & 1 & 1 & -1 & - \\ \hline z & 0 & 0 & 0 & 1 & -1 & \end{array}

We are not trapped in the degenerate solution. Therefore, $x$ leaves the basis and $\color{red}{s_2}$ enters the basis.

basis = [4 2]; B = A(:,basis); cB = c(basis);
T = [B\b B\A; cB'*(B\b) cB'*(B\A) - c']
T =

1          1          0          0          1
1          1          1          1          0
1          1          0          1          0

Therefore, the simplex tableau is

\begin{array}{c|r|rrrr|r} B & b & x & y & s_1 & s_2 & \theta \\ \hline s_2 & 1 & 1 & 0 & 0 & 1 & - \\ y & 1 & 1 & 1 & 1 & 0 & - \\ \hline z & 1 & 1 & 0 & 1 & 0 & \end{array}

I don't think that "replacing $0$ of the basic variable by $\epsilon > 0$" is meaningful.

Suppose that "the initial solution is degenerate, then we might have had $x_{10}=0, x_{1j}>0$, thus $\theta_0=0$ and therefore $x_1=x_0$ and $z_1=z_0$, i.e. that the solution can't be improved". Then congratulations, you have found the optimal degenerate solution, if you aren't trapped in a cycle with negative entries on the last row.

I'll now demonstrate "turning the degenerate solution to non-degenerate, replacing $0$ of the basic variable by $\epsilon>0$".

We have this simplex tableau.

\begin{equation*} \begin{array}{c|r|r|rr|r} B & c_B & b & P_1 & P_j & \theta \\ \hline P_1 & * & 0 & \cdots & x_{1j} (> 0) & 0 \\ \vdots & \vdots & \vdots & \ddots & \vdots & \vdots \\ \hline & z & \star & \cdots & z_j - c_j & \end{array} \end{equation*}

Replace $0$ by $\epsilon$.

\begin{equation*} \begin{array}{c|r|r|rr|r} B & c_B & b & P_1 & P_j & \theta \\ \hline P_1 & * & \epsilon & \cdots & x_{1j} (> 0) & \epsilon/x_{1j} \\ \vdots & \vdots & \vdots & \ddots & \vdots & \vdots \\ \hline & z & \star & \cdots & z_j - c_j & \end{array} \end{equation*}

Therefore, the new objective function value is

\begin{equation*} z' = \star - \frac{z_j - c_j}{x_{1j}} \epsilon. \end{equation*}

If $z_j - c_j < 0$, you've improved the objective function value by $\dfrac{c_j - z_j}{x_{1j}} \color{red}{\epsilon}$, and you still have an arbitrarily small number $\epsilon/x_{1j}$ in the first row. Therefore, I don't understand what's the point of replacing $0$ by $\epsilon$.

• Is it necessary to replace $0$ of the basic variable by $\epsilon>0$ ?  Also I get other values of $z_0-c_0$. At the last tableau, I get it equal to $-7$ and at the other previous one $-\frac{56}{3}$. Am I wrong? – Evinda Jan 9 '16 at 13:03
• @Evinda I'm editing my answer. <del>Which one are you referring to? Where you do get $-\frac{56}{3}$?</del> You're right for the values of $z_0 - c_0 = c_B x_B$, which can be calculated by cB'*(B\b). Btw, I don't understand the point of "replacing $0$ of the basic variable by $\epsilon > 0$. – GNUSupporter 8964民主女神 地下教會 Jan 9 '16 at 16:17
• The last tableau that I got is this one: $\begin{matrix} B & c_B & b & P_1 & P_2 & P_3 & P_4 & P_5 & P_6 & \theta & \\ P_1 & -1& 7 & 1 & 0 & \frac{3}{4} & 0 & \frac{1}{4} & -\frac{1}{2} & & L_1''''=L_1''' \frac{3}{4}\\ P_2 & 2 & 0 & 0 & 1 & -\frac{1}{2} & 0 & \frac{1}{2} & 0 & & L_2''''=L_2'''-\frac{2}{3}L_1''''\\ P_4 & 0 & 4 & 0 & 0 & 0 &1 & 0 & \frac{1}{2} & & L_3''''=L_3'''\\ & z & -7 & 0 & 0 & \frac{5}{4} & 0 & \frac{3}{4} & \frac{1}{2} & & L_4''''=L_4'''+\frac{5}{3}L_1'''' \end{matrix}$  and the previous tableau at the last row where -7 is, I had found $-\frac{56}{3}$. – Evinda Jan 9 '16 at 16:26
• According to the textbook, we can replace at the intitial solution $(11,0,0,0,0,8)$ , the $0$ that corresponds to $x_5$ by $\epsilon$ and then apply the simplex method. Then at the last tableau we replace $\epsilon$ by $0$ and get the solution. But this isn't necessary, isn't it? – Evinda Jan 9 '16 at 16:29
• @Evinda Yes, your tableau is correct. I'm going to edit my answer again and explain why "replacing $\epsilon$ by $0$" is pointless. – GNUSupporter 8964民主女神 地下教會 Jan 9 '16 at 16:33


text:
# Coronographie à masque de phase: applications aux télescopes et interféromètres au sol et dans l'espace

Abstract : In this thesis work a complete study of the four quadrants coronagraph is presented. This new coronagraph offers performances of extremely large theorical rejection factor (>10^6). These performances open possibilities of exo-planets detection around nearby stars. The first laboratory results gave an attenuation rate higher than 90000 on the central peak. Thus, I naturally studied all the possible uses of this coronagraph on ground-based as well as on the space telescopes. On the former, this device can be used, coupled with high order adaptive optics for detection of brown dwarves and hot giant planets around the closest stars. On the latter, I focalized my interest on the MIRI instrument (Mid InfraRed Imager) for the NGST telescope, and on a space interferometer (for Terrestrial Planet Finder'') of NASA. Numerical simulations are presented within the framework of the Planet Finder'' ESO project. Nevertheless, an important problem must be solved: the chromatism of \pi phase-shift. Thus, I will present three solutions leading to the achromatisation of the coronagraph.
Mots-clés :
Document type :
Theses

https://tel.archives-ouvertes.fr/tel-00004234
Contributor : Pierre Riaud <>
Submitted on : Tuesday, January 20, 2004 - 3:27:52 PM
Last modification on : Thursday, December 10, 2020 - 11:09:09 AM
Long-term archiving on: : Monday, September 17, 2012 - 3:20:47 PM

### Identifiers

• HAL Id : tel-00004234, version 1

### Citation

Pierre Riaud. Coronographie à masque de phase: applications aux télescopes et interféromètres au sol et dans l'espace. Astrophysique [astro-ph]. Université Pierre et Marie Curie - Paris VI, 2003. Français. ⟨tel-00004234⟩

Record views


text:
Operating Systems Basics

As is known to the reader, an operating system is a vital component of almost every computer system.

The user is interested in using application software on his hardware. The operating system acts as a mediating layer between the application and the hardware layer. Which brings us to our first problem. How exactly is operating system defined?

Introduction

What is an operating system?

In general, there are two approaches to defining what an operating system (OS) is.

• The top-down approach
• The bottom-up approach

The Top-Down Approach

The first approach is the approach of a software developer. An average software developer does not want to be concerned with hardware intricacies, such as the rotating speed or the position of the reading head of his hard drive. And what about floppy disks that function entirely different? Here the operating system comes to the rescue and provides an easy enough to understand interface abstracting physical hardware. Eventually, the programmer has access to a set of convenient functions that allow read, write and executive access to files on the physical drive. In a sense, we can say that the operating system is a providing a virtual machine to the programmer/user that he can work on.

The Bottom-Up Approach

The second approach is the approach of a hardware manufacturer. The operating system is the layer, that manages all of the available resources. These include, but are not limited to the processor, memory, timers, network interfaces, printers and a huge variety of other peripherals. Ideally, the OS should manage the resources in the most ordered, controlled and efficient way possible. What exactly and how it manages the resources is depicted in a later section.

Design Goals of an OS

Let’s first summarize the purposes of an operating system:

• providing reasonable programming interfaces for programmers
• process management
• hardware organization
• resource management

Criterions to evaluate the quality of an operating system:

• efficiency (memory usage and time efficiency)
• reliability
• maintainability
• portability

Thus, in order to successfully realize a high-quality operating system, the following design goals are definded:

• layered structure (same reasoning as for networks)
• component interfaces universal and complete
• insensitive to machine and user errors
• consequences of errors local
• no loss of persistently saved data
• adaptable to environmental changes (new hardware, new programming languages)
• adaptable to different hardware and sofware choices

OS Types

Protection rings
One way to increase system stability and security is to prevent “misleading user behavior”, which includes both erroneous user behavior and attacks from the outside. Without delving too deep into the matter this means, that different privilege levels so-called rings are introduced. The innermost ring (ring 0) is the most privileged ring, the kernel ring. The farther a ring is away from the center, the less privileged it is.

Simplifying this model we have two privilege levels. One is the supervisor or kernel mode, the other is the user mode. This supervisor mode is a hardware-mediated flag, which is set by system-level code. Programs running in supervisor mode are trusted never to fail, which could lead to a system crash. Programs running in user mode are less privileged, but can still indirectly have access to system functions through system calls. These lines of trusted code run in supervisor mode and then return the result to the program, which continues to run in user mode. This way both security and stability of the system can be increased.

OS structures

Monolithic operating systems
Monolithic operating systems have no real structure. They consist of a set of procedures that can call each other. These procedures are linked to one big program. As a result, there’s no information hiding. However, there’s still some structure in monolithic operating systems. In order to call system functions, programs must put specific pre-defined parameters into specific places, such as the stack or processor registers. Then the user has to make a system call (also called supervisor call) with the TRAP (assembly language) command. By calling the TRAP command the desired service routine is looked up in a dispatch table. Eventually, the service routine returns a value and hands the control back to the user program. The combination of high-level languages and compilers provide convenient interfaces so that as a programmer you don’t really need to worry about this, though.

Layered systems
A more abstract, generalized version of the aforementioned monolithic approach is the layered systems approach. Similar to OSI-reference model for networks, a hierarchy of functional levels is established. E.g. if the lowest level is responsible for scheduling the processor, the levels above can act as if they were the exclusive user of the processor.

Client-Server system
Modern operating systems take a slightly different approach. User programs run as client processes in the user mode. These client processes now request service processes, which also run in user mode and return the desired results. In this model, the operating system is solely responsible for the communication between clients and servers. Advantages include the reduced scope of errors. An error in the file system server does now not affect the memory server. Another advantage is the easy adaptability to distributed systems. A client whether his message is processed on a local or a remote server.

Resource and Process Management

Motivation

The purpose of resource and process management should be quite obvious at this point. The most basic goal is to make multitasking possible. Basically, this means that you progress a process for a certain little amount of time and then go on to the next. This creates the illusion of multiple processes conducted at the same time. Enabling multitasking, you also want great sturdiness, high efficiency, approximative fairness and avoidance of errors (which otherwise can be costly).

Sturdiness in this context means, that you do not want your system to freeze if a single task fails. Our keywords here are preemption, and (the avoidance and prevention – yes there is a nuance here – of) deadlocks. Both topics are discussed in a later section.

Each time we switch the context – which is a fancy way of saying hopping between the processes that currently get processing time on the processor – we inevitably lose a small amount of time, which is required to conduct said context switch. Obviously, we can avoid these time losses by reducing the number of context switches, thereby increasing the overall efficiency – or not? It depends on how you define efficiency. This is discussed in the scheduling algorithms section.

If you did no context switching you would not be able to do multitasking at all. If you switch between processes at a very low rate or prefer specific tasks over the other that might not be fair. However, you might want to prioritize some process over another, e.g. if someone is willing to pay more for a higher priority. On the other hand, you would not want your free customers to “starve”. The intricacies of this subject are discussed in the section about priority based scheduling algorithms.

Resources and Their Classification

A resource is virtually any hardware or software “component” that is needed to complete an action. Example: If you want to print a digital file, the printer would be a hardware resource, whereas the file would be a software resource.

Here is an incomplete table of hardware and software resources:

Overview Hardware and Software Resources
Hardware Resources Software Resources
CPU, GPU, RAM, printer and other devices compiler, semaphore, process control blocks (PCB), files

These resources can be further classified as demonstrated next:

Resource Classification
Classification (2)1 Alternative 1 Alternative 2
(non-)reusable resources memory segments, datasets printing paper, writing on ROM
(in)alienable resources processors, memory, I/O devices printers
(non-)exclusive (access) resources printers reading access on files

Explanation

reusable resource: resource that can be used more than once
alienable resource: resource that can be taken away from the process
exclusive resource: resource that can only be used by one process at a time

Resource Scheduling

Resource Scheduling Goals

The most basic goal that comes to mind would be the avoidance of conflicts. For instance, you probably would not want multiple processes to access the same file and write in the same lines in unpredictable ways. Here is a more or less comprehensive list:

• avoidance of conflicts
• correct execution in limited time
• high performance (jobs/time unit)
• low dwell time (time units)
• high reliability/availability (productive time/total time)

Resource Scheduling Responsibilities

• manage existing resources
• bookkeep occupied resources
• queue resource requests
• check access permissions to resources
• select and coordinate resource scheduling strategy
• allot resources
• withdraw resources
• release unused resources

Scheduling Algorithms

Alienable resources, such as the prominent CPU can be scheduled in a way, such that different processes get some attention for a defined period of time before the previously active one is finished.

There are different scheduling algorithms available, to realize the different goals mentioned earlier.

Scheduling algorithms
abbreviation name efficient (jobs/time)? efficient (dwell time)? fair? reliable? comments
FIFO/FCFS first in, first out / first come, first serve no no partially yes easiest to implement, fair in terms of FCFS
SJF shortest job first yes yes no yes
RR Round Robin with “little” $\delta$ no no yes yes very inefficient, due to large overhead compared to processing time
RR Round Robin with “big” $\delta$ no no yes yes smaller overhead compared to RR with “little” $\delta$
priority based no no no yes
priority weight based no no partially yes makes paying for faster processing time possible, degree of fairness depends on weights

Explanation

FIFO/FCFS means that the job that comes first is processed first. This is a queue concept, which is fair in so far, that whoever comes first is served first. However, if you define fair in a way such that the every process waiting for processing time gets served every once in a while – no “starvation” – then this algorithm is very unfair. Implementation is very easy, all that is needed is a queue.

SJF will always process the shortest job available , but it will not switch to another when a shorter job comes in. This is very efficient in terms of jobs per time. If your jobs are all equally important this might be a good choice. Since it also optimizes (reduces) the number of jobs waiting for processing time it is also efficient in terms of a low dwell time.

The Round Robin algorithm is fair in so far that it periodically switches between processes. As mentioned above this is called context switching. Each time the context is switched there’s a little loss of time. The period $\delta$ determines after how many clock cycles / how much time these switches take place. A smaller $\delta$ means more context switches, thus dropping performance. On the other hand it is “more fair” due to lower “starvation times”

In case you have (time) critical processes (e.g. in nuclear power plants) you would always want to execute them first. This can be achieved by priority-based scheduling. Naturally, this scheduling algorithm cannot be considered fair.

If you want to prioritize important jobs, but still do not want to starve the others you could introduce weights and penalties to the algorithm. The scheme presented in the lecture works as following. Each job has a base priority, e.g 10. The process with the lowest (in this example the process with the lowest priority is the most important) priority gets a defined time slot of computing time. After that, a penalty of e.g. 60 points is added to only that process. Then, all processes get their priority halfed. The last step is adding the base priority to all processes.


text:
# Hat is the charge of the ion most likely to be formed from Magn to achieve noble gas notation?

Dec 5, 2016

The magnesium ion will have a charge of ${2}^{+}$ and will have the electron configuration of the noble gas neon.

#### Explanation:

Magnesium ions have a charge of ${2}^{+}$, due to the loss of its two valence electrons, giving it the noble gas configuration of neon, $\text{Ne}$.

The electron configuration of the noble gas neon, $\text{Ne}$ is $\text{1s"^2"2s"^2"2p"^6}$.

$\text{Mg}$: A neutral magnesium atom has 12 electrons arranged in the following ground state configuration: $\text{1s"^2"2s"^2"2p"^6"3s"^2}$>

${\text{Mg}}^{2 +} :$ A magnesium ion has 10 electrons arranged in the following ground state configuration: $\text{1s"^2"2s"^2"2p"^6}$.

This only occurs if magnesium is involved in ionic bonding , in which one or more electrons are transferred to one or more negatively charged ions.


text:
# Definition of Multiplication Tables

A multiplication table is a table that shows you all the results of multiplying two of the integers from $1$ to $12$.

One number is taken from a row, and the other is taken from the column. The product is found in the square where the row and column overlap.

For example, the two times multiplication table includes: $2 \times 1 = 2$, $2 \times 2 = 4$, $2 \times 3 = 6$ and so on.

Other terms for Multiplication Tables include "Times Tables" and, simply, "Tables"

### Description

The aim of this dictionary is to provide definitions to common mathematical terms. Students learn a new math skill every week at school, sometimes just before they start a new skill, if they want to look at what a specific term means, this is where this dictionary will become handy and a go-to guide for a student.

### Audience

Year 1 to Year 12 students

### Learning Objectives

Learn common math terms starting with letter M

Author: Subject Coach


text:
# survey data analysis (discrete data)

I did small survey and get such data:

|-------------| Yes | No | Dont_Know |
|-------------|     |    |           |
| Employee    | 60  | 5  | 5         |
| Workers     | 17  | 0  | 1         |
| Businessmen | 71  | 5  | 10        |
| Jobless     | 4   | 30 | 0         |

R code

dt <- data.frame(workers = c("Employee",
"Workers",
"Jobless"),
yes = c(60,17,71,4),
no = c(5,0,5,30),
dont_know = c(5,1,10,0)
)

1. What kind of test I must do, if I want to show, that the Jobless people are often choosing No answer?


text:
# Revision history [back]

Found an example in the docs after posting this... Solving Differential Equations

Found I found an example in the docs after posting this... asking: Solving Differential Equations

Applying to the example given in the question though:

var("x")
f = function("f")(x)
g = x * f
g.diff(x)

Output:

x*diff(f(x), x) + f(x)


text:
[EP-tech] EPrints Security Announcement - February 2020

I would definitely use mathjax over the cgi route.

Our server has the js added to? cfg/lang/en/templates/default.xml

<script type="text/x-mathjax-config">
MathJax.Hub.Config({tex2jax: {inlineMath: [['$','$'], ['\$','\$']]}});
</script>
<script type="text/javascript" async="async"
</script> And nothing else. Maybe that's enough to get it to work?

On 24/02/2021 14:35, John Salter via Eprints-tech wrote:
> *CAUTION:* This e-mail originated outside the University of Southampton.
> I was wondering if anyone had integrated any javascript libraries
> achieve something similar to this?
>
> Cheers,
> John
> ------------------------------------------------------------------------
> *From:* eprints-tech-bounces at ecs.soton.ac.uk
> <eprints-tech-bounces at ecs.soton.ac.uk> on behalf of Alan.Stiles via
> Eprints-tech <eprints-tech at ecs.soton.ac.uk>
> *Sent:* 24 February 2021 14:03
> *To:* eprints-tech at ecs.soton.ac.uk <eprints-tech at ecs.soton.ac.uk>
> *Subject:* Re: [EP-tech] EPrints Security Announcement - February 2020
> *CAUTION:* This e-mail originated outside the University of Southampton.
>
> The patch does leave latex2png empty.
>
> We still use this to include e.g. mathematical symbology in item
> abstracts so we have added some sanitisation to the input parameter in
> that cgi script rather than removing the function completely (3.3.15
> or 16 here).
>
> Alan
>
> *From: *<eprints-tech-bounces at ecs.soton.ac.uk> on behalf of
> "eprints-tech at ecs.soton.ac.uk" <eprints-tech at ecs.soton.ac.uk>
> *Reply to: *"eprints-tech at ecs.soton.ac.uk"
> <eprints-tech at ecs.soton.ac.uk>, James Kerwin <jkerwin2101 at gmail.com>
> *Date: *Wednesday, 24 February 2021 at 13:41
> *To: *"eprints-tech at ecs.soton.ac.uk" <eprints-tech at ecs.soton.ac.uk>,
> David R Newman <drn at ecs.soton.ac.uk>
> *Subject: *Re: [EP-tech] EPrints Security Announcement - February 2020
>
> CAUTION: This mail comes from outside the University. Please consider
> this before opening attachments, clicking links, or acting on the
> content.
>
> *CAUTION:*This e-mail originated outside the University of Southampton.
>
> Hi David,
>
> Thank you very much for bringing this to our attention and providing
> the solutions.
>
> Shamefully, we are still on 3.3.14 (I promise we are upgrading this
> year). The patch mentioned works on 3.3.16 and the page says it might
> work on earlier versions (a brief look through two of the files
> suggests they're more or less the same as those for 3.3.16)
>
> In my attempt to avoid any problems that could result from "might" are
> these the files that need altering if I were to do it manually:
>
> ?/cgi/ajax/phrase : CVE-2021-26703
>
> /cgi/latex2png : CVE-2021-3342
>
> /cgi/toolbox/toolbox : CVE-2021-26704
>
> There also appears to be some changes to be made to XML.pm
>
> Am I interpreting it correctly where it looks as though latex2png will
> be left as an empty file (deleted) by the end?
>
> I think the page makes it very clear that these are the files that are
> affected, but I just want to check there aren't any others that the
> patch addresses. I have looked at the patch, but I try not to
> underestimate my ability to totally misunderstand the most obvious of
> things.
>
> My plan is to try the command first on a test EPrints server and if it
> doesn't?work, do it manually.
>
> Thanks,
>
> James
>
> On Wed, Feb 24, 2021 at 9:27 AM David R Newman via Eprints-tech
> <eprints-tech at ecs.soton.ac.uk <mailto:eprints-tech at ecs.soton.ac.uk>>
> wrote:
>
>     Hi all,
>
>     EPrints Services was recently made aware of a small number of
>     security vulnerabilities within the EPrints codebase, affecting
>     both EPrints 3.4 and EPrints 3.3.
>
>     I have created two patch files to fix the vulnerabilities and
>
>
>
>
>     The former fixes the EPrints 3.4.2 release and the latter fixes
>     EPrints 3.3 (based on the current HEAD of
>     These links also provide instructions on how to apply the patch
>     file and some more details on the affected files.? There are
>     references to the Common Vulnerabilities and Exposure (CVE) IDs
>     but as of now these are yet to be published.? All the
>     vulnerabilities identified relate to either Cross-Site Scripting
>     (XSS) or Remote Code Execution (RCE) vulnerabilities. All of these
>     vulnerabilities would require analysis of the codebase to
>     determine an exploit.? It is very unlikely that generic tools used
>     to identify vulnerabilities would discover these, as specific
>     knowledge is required.
>
>     I have also updated to patch these vulnerabilities on both the
>     eprints and eprints3.4 GitHub repositories for the eprints
>     The next release of EPrints 3.4 (3.4.3) will have these security
>     fixes in place.
>
>     EPrints Services customers both those who EPrints Services host
>     and those that self-host have either been patched or where this
>     has not been possible, informed of the vulnerabilities and how
>     they can be fixed.
>
>     If you have any follow-up questions please feel free to ask.
>     Hopefully, the CVEs will be published shortly for those interested
>     in more detail.? However, they were raised by a third party, who I
>     have only just given go-ahead to make these public.
>
>     Regards
>
>     David Newman
>
>     Image removed by sender.
>
>
>
>
>
>     *** Options:
>     http://mailman.ecs.soton.ac.uk/mailman/listinfo/eprints-tech
>     <http://mailman.ecs.soton.ac.uk/mailman/listinfo/eprints-tech>


text:
## Tuesday, July 16, 2013

### A computational problem

There have been a good few interesting talks at GAIA2013 dealing with computation. Peter Jipsen's talk was, as usual, both fun and stimulating; today's talk by David Clark was also well worth listening to.

Talking to Peter today in one of the coffee breaks, I asked him what computational tool is currently unimplemented for universal algebraists that he would like to see. His answer: finite presentations.

To wit: if you are a group theorist, you can use GAP to investigate the following problem: consider the group$\langle X \| \mathcal{R} \rangle$given by a finite presentation. Now, it is a famous theorem that there does not exist an algorithm to determine whether two words in the generators evaluate to the same element; however, GAP tries to get "close" to this, whatever that might precisely mean.

This undecidability comes from the fact that a finitely presented group can still be infinite. But what if we're universal algebraists and we want to look at presentations with respect to a background theory strong enough to guarantee that finitely generated things are finite? Wait! We are universal algebraists, and such theories (called locally finite) are a stock in trade! In particular, presentations are the right tool for theories, like the theory of groups, which are axiomatized by equations.

Problem 1: Given a locally finite variety $$\mathcal{V}$$ in a finite language, to produce an algorithm which efficiently solves the word problem with respect to finite presentations.
INPUT:
• a finite set $$X = \{x_1, \ldots, x_n \}$$ of generators
• a finite set $$\mathcal{R} = \{ u_1 = v_1, \ldots, u_\ell = v_\ell \}$$ of relators. (Each $$u_i$$ and $$v_i$$ is a term built from the variables in $$X$$.)
• two more terms $$a,b$$ built from $$X$$.
OUTPUT:
• "True" if, whenever $$\mathbf{A}$$ is an algebra in $$\mathcal{V}$$ generated by elements $$x_1, \ldots, x_n$$ such that the terms $$u_i = v_i$$ in $$\mathbf{A}$$, the elements $$a(X)$$ and $$b(X)$$ are equal also,
• "False" otherwise.
However, as stated, I don't know of even a brute-force algorithm that solves Problem 1.

I left something out of the description of this problem, however: how we even told the computer enough about $$\mathcal{V}$$ to make sense of the problem. I'm not going to remedy that: I don't know what you'd want to give the computer, or what would be reasonable to expect for an arbitrary locally finite variety.

Exercise for the reader 2: Give an algorithm solving problem 1 in the case where we have an algorithm which, given an input $$n$$, computes the word problem for the free algebra in $$\mathcal{V}$$ on $$n$$ generators.

The situation really isn't so bad, however, because we are very seldom actually interested in a completely arbitrary locally finite variety; instead, we're usually interested in varieties of the form $\mathcal{V} = \mathrm{HSP}(\mathbf{A})$that is, algebras $$\mathbf{B}$$ which can be obtained by starting from some fixed finite algebra $$\mathbf{A}$$, taking some direct power, then taking a subalgebra, then taking a homomorphic image. (Such varieties are called finitely generated.)

Theorem 3: The free algebra on $$n$$ generators in $$\mathrm{HSP}(\mathbf{A})$$ can be found as a subalgebra of $$\mathbf{A}^{(A^n)}$$ in an algorithmic way.

Sketch of proof: Elements of $$\mathbf{A}^{(A^n)}$$ should be thought of as functions $$f: A^n \rightarrow \mathbf{A}$$. The generators we want to use are the functions $\pi_i: \begin{bmatrix} a_1 \\ \vdots \\ a_n \end{bmatrix} \mapsto a_i \qquad (1 \leq i \leq n)$. Now we just have to prove that this works.

Together, Exercise 2 and Theorem 3 give a brute-force algorithm solving Problem 1 if $$\mathcal{V}$$ is finitely generated. However, it's not a good algorithm.

Problem 1': Give a better algorithm.

Problem 4: Find conditions under which the word problem for $$\mathcal{V}$$-presentations is polynomial-time. Then find the actual algorithm that works.

Problem 4': Find conditions under which the word problem for $$\mathcal{V}$$-presentations is in NP.

Problem 5: Implement these algorithms in Python, in UACalc, or in other systems.


text:
# How to use Cambria with MiKTeX 2.8?

I'm trying to use Cambria in an article; I'm using Windows (XP, but I have the font installed), MiKTeX 2.8 and WinEdt 6.0. (Yes, I know Cambria is not available on all machines, but it's not a concern this time, it's a document for only a few people).

I have

\usepackage{fontspec}
\setmainfont{Cambria}

in my article, but receive the following error.

("C:\Program Files\MiKTeX 2.8\tex\xelatex\fontspec\fontspec.sty"
("C:\Program Files\MiKTeX 2.8\tex\latex\ifxetex\ifxetex.sty")
!
********************************************
* XeTeX is required to compile this document.
* Sorry!
********************************************.
\RequireXeTeX ...********************************}
\endgroup \fi
l.18 \RequireXeTeX

?

I've also tried to load the font without fontspec as seen at http://tex.stackexchange.com/a/10714

\font\x="Cambria" at 12 pt \x hello

Running miktex-makemf.exe...
miktex-makemf: The Cambria source file could not be found.

Running hbf2gf.exe...

hbf2gf (CJK ver. 4.8.0)

Couldn't find `Cambr.cfg'

miktex-maketfm: No creation rule for font Cambria.

l.9 \font\x="Cambria" at 12 pt
-
Have you tried to compile with XeLaTeX? – Peter Grill Nov 19 '12 at 1:45
May I remark: the actual versions are MiKTeX 2.9 and WinEdt 7. – Kurt Nov 19 '12 at 1:55
@PeterGrill: Oops, I didn't (I hadn't used it in more than a year and forgot it could be there). OK, so this time it worked but, interestingly, the output is hyphenated (it wasn't before) and in one case one letter bled over beyond the right margin. I shouldn't go astray with this different problem in the comment thread, however (if you happen to have an insight into the different output, please follow-up). If you post your suggestion as an answer, I'll accept it. Thanks again. – wishihadabettername Nov 19 '12 at 2:01
@wishihadabettername: Answer posted as requested. Sorry, I don't know about the hyphenating, My ingenious insight was solely based on the error message. :-) And, I know how you feel given that I have asked a few embarrassing questions myself. Funny thing is that I can sort of recognize beforehand when they are going to be embarrassing and on the last two I correctly predicted that the answer was going to be embarrassing. – Peter Grill Nov 19 '12 at 2:10


text:
# Finding Impulse using Momentum Principle

I am currently doing my Physics homework and am stuck on a problem that is giving me an issue.

A tennis ball has a mass of 0.057 kg. A professional tennis player hits the ball hard enough to give it a speed of 46 m/s (about 103 miles per hour.) The ball moves toward the left, hits a wall and bounces straight back to the right with almost the same speed (46 m/s). As indicated in the diagram below, high-speed photography shows that the ball is crushed about d = 2.7 cm at the instant when its speed is momentarily zero, before rebounding.

Making the very rough approximation that the large force that the wall exerts on the ball is approximately constant during contact, determine the approximate magnitude of this force.

After solving a very similar problem correctly, I took the following approach to solve it.

$$|\vec{v_{avg}}|=\frac{46}{2}=23 \frac{m}{s}$$ This is the average velocity from the time that the ball first makes contact with the wall until it comes to rest, and this answer was marked as correct.

The next step I took was to find the time it took from first contact with the wall until the ball came to rest. The question this was intended to answer was

How much time elapses between first contact with the wall, and coming to a stop?

$$t=\frac{|\vec{r}|}{|\vec{v_{avg}}|}=\frac{0.027 m}{23 \frac{m}{s}}=.0012\, s$$ However, this answer was marked incorrect, along with my answer that followed which found the total force exerted on the ball by the wall using this time.

$$\Delta\vec{p}=\vec{F_{net}}\Delta t$$ $$|\vec{F_{net}}|=\frac{\Delta|\vec{p}|}{\Delta t}=\frac{2.622 \frac{kgm}{s}}{.0012\,s}=2185\,N$$

Where did I go wrong in solving this problem?

• Note, your expression for time is incorrect. The expression should give a vector as the result, but doesn't. You are missing the numerical symbols on $\vec r$. And again on force and momentum, you are finding the magnitudes, not the vectors, even though they are written with vector arrows. – Steeven Aug 28 '16 at 22:09
• @Steeven Thank you, I believe I edited it correctly. – Chris Loonam Aug 28 '16 at 22:12
• Your calculations are correct. Where you went wrong, it seems, is in choice of teacher. – Ben51 Jan 20 '18 at 17:10

The time can be found out by using the equation $\vec{s}= \vec{u}t + \frac{1}{2}\vec{a}t^2$ where $s$ is displacement in time $t$, $u$ is the initial velocity and $a$ is acceleration :- $$0.027=46t - \frac{1}{2}at^2 ; a=\frac{46}{t}$$ Therefore $$0.027=46t - 23t$$ Thus $$t=\frac{0.027}{23}$$ So the answer is coming out to be the same.


text:
## The Annals of Probability

### Coverage processes on spheres and condition numbers for linear programming

#### Abstract

This paper has two agendas. Firstly, we exhibit new results for coverage processes. Let p(n, m, α) be the probability that n spherical caps of angular radius α in Sm do not cover the whole sphere Sm. We give an exact formula for p(n, m, α) in the case α∈[π/2, π] and an upper bound for p(n, m, α) in the case α∈[0, π/2] which tends to p(n, m, π/2) when απ/2. In the case α∈[0, π/2] this yields upper bounds for the expected number of spherical caps of radius α that are needed to cover Sm.

Secondly, we study the condition number ${\mathscr{C}}(A)$ of the linear programming feasibility problem ∃x∈ℝm+1Ax≤0, x≠0 where A∈ℝn×(m+1) is randomly chosen according to the standard normal distribution. We exactly determine the distribution of ${\mathscr{C}}(A)$ conditioned to A being feasible and provide an upper bound on the distribution function in the infeasible case. Using these results, we show that $\mathbf{E}(\ln{\mathscr{C}}(A))\le2\ln(m+1)+3.31$ for all n>m, the sharpest bound for this expectancy as of today. Both agendas are related through a result which translates between coverage and condition.

#### Article information

Source
Ann. Probab., Volume 38, Number 2 (2010), 570-604.

Dates
First available in Project Euclid: 9 March 2010

https://projecteuclid.org/euclid.aop/1268143527

Digital Object Identifier
doi:10.1214/09-AOP489

Mathematical Reviews number (MathSciNet)
MR2642886

Zentralblatt MATH identifier
1205.60027

#### Citation

Bürgisser, Peter; Cucker, Felipe; Lotz, Martin. Coverage processes on spheres and condition numbers for linear programming. Ann. Probab. 38 (2010), no. 2, 570--604. doi:10.1214/09-AOP489. https://projecteuclid.org/euclid.aop/1268143527

#### References

• [1] Agmon, S. (1954). The relaxation method for linear inequalities. Canad. J. Math. 6 382–392.
• [2] Bürgisser, P. and Amelunxen, D. (2008). Uniform smoothed analysis of a condition number for linear programming. Accepted for Math. Program. A. Available at arXiv:0803.0925.
• [3] Cheung, D. and Cucker, F. (2001). A new condition number for linear programming. Math. Program. 91 163–174.
• [4] Cheung, D. and Cucker, F. (2002). Probabilistic analysis of condition numbers for linear programming. J. Optim. Theory Appl. 114 55–67.
• [5] Cheung, D., Cucker, F. and Hauser, R. (2005). Tail decay and moment estimates of a condition number for random linear conic systems. SIAM J. Optim. 15 1237–1261.
• [6] Cucker, F. and Peña, J. (2002). A primal-dual algorithm for solving polyhedral conic systems with a finite-precision machine. SIAM J. Optim. 12 522–554.
• [7] Cucker, F. and Wschebor, M. (2002). On the expected condition number of linear programming problems. Numer. Math. 94 419–478.
• [8] Dunagan, J., Spielman, D. A. and Teng, S.-H. (2009). Smoothed analysis of condition numbers and complexity implications for linear programming. Math. Programming. To appear. Available at http://arxiv.org/abs/cs/0302011v2.
• [9] Dvoretzky, A. (1956). On covering a circle by randomly placed arcs. Proc. Natl. Acad. Sci. USA 42 199–203.
• [10] Gilbert, E. N. (1965). The probability of covering a sphere with N circular caps. Biometrika 52 323–330.
• [11] Goffin, J.-L. (1980). The relaxation method for solving systems of linear inequalities. Math. Oper. Res. 5 388–414.
• [12] Hall, P. (1985). On the coverage of k-dimensional space by k-dimensional spheres. Ann. Probab. 13 991–1002.
• [13] Hall, P. (1988). Introduction to the Theory of Coverage Processes. Wiley, New York.
• [14] Hauser, R. and Müller, T. (2009). Conditioning of random conic systems under a general family of input distributions. Found. Comput. Math. 9 335–358.
• [15] Janson, S. (1986). Random coverings in several dimensions. Acta Math. 156 83–118.
• [16] Kahane, J.-P. (1959). Sur le recouvrement d’un cercle par des arcs disposés au hasard. C. R. Math. Acad. Sci. Paris 248 184–186.
• [17] Miles, R. E. (1968). Random caps on a sphere. Ann. Math. Statist. 39 1371.
• [18] Miles, R. E. (1969). The asymptotic values of certain coverage probabilities. Biometrika 56 661–680.
• [19] Miles, R. E. (1971). Isotropic random simplices. Adv. in Appl. Probab. 3 353–382.
• [20] Moran, P. A. P. and Fazekas de St. Groth, S. (1962). Random circles on a sphere. Biometrika 49 389–396.
• [21] Motzkin, T. S. and Schoenberg, I. J. (1954). The relaxation method for linear inequalities. Canad. J. Math. 6 393–404.
• [22] Reitzner, M. (2002). Random points on the boundary of smooth convex bodies. Trans. Amer. Math. Soc. 354 2243–2278.
• [23] Rosenblatt, F. (1962). Principles of Neurodynamics. Perceptrons and the Theory of Brain Mechanisms. Spartan Books, Washington, DC.
• [24] Santaló, L. A. (1976). Integral Geometry and Geometric Probability. Addison-Wesley, Reading, MA.
• [25] Siegel, A. F. (1979). Asymptotic coverage distributions on the circle. Ann. Probab. 7 651–661.
• [26] Siegel, A. F. and Holst, L. (1982). Covering the circle with random arcs of random sizes. J. Appl. Probab. 19 373–381.
• [27] Solomon, H. (1978). Geometric Probability. SIAM, Philadelphia, PA.
• [28] Stevens, W. L. (1939). Solution to a geometrical problem in probability. Ann. Eugenics 9 315–320.
• [29] Wendel, J. G. (1962). A problem in geometric probability. Math. Scand. 11 109–111.
• [30] Whitworth, W. A. (1965). DCC Exercises in Choice and Chance. Dover, New York.
• [31] Zähle, M. (1990). A kinematic formula and moment measures of random sets. Math. Nachr. 149 325–340.


text:
Analyzing a wave

1. Sep 2, 2007

aliaze1

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

http://photo.ringo.com/230/230997145O975467146.jpg [Broken]

http://photo.ringo.com/230/230997145O975467146.jpg [Broken]

2. Relevant equations

λ=v/f
k=2π/λ
ω=2πf
T=1/f
v=ω/k
V(string) = √(Tension of string/μ), where μ = denisty

D(x,t) = A sin (kx - ωt + Φ)

3. The attempt at a solution

I found the maximum displacement as 2, found from the given equation

The third part seemed to be the next simplest, so using v=ω/k, I calculated 638/12.57 as the speed, which was incorrect

To my knowledge, this speed is needed to calculate the tension of part 1

Last edited by a moderator: May 3, 2017
2. Sep 2, 2007

3. Sep 2, 2007

aliaze1

i guess it is possible, but that is the exact question copied word-for-word

4. Sep 2, 2007

rootX

what does d(D(x,y))/dt means?

5. Sep 23, 2007

aliaze1

I just realized that the third part is not required to complete the first part,

using v=ω/k, and plugging this v into

V(string) = √(Tension of string/μ), where μ = denisty;

I get an answer of 12880.7, which is essentially 12.9 *103....the answer however is simply 12.9....

where am i going wrong?

thanks

6. Sep 23, 2007

aliaze1

I thought it had to do with the amplitude being in centimeters, so I divided by 100, but that still is 129 not 12.9...

7. Sep 23, 2007

Staff: Mentor

Make sure all units are consistent. The linear mass density is 5 g/ m as opposed to 0.05 g /cm or 0.005 kg/m. Perhaps that is where one is off by 3 or 1 order of magnitude depending on the values one uses. Tension should be in Newtons (for SI/mks).

Last edited: Sep 23, 2007
8. Sep 23, 2007

aliaze1

thanks!, makes sense