text:
## Third collection

5. Let $p$ and $q$ be relatively prime. A standard Seifert fibering of $S^1\times D^2$ corresponding to $(p,q)$ is a decomposition of $S^1\times D^2$ into disjoint circles (fibers), constructed as follows. Starting with $[0,1]\times D^2$, decomposed into the segments $[0,1]\times \{x\}$, identify the disks $\{0\}\times D^2$ and $\{1\}\times D^2$ via a $2\pi p/q$ rotation. Note that the single segment $[0,1]\times\{0\}$ becomes a (middle) fiber $S^1\times\{0\}$, while every other fiber in $S^1\times D^2$ is made from $q$ segments $[0,1]\times\{x\}$. If $q=1$ the middle fiber is called ordinary, while if $q>1$ the middle fiber is called exceptional.

A manifold $M$ is a Seifert manifold if it admits a decomposition into disjoint circles (fibers), such that each fiber has a neighborhood fibers-preservingly diffeomorphic to a neighborhood of a fiber in some standard Seifert fibering of $S^1\times D^2$. A compact Seifert fiber space has only a finite number of exceptional fibers. A Seifert manifold is often allowed to have a boundary (also fibered by circles, so it is a union of tori). Any 3-manifold supporting a foliation by circles is Seifert.

The set of fibers forms a $2$-dimensional orbifold, denoted by $B$ and called the base–also called the orbit surface–of the fibration. It has an underlying $2$-dimensional surface $B_0$, but may have some special orbifold points corresponding to the exceptional fibers.

4. If smooth distribution is non-integrable, then it cannot be $C^0$ approximated by integrable ones.

Sergei Ivanov gave the following argument for exmaple (here): consider the following 2-dimensional distribution in $\mathbb{R}^3$: the plane at $(x,y,z)\in\mathbb{R}^3$ is spanned by vectors $(1,0,0)$ and $(0,1,x)$. Perturb this distribution within a small $C^0$ distance $\epsilon\ll1$. Consider the square in $\mathbb{R}^2$ with vertices $(0,0)$, $(1,0)$, $(1,1)$, $(0,1)$ and let $\gamma$ be its boundary (counter-clockwise). This curve has a `lift’, which is a curve $\tilde{\gamma}$ in $\mathbb{R}^3$ tangent to the distribution and projects to $\gamma$. The lift is found by solving an o.d.e., so it is unique if the distribution is smooth but may be non-unique if it is only $C^0$. In the non-perturbed case, the unique lift ends at $(0,0,1)$, hence in the perturbed case all lifts from $(0,0,0)$ will end near $(0,0,1)$. This implies that the distribution is not integrable: if it was integrable, there would be at least one lift (the one contained in a leaf of a foliation) that ends near the origin.

The proof in the general case is similar.

3. Let $\phi:\Sigma_A\to\mathbb{R}$ be a continuous function, $L_\phi:C(\Sigma_A,\mathbb{R})\to C(\Sigma_A,\mathbb{R})$ be a linear operator defined as $L_\phi(f)(x)=\sum_{\sigma y=x}e^{\phi(y)}f(y)$.

Ruelle‘s Perron-Frobenius theorem: If $\phi$ is Holder continuous, then there are a positive number $\lambda>0$, a positive function $h\in C(\Sigma_A,\mathbb{R})$ and a probability measure $\nu\in\mathcal{M}(\Sigma_A)$ such that
$L_\phi(h)=\lambda\cdot h$, $L_\phi^*\nu=\lambda\cdot \nu$, $\nu(h)=1$ and the suprenorm

(*) $\|\lambda^{-m}L^m_\phi g-\nu(g)\cdot h\|\to 0$ for all Holder continuous function $g:\Sigma_A\to\mathbb{R}$.

Moreover the pressure $P(\sigma,\phi)=\log\lambda$ and $\mu=h\cdot\nu$ is the unique equilibrium state (or Gibbs measure) of $\phi$.

A priori, the existence of $\nu$ and $h$ follow from Schauder–Tychonoff theorem (so the uniqueness is not automatically guaranteed). The last characterization (*) gives us the uniqueness of the pari $(\nu,h)$: if there is another pair $(\mu,f)$, then $\mu(g)\cdot f=\nu(g)\cdot h$ for all Holder $g$. Since both $\mu$ and $\nu$ are probabilities, $\mu(1)=\nu(1)=1$ and hence $f=h$. Now $\mu=\nu$ since $\mu(g)=\nu(g)$ for all Holder $g$.

2. Let $[0,1]$ be the unit interval equiped with Lebesgue measure $m$, $(\{0,1\}^{\mathbb{N}},\sigma)$ be the full-shift and $\mu_p=(p,1-p)^{\mathbb{N}}$ be the standard ergodic measure for all $p\in(0,1)$. Let $\pi:\{0,1\}^{\mathbb{N}}\to[0,1]$ be the standard binary representation.

Since $\pi$ is almost one-to-one and $\mathrm{supp}(\mu_p)=\{0,1\}^{\mathbb{N}}$ for each $p\in(0,1)$, we see $\mathrm{supp}(m_p)=[0,1]$. But we also know $\pi(\mu_{0.5})=m$ and all other $m_p=\pi(\mu_p)$ are singular with respect to $m$. So $m_p$ (for each $p\in(0,0.5)$) is a singular continuous measure whose support is all of $[0,1]$.

1. By the local genericity of the universal dynamics at a wild homoclinic class, any diffeomorphism having a wild homoclinic class can be $C^1$–approximated by a diffeomorphism with a hyperbolic basic set that exhibits $C^1$–persistent homoclinic tangency.


text:
## Saturday, December 17, 2011

### Notes on classical sequent calculi (1/2)

These are some notes I made to try to help me understand Noam's focused presentation of classical logic in Polarity and the Logic of Delimited Continuations. I hope these notes coud be useful to others.

## Sequent presentations of classical logic

I write, inevitably, from the perspective of an intuitionistic proof theorist, so let's start there. In intuitionistic logics, sequents have the form $$\Gamma \vdash A~\mathit{true}$$, where the hypothetical context $$\Gamma$$ has the form $$A_1~\mathit{true}, \ldots, A_n~\mathit{true}$$. The whole sequent $$\Gamma \vdash A~\mathit{true}$$ is read as "assuming the truth of all the things in $$\Gamma$$, we know $$A~\mathit{true}$$." Let's look at a couple of ways of presenting sequent calculi for classical logic.

### Two-sided judgmental classical sequent calculi

One way of presenting a classical sequent calculus is to give a two-sided sequent, $$\Gamma \vdash \Delta$$. As before, $$\Gamma$$ has the form $$A_1~\mathit{true}, \ldots, A_n~\mathit{true}$$, but $$\Delta$$ has the form $$B_1~\mathit{false}, \ldots, A_m~\mathit{false}$$, and the whole sequent is read as "taken together, if all the things judged true in $$\Gamma$$ are true and all the things judged false in $$\Delta$$ are false, then there is a contradiction."

The language of propositions is $$A ::= P \mid \neg A \mid A \wedge B \mid A \vee B$$ - uninterpreted atomic propositions $$P$$, negation, conjunction ("and"), and disjunction ("or"). I'm leaving out truth $$\top$$ and falsehood $$\bot$$ because they're boring. The rules for these two-sided classical sequent calculi look like this:

$\infer {\Gamma, ~ P~\mathit{true} \vdash \Delta, ~ P~\mathit{false}} {}$ $\infer {\Gamma \vdash \Delta, ~ \neg A~\mathit{false}} {\Gamma, ~ A~\mathit{true} \vdash \Delta} \qquad \infer {\Gamma, ~ \neg A~\mathit{true} \vdash \Delta} {\Gamma \vdash \Delta, ~ A~\mathit{false}}$ $\infer {\Gamma \vdash \Delta, ~ A \wedge B~\mathit{false}} {\Gamma \vdash \Delta, ~ A~\mathit{false} &\Gamma \vdash \Delta, ~ B~\mathit{false}} \qquad \infer {\Gamma, ~ A \wedge B~\mathit{true} \vdash \Delta} {\Gamma, ~ A~\mathit{true}, ~ B~\mathit{true} \vdash \Delta}$ $\infer {\Gamma \vdash \Delta, ~ A \vee B~\mathit{false}} {\Gamma \vdash \Delta, ~ A~\mathit{false}, ~ B~\mathit{false}} \qquad \infer {\Gamma, ~ A \vee B~\mathit{true} \vdash \Delta} {\Gamma, ~ A~\mathit{true} \vdash \Delta &\Gamma, ~ B~\mathit{true} \vdash \Delta}$

Two asides. First, in presentations that do not emphasize the fact that $$A_i~\mathit{false}$$ and $$B_j~\mathit{true}$$ are judgments and not propositions, there is another reading of the two-sided sequent $A_1,\ldots,A_n \vdash B_1,\ldots,B_m$ This interpretation is that the truth of all of the $$A_i$$ implies the truth of one of the $$B_j$$ - this reading suggests a reading of any intuitionistic sequent proof as a classical sequent proof with one conclusion. You should convince yourself that this interpretation is equivalent to the interpretation above (hint: it's just a mode of use of De Morgan's laws).

Second aside: your rules may differ. I'm using a style of presentation where every connective is broken down by a unique connective and, from the perspective of bottom-up proof search, it's never a mistake to apply any rule, because the conclusion implies all of the premises (a property called invertibility). The "true" (or left) rule for conjunction (that is, "and" or $$\wedge$$) and the "false" (or right) rule for disjunction (that is, "or" or $$\vee$$) both have a different, non-invertible presentation. In the case of conjunction, it's this pair of rules: $\infer {\Gamma, ~ A \wedge B~\mathit{true} \vdash \Delta} {\Gamma, ~ A~\mathit{true} \vdash \Delta} \qquad \infer {\Gamma, ~ A \wedge B~\mathit{true} \vdash \Delta} {\Gamma, ~ B~\mathit{true} \vdash \Delta}$ You could "make a mistake" applying these rules in bottom-up proof search: just because there is a proof of $$\Gamma, ~ A \wedge B~\mathit{true} \vdash \Delta$$ does not mean that there is a proof of $$\Gamma, ~ A~\mathit{true} \vdash \Delta$$.

### One-sided judgmental sequent sequent calculi

Of course, hypotheses are just hypotheses, there's no a priori reason why we need to separate the true ones and the false ones into separate contexts. Let's use a unified context and call it $$\Psi$$. $\Psi ::= \cdot \mid \Psi, A~\mathit{true} \mid \Psi, A~\mathit{false}$ Then, we can have the sequent form $$\Psi \vdash \#$$, which we read as "all the assumptions in $$\Psi$$ together imply a contradiction" - we pronounce $$\#$$ as "contradiction." We'll need rewrite all of our rules:

$\infer {\Psi, ~ P~\mathit{true}, ~ P~\mathit{false} \vdash \#} {}$ $\infer {\Psi, ~ \neg A~\mathit{false} \vdash \#} {\Psi, ~ A~\mathit{true} \vdash \#} \qquad \infer {\Psi, ~ \neg A~\mathit{true} \vdash \#} {\Psi, ~ A~\mathit{false} \vdash \#}$ $\infer {\Psi, ~ A \wedge B~\mathit{false} \vdash \#} {\Psi, ~ A~\mathit{false} \vdash \# &\Psi, ~ B~\mathit{false} \vdash \#} \qquad \infer {\Psi, ~ A \wedge B~\mathit{true} \vdash \#} {\Psi, ~ A~\mathit{true}, ~ B~\mathit{true} \vdash \#}$ $\infer {\Psi, ~ A \vee B~\mathit{false} \vdash \#} {\Psi, ~ A~\mathit{false}, ~ B~\mathit{false} \vdash \#} \qquad \infer {\Psi, ~ A \vee B~\mathit{true} \vdash \#} {\Psi, ~ A~\mathit{true} \vdash \# &\Psi, ~ B~\mathit{true} \vdash \#}$

Hopefully you'll agree that this is "obviously the same" as the first presentation.

### One-sided, truth-oriented sequent calculi

But wait! The "false" rule for conjunction looks just like the "true" rule for disjunction, and the "true" rule for conjunction looks just like the "false" rules for disjunction. Can we simplify these rules?

The usual answer is that you can, indeed, do with fewer rules and without a false judgment at all. However, we need two twists to deal with the rules that involved both the true and false judgments. First, we need to let every atomic proposition come in two flavors, the "regular" flavor $$P$$ and the "negated" flavor $$\overline{P}$$. Then, the rule dealing with atomic propositions looks like this: $\infer {\Gamma, ~ P~\mathit{true}, ~ \overline{P}~\mathit{true} \vdash \#} {}$ Second, instead of negation being a proposition $$\neg A$$, we define a negation function, which I will write as $$(A)^\bot$$ to distinguish it from the propositional negation $$\neg A$$. W. The negation function is defined as follows: \begin{align} {(P)^\bot} = & \overline{P}\\ {(\overline{P})^\bot} = & P\\ {(A \wedge B)^\bot} = & {(A)^\bot} \vee {(B)^\bot}\\ {(A \vee B)^\bot} = & {(A)^\bot} \wedge {(B)^\bot}\\ \end{align} With this definition, we can eliminate the negation proposition altogether - the negation function just applies De Morgan laws all the way down to atomic propositions. We now get our sequent calculus for "half off" - there's no more official negation, and we don't need the false judgment at all anymore. We only need two more rules (for a total of three)!

$\infer {\Psi, ~ A \wedge B~\mathit{true} \vdash \#} {\Psi, ~ A~\mathit{true}, ~ B~\mathit{true} \vdash \#} \qquad \infer {\Psi, ~ A \vee B~\mathit{true} \vdash \#} {\Psi, ~ A~\mathit{true} \vdash \# &\Psi, ~ B~\mathit{true} \vdash \#}$

It would also be possible to play this game the other way around: gather everything on the right-hand side, bias the whole thing towards the "false" judgment, and basically get the "other half" of the two-sided sequent calculi. This ability to play the game equally well either way is part of what people mean when they say that classical logic is "very symmetric."

However, given that it's all the same, why not reason about truth and not falsehood? I've never understand why classical linear logic (in particular) always seems to bias itself towards one-sided sequent calculi on the right. There are important differences in what it means to think like a classical linear logician and what it means to think like an intuitionistic linear logician, but I really think that it unnecessarily exacerbates this divide when we have to turn all of our $$\oplus$$es to $$\&$$s and $$\otimes$$es to pars in order to talk to one another.

## Polarized presentations of classical logic

Now for the real purpose of this note: writing out the review of classical logic that Noam gives in "Polarity and the Logic of Delimited Continuations." This discussion is a synthesis of that presentation and a little bit of "On the unity of duality."

### Two for the price of two

Fundamentally, the observation Noam is making is that the one-sided truth-oriented sequent calculus goes too far - really, there are two kinds of disjunction, and two kinds of conjunction, which is why it seemed like the original calculus seemed to have redundancies. The third system above (the one-sided, truth-oriented sequent calculus) made it look like we were getting our logic for "half-off" - but really that's because the first two judgmental presentations were defining twice as many connectives as appeared to the naked eye. (As an aside, if you study classical linear logic, you're forced into the same conclusion for different reasons.)

Jason Reed taught me that, if you have two different judgments in a logic, it's worth seeing what happens if you syntactically differentiate the things you judge to be true and the things you judge to be false. Let's go ahead and "guess the right answer" - I'm going to call the things we judge to be true positive, and that the things we judge to be false negative. There's more than one of everything! \begin{align} A^- = & \neg^- A^+ \mid P^- \mid A^- \wedge^- B^- \mid A^- \vee^- B^-\\ A^+ = & \neg^+ A^- \mid P^+ \mid A^+ \wedge^+ B^+ \mid A^+ \vee^+ B^+ \end{align} Here are a bunch of rules: note that the fact that the two negations change the polarity of the propositions; the rules make it evident that this is the right thing to do, because we have (for example) $$\neg^+ A^- ~\mathit{true}$$ but $$A^- ~\mathit{false}$$:

$\infer {\Psi, ~ \neg^- A^+~\mathit{false} \vdash \#} {\Psi, ~ A^+~\mathit{true} \vdash \#} \qquad \infer {\Psi, ~ \neg^+ A^-~\mathit{true} \vdash \#} {\Psi, ~ A^-~\mathit{false} \vdash \#}$ $\infer {\Psi, ~ A^- \wedge^- B^-~\mathit{false} \vdash \#} {\Psi, ~ A^-~\mathit{false} \vdash \# &\Psi, ~ B^-~\mathit{false} \vdash \#} \qquad \infer {\Psi, ~ A^+ \wedge^+ B^+~\mathit{true} \vdash \#} {\Psi, ~ A^+~\mathit{true}, ~ B^+~\mathit{true} \vdash \#}$ $\infer {\Psi, ~ A^- \vee^- B^-~\mathit{false} \vdash \#} {\Psi, ~ A^-~\mathit{false}, ~ B^-~\mathit{false} \vdash \#} \qquad \infer {\Psi, ~ A^+ \vee^+ B^+~\mathit{true} \vdash \#} {\Psi, ~ A^+~\mathit{true} \vdash \# &\Psi, ~ B^+~\mathit{true} \vdash \#}$

So, are we good? Well, no, not really. The problem is that the "+" or "-" stuck to an atomic proposition isn't an annotation or modifier the way the overbar was in the one-sided, truth-oriented sequent calculus above. $$P^+$$ and $$P^-$$ are different atomic propositions, and it wouldn't be right to given an inference rule that had, as its conclusion, $$\Psi, ~ P^+~\mathit{true}, ~ P^-\mathit{false}$$. Why? Well, for now let's go with "because I said so." The argument I have for this point isn't bulletproof, and it has to do with the role of atomic propositions as stand-ins for other propositions.

However, if you accept my argument from authority, we are left no way to prove, or even to state, anything equivalent to the classical $$P \vee \neg P$$ into polarized logic, since any way we try to polarize this formula will lead to $$P$$ needing to be both positive and negative. We're going to need some way, different from negation, of including positive propositions in negative ones.

These inclusions of positive propositions into negative ones (and vice versa) are called shifts - $${\downarrow}A^-$$ is a positive proposition and $${\uparrow}A^+$$ is a negative proposition. We could just add these two rules and call it a day... $\infer {\Psi, ~ P^+~\mathit{true}, ~ {\uparrow}P^+~\mathit{false} \vdash \#} {} \qquad \infer {\Psi, ~ P^-~\mathit{false}, ~ {\downarrow}P^-~\mathit{true} \vdash \#} {}$ ...but this is hardly general: the rules above should be derivable; this rule should be derivable as well: $\infer {\Psi, ~ P^+~\mathit{true}, ~{\uparrow}(P^+ \vee^+ Q^+)~\mathit{false} \vdash \#} {}$ All three of these derivable rules share a common property: in an on-paper proof, we would say that the contradiction is "trivial." The hypothesis $${\uparrow}P^+~\mathit{false}$$ is trivial due to the fact that $$P^+$$ is true by a different hypothesis, and because the truth of $$P^+$$ allows us to trivially conclude that $$P^+ \vee^+ Q^+$$ is true, $$P^+ \vee^+ Q^+$$ is trivially contradictory as well.

This idea is encoded in two rules which capture proof-by-contradiction. One way we establish a contradiction is by showing that $$A^+$$ is both false (by assumption) and trivial (by direct proof). The other way we establish a contradiction is by showing that $$A^-$$ is both true (by assumption) and false (by direct proof of absurdity). These are embodied in the following two rules:

$\infer {\Psi \vdash \#} {{\uparrow}A^+~\mathit{false} \in \Psi &\Psi \vdash A^+~\mathit{trivial}} \qquad \infer {\Psi \vdash \#} {{\downarrow}A^-~\mathit{true} \in \Psi &\Psi \vdash A^-~\mathit{absurd}}$

Now, of course, we need to give a bunch more rules to describe how to prove positive propositions trivial and negative propositions absurd.

$\infer {\Psi, ~ P^+~\mathit{true} \vdash P^+~\mathit{trivial}} {} \qquad \infer {\Psi, ~ P^-~\mathit{false} \vdash P^-~\mathit{absurd}} {}$ $\infer {\Psi \vdash \neg^+ A^-~\mathit{trivial}} {\Psi \vdash A^-~\mathit{absurd}} \qquad \infer {\Psi \vdash \neg^- A^+~\mathit{absurd}} {\Psi \vdash A^+~\mathit{trivial}}$ $\infer {\Psi \vdash A^+ \wedge^+ B^+~\mathit{trivial}} {\Psi \vdash A^+~\mathit{trivial} &\Psi \vdash B^+~\mathit{trivial}} \qquad \infer {\Psi \vdash A^+ \vee^+ B^+~\mathit{trivial}} {\Psi \vdash A^+~\mathit{trivial}} \qquad \infer {\Psi \vdash A^+ \vee^+ B^+~\mathit{trivial}} {\Psi \vdash B^+~\mathit{trivial}}$ $\infer {\Psi \vdash A^- \wedge^- B^-~\mathit{absurd}} {\Psi \vdash A^-~\mathit{absurd}} \qquad \infer {\Psi \vdash A^- \wedge^- B^-~\mathit{absurd}} {\Psi \vdash B^-~\mathit{absurd}} \qquad \infer {\Psi \vdash A^- \vee^- B^-~\mathit{absurd}} {\Psi \vdash A^-~\mathit{absurd} &\Psi \vdash B^-~\mathit{absurd}}$

Even yet, we are not done! We need to deal with the shifts, which embody another form of proof-by-contradiction: to prove that $$A^-$$ holds trivially, assume it's false and derive a contradiction.

$\infer {\Psi \vdash {\downarrow}A^-~\mathit{trivial}} {\Psi, A^-~\mathit{false} \vdash \#} \qquad \infer {\Psi \vdash {\uparrow}A^+~\mathit{absurd}} {\Psi, A^+~\mathit{true} \vdash \#}$

The thing that we've come up with by this process is what I've been calling a "weakly focused" version of classical logic. If we wanted to turn it into a "fully focused" presentation of classical logic, we'd only need to make one change: the first "proof by contradiction" rules, which we call "focusing" rules, would need to require that the context $$\Psi$$ includes only judgments of the form $$P^+~\mathit{true}$$, $${\downarrow}A^-~\mathit{true}$$, $$P^-~\mathit{false}$$, and $${\uparrow}A^+~\mathit{false}$$. A context including only judgments of these four forms is called stable. To get full focusing, we would modify the "trivial focus" rule like this (a similar modification would be made to the "absurd focus" rule):

$\infer {\Psi \vdash \#} {{\uparrow}A^+~\mathit{false} \in \Psi &\Psi \vdash A^+~\mathit{trivial} &\Psi~\mathit{stable}}$

Thinking about the sequent calculus as a bottom-up proof search procedure, if we are looking for a proof of a non-stable sequent, we can use our original, invertible rules to break down the connectives in the contexts until we have only stable sequents, at which point we can apply a focusing rule.

### Until next time...

I haven't quite had time to do the thing I originally set out to do, which was to work through the notation in "Polarity and the logic of delimited continuations" better. But I will save that for another time. The motivation is the same as the one from before: it seems like we're almost certainly duplicating work. Is it possible to give the presentation of polarized classical logic from the previous section using about half as many rules?

format: plain


text:
# 1267 search results for "LaTeX"

## BCEs0

July 20, 2013
By

BCEs0 is the new R package I've written \$-\$ well, nearly finished to, anyway; it should be ready in version 1.0 in the next few days. The acronym stands for Bayesian models for Cost-Effectiveness with structural 0s, and it basically implements the mode...

## Optimising a Noisy Objective Function

July 16, 2013
By
$Optimising a Noisy Objective Function$

I am busy with a project where I need to calibrate the Heston Model to some Asian options data. The model has been implemented as a function which executes a Monte Carlo (MC) simulation. As a result, the objective function is rather noisy. There are a number of algorithms for dealing with this sort of problem, and

## Getting Started with Reproducible Research: A chapter from my new book

July 15, 2013
By

(This article was first published on Christopher Gandrud (간드루드 크리스토파), and kindly contributed to R-bloggers) This is an abridged excerpt from Chapter 2 of my new book Reproducible Research with R and RStudio. It’s published by Chapman & Hall/CRC Press. You can purchase it on Amazon. “Search inside this book” includes a complete table of contents. Researchers often start...

## Exploratory Data Analysis: Conceptual Foundations of Histograms – Illustrated with New York’s Ozone Pollution Data

Introduction Continuing my recent series on exploratory data analysis (EDA), today’s post focuses on histograms, which are very useful plots for visualizing the distribution of a data set.  I will discuss how histograms are constructed and use histograms to assess the distribution of the “Ozone” data from the built-in “airquality” data set in R.  In

## Veterinary Epidemiologic Research: Modelling Survival Data – Parametric and Frailty Models

July 5, 2013
By
$Veterinary Epidemiologic Research: Modelling Survival Data – Parametric and Frailty Models$

Last post on modelling survival data from Veterinary Epidemiologic Research: parametric analyses. The Cox proportional hazards model described in the last post make no assumption about the shape of the baseline hazard, which is an advantage if you have no idea about what that shape might be. With a parametric survival model, the survival time

## A Brief Look at Mixture Discriminant Analysis

July 2, 2013
By

Lately, I have been working with finite mixture models for my postdoctoral work on data-driven automated gating. Given that I had barely scratched the surface with mixture models in the classroom, I am becoming increasingly comfortable with them. With this in mind, I wanted to explore their application to classification because there are times when a single class is clearly made up of...

## Some Common Approaches for Analyzing Likert Scales and Other Categorical Data

July 1, 2013
By
$Some Common Approaches for Analyzing Likert Scales and Other Categorical Data$

Analyzing Likert scale responses really comes down to what you want to accomplish (e.g. Are you trying to provide a formal report with probabilities or are you trying to simply understand the data better). Sometimes a couple of graphs are sufficient and a formalize statistical test isn’t even necessary. However, with how easy it is

## Learning R: Parameter Fitting for Models Involving Differential Equations

June 30, 2013
By
$Learning R: Parameter Fitting for Models Involving Differential Equations$

It looks like MATLAB, Octave and Python seem to be the preferred tools for scientific and engineering analysis (especially those involving physical models with differential equations). However as part of my learning R experience, I wanted to check out some … Continue reading →

## R snippets for vim-SnipMate

June 26, 2013
By

Vim is my editor of choice, reasonable so, whether it be for coding C++, LaTeX or even R. I’ve used RStudio, which even has a Vim-Mode, but I still prefer to use Vim. Vim has it’s own R plugin, namely Vim-R-plugin, but this post is about snippets. SnipMate is an awesome auto-completion plugin for Vim The post R...

## Exploratory Data Analysis: 2 Ways of Plotting Empirical Cumulative Distribution Functions in R

$Exploratory Data Analysis: 2 Ways of Plotting Empirical Cumulative Distribution Functions in R$

Introduction Continuing my recent series on exploratory data analysis (EDA), and following up on the last post on the conceptual foundations of empirical cumulative distribution functions (CDFs), this post shows how to plot them in R.  (Previous posts in this series on EDA include descriptive statistics, box plots, kernel density estimation, and violin plots.) I


text:
Functions of Lebesgue Integrals

# Functions of Lebesgue Integrals

Let $f$ be a Lebesgue integrable function on $[a, b]$. We will define a new function $F : [a, b] \to \mathbb{R}$ for all $x \in [a, b]$.

(1)
\begin{align} \quad F(x) = \int_a^x f(t) \: dt \end{align}

Note that $\displaystyle{F(A) = \int_a^a f(t) \: dt = \int_{\{a\}} f(t) \: dt = 0}$.

We now state two important properties of $F$.

 Theorem 1: If $f$ is Lebesgue integrable on $[a, b]$ and $\displaystyle{F(x) = \int_a^x f(t) \: dt}$ then $F$ is uniformly continuous on $[a, b]$ and hence continuous on $[a, b]$.
• Proof: Let $\epsilon > 0$ be given. Then there exists a $\delta > 0$ such that if $E$ is a Lebesgue measurable set in $[a, b]$ with $m(E) < \delta$ then:
(2)
\begin{align} \quad \int_E |f(x)| < \epsilon \end{align}
• So for $\epsilon > 0$ if $x_1, x_2 \in [a, b]$ are such that $|x_2 - x_1| < \delta$ (and assuming without loss of generality that $x_1 < x_2$ then:
(3)
\begin{align} \quad | F(x_2) - F(x_1) | &= \biggr \lvert \int_a^{x_2} f(t) \: dt - \int_a^{x_1} f(t) \: dt \biggr \rvert \\ & \leq \biggr \lvert \int_{x_1}^{x_2} f(t) \: dt \biggr \rvert \\ & \leq \biggr \lvert \int_{[x_1, x_2]} f(t) \: dt \biggr \rvert \\ & \leq \int_{[x_1, x_2]} | f(t) | \: dt \\ & < \epsilon \end{align}
• So for all $\epsilon > 0$ there exists a $\delta > 0$ such that if $x_1, x_2 \in [a, b]$ and $| x_2 - x_1 | < \delta$ then $| F(x_2) - F(x_1) | < \epsilon$, so $F$ is uniformly continuous on $[a, b]$ and hence $F$ is continuous on $[a, b]$. $\blacksquare$
 Theorem 2: If $f$ is Lebesgue integrable on $[a, b]$ and $\displaystyle{F(x) = \int_a^x f(t) \: dt}$ then $F$ is of bounded variation on $[a, b]$.
• Proof: Let $P = \{ a = x_0, x_1, ..., x_n = b \}$ be a partition on $[a, b]$ and let $\displaystyle{M = \int_a^b | f(t) | \: dt}$. Since $f$ is Lebesgue integrable on $[a, b]$ we have that $M < \infty$ by definition. Now consider the variation of $f$ associated with $P$:
(4)
\begin{align} \quad V(P, F) &= \sum_{i=1}^{n} | F(x_i) - F(x_{i-1}) | \\ &= \sum_{i=1}^{n} \biggr \lvert \int_a^{x_i} f(t) \: dt - \int_a^{x_{i-1}} f(t) \: dt \biggr \rvert \\ &= \sum_{i=1}^{n} \biggr \lvert \int_{x_{i-1}}^{x_i} f(t) \: dt \biggr \rvert \\ & \leq \sum_{i=1}^{n} \int_{x_{i-1}}^{x_i} |f(t)| \: dt \\ & \leq \int_a^b | f(t) | \: dt \\ & \leq M \end{align}
• So for all partitions $P \in \mathscr{P}[a, b]$ we have that $V(P, F) \leq M$ and so $F$ is of bounded variation on $[a, b]$. $\blacksquare$


text:
## Thinking Mathematically (6th Edition)

If $A\subseteq B$, then $A\cap B = B$ This is false. $B$ is a superset of $A$ and can have elements that aren't in $A$. Those elements can't be in $A\cap B$, which means $A\cap B \neq B$. It should say: If $A\subseteq B$, then $A\cap B = A$


text:
preliminaries.tex 8.69 KB
 Paul Fiterau Brostean committed Feb 17, 2017 1 \section{Model learning}\label{sec:prelims}  Paul Fiterau Brostean committed Jan 20, 2017 2 \subsection{Mealy machines} \label{ssec:mealy}  Frits Vaandrager committed Feb 17, 2017 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 A \emph{Mealy machine} is a tuple $\M = (I, O, Q, q_0, \delta, \lambda)$, where $I$ is a finite set of inputs, $O$ is a finite set of outputs, $Q$ is a finite set of states, $q_0 \in Q$ is the initial state, $\delta: Q \times I \rightarrow Q$ is a transition function, and $\lambda: Q \times I \rightarrow O$ is an output function. % Output function $\lambda$ is extended to sequences of inputs by defining, for all $q \in Q$, $i \in I$ and $\sigma \in I^{\ast}$, %\begin{eqnarray*} %\delta(q, \epsilon) = q &,~~~ & \delta(q, i \sigma) = \delta(\delta(q,i), \sigma),\\ $\lambda(q, \epsilon) = \epsilon$ and $\lambda(q, i \sigma) = \lambda(q, i) \lambda(\delta(q, i), \sigma)$. %\end{eqnarray*} % The behavior of Mealy machine $\M$ is defined by function $A_\M : I^{\ast} \rightarrow O^{\ast}$ with $A_\M(\sigma) = \lambda (q^0, \sigma)$, for $\sigma \in I^{\ast}$. Mealy machines $\M$ and $\N$ are \emph{equivalent}, denoted $\M \approx \N$, iff $A_{\M} = A_{\N}$. Sequence $\sigma \in I^{\ast}$ \emph{distinguishes} $\M$ and $\N$ if and only if $A_{\M}(\sigma) \neq A_{\N}(\sigma)$.  Paul Fiterau Brostean committed Jan 18, 2017 23 24   Paul Fiterau Brostean committed Jan 20, 2017 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 %\begin{figure}[h] %\centering %\begin{tikzpicture}[>=stealth',shorten >=1pt,auto,node distance=2.8cm] % \node[initial,state] (q0) {$q_0$}; % \node[state] (q1) [right of=q0] {$q_1$}; % \node[state] (q2) [right of=q1] {$q_2$}; % % \path[->] (q0) edge node [below] {X/A} (q1); % \path[->] (q1) edge node [below] {Y/B} (q2); % \path[->] (q0) edge [bend left] node {Y/A} (q2); % \path[->] (q2) edge [loop right] node [text centered, text width = 1cm] {X/A \\ Y/B} (q2); % \path[->] (q1) edge [loop below] node {X/B} (q1); %\end{tikzpicture} %\caption{Example Mealy machine} %\label{mealy example} %\end{figure}  Paul Fiterau Brostean committed Jan 18, 2017 41   Paul Fiterau Brostean committed Jan 20, 2017 42 \subsection{MAT Framework} \label{ssec:mat}  Erik Poll committed Feb 15, 2017 43   Paul Fiterau Brostean committed May 16, 2017 44 The most efficient algorithms for model learning (see~\cite{Isberner2015} for a recent overview) all follow  Paul Fiterau Brostean committed Jan 29, 2017 45 the pattern of a \emph{minimally adequate teacher (MAT)} as proposed by Angluin~\cite{Angluin1987Learning}.  Paul Fiterau Brostean committed Feb 18, 2017 46 Here learning is viewed as a game in which a \textit{learner} has to infer an unknown automaton by asking queries to a teacher. The teacher knows the automaton, which in our setting is a Mealy machine $\M$,  Erik Poll committed Feb 15, 2017 47 also called the System Under Learning ({\dsut}).  Paul Fiterau Brostean committed Feb 18, 2017 48 Initially, the {\dlearner} only knows the input alphabet $I$ and output alphabet $O$ of $\M$.  Paul Fiterau Brostean committed Feb 18, 2017 49 The task of the {\dlearner} is to learn $\M$ via two types of queries:  Paul Fiterau Brostean committed Jan 20, 2017 50 51 \begin{itemize} \item  Paul Fiterau Brostean committed Feb 18, 2017 52 With a \emph{membership query}, the {\dlearner} asks what the response is to an input sequence $\sigma \in I^{\ast}$.  Paul Fiterau Brostean committed Jan 20, 2017 53 54 The teacher answers with the output sequence in $A_{\M}(\sigma)$. \item  Paul Fiterau Brostean committed Feb 14, 2017 55 With an \emph{equivalence query}, the {\dlearner} asks whether a hypothesized Mealy machine $\CH$ is correct, that is,  Paul Fiterau Brostean committed Jan 20, 2017 56 57 58 59 whether $\CH \approx \M$. The teacher answers \emph{yes} if this is the case. Otherwise it answers \emph{no} and supplies a \emph{counterexample}, which is a sequence $\sigma \in I^{\ast}$ that triggers a different output sequence for both Mealy machines, that is, $A_{\CH}(\sigma) \neq A_{\M}(\sigma)$. \end{itemize}  Erik Poll committed Feb 15, 2017 60 %  Frits Vaandrager committed Feb 17, 2017 61 62 The MAT framework can be used to learn black box models of software. If the behavior of a software system, or System Under Learning ({\dsut}), can be described by some unknown Mealy machine $\M$,  Paul Fiterau Brostean committed Feb 17, 2017 63 then a membership query can be implemented by sending inputs to the {\dsut} and observing resulting outputs.  Frits Vaandrager committed Feb 17, 2017 64 65 66 An equivalence query can be approximated using a conformance testing tool \cite{LeeY96} via a finite number of \emph{test queries}. A test query consists of asking the {\dsut} for the response to an input sequence $\sigma \in I^{\ast}$, similar to a membership query.  Paul Fiterau Brostean committed Feb 17, 2017 67 Note that this cannot rule out that there is more behavior  Paul Fiterau Brostean committed May 16, 2017 68 that has not been discovered.  Paul Fiterau Brostean committed Jan 18, 2017 69   Paul Fiterau Brostean committed Jan 20, 2017 70 71 72 \subsection{Abstraction} \label{ssec:mappers} Most current learning algorithms are only applicable to Mealy machines with small alphabets comprising abstract messages. Practical systems typically have parameterized input/output alphabets, whose application triggers updates on the system's state variables. To learn  Paul Fiterau Brostean committed Feb 18, 2017 73 these systems we place a \emph{mapper} between the {\dlearner} and the {\dsut}. The {\dmapper} is a transducer which translates  Frits Vaandrager committed Feb 17, 2017 74 concrete inputs to abstract inputs and concrete outputs to abstract outputs. For a thorough discussion of mappers, we refer to \cite{AJUV15}.  Paul Fiterau Brostean committed Jan 18, 2017 75   Paul Fiterau Brostean committed Jan 20, 2017 76 %Perhaps some explanation  Paul Fiterau Brostean committed Jan 18, 2017 77   Paul Fiterau Brostean committed Jan 20, 2017 78 79 80 81 82 83 84 85 86 87 88 89 90 91 92 93 94 95 96 97 98 99 100 101 102 103 104 105 106 107 108 109 110 111 112 113 114 115 116 117 118 119 120 % % %concrete inputs in $I$ to abstract inputs in $X$, %concrete outputs in $O$ to abstract outputs in $Y$, and vice versa. %alphabets % % %Starting from Angluin's seminal $L^{\ast}$ algorithm \cite{Ang87}, many %algorithms have been proposed for learning finite, deterministic Mealy machines %via a finite number of queries. We refer to \cite{Isberner2015} for recent overview. %In applications in which one wants to learn a model of a black-box reactive system, the teacher typically %consists of a System Under Learning (\dsut) that answers the membership queries, and a conformance %testing tool \cite{LeeY96} that approximates the equivalence queries using a set %of \emph{test queries}. A test query consists of asking to the \dsut for the response to an input sequence %$\sigma \in I^{\ast}$, similar to a membership query. % % % % % % %In protocol state fuzzing, a state machine of the implementation is learned by actively sending and receiving messages to and from the System Under Learning (SUL). A single message is called a \textit{query}, and a sequence of messages a \textit{trace}. The state machine is formed by sending consecutive traces to the SUL and analyzing the output. The trace to be sent is, in our case, determined by the L*-algorithm. When a trace is completed the SUL has to be reset to its initial state. % %After sending a sufficient amount of traces, a hypothesis is formed. The hypothesis is checked for by a testing oracle. Said oracle sends traces to the SUL, predicts an output through the hypothesis, and compares this output to the actual output received by the SUL. % %In our case, sending messages to the SUL means sending packets that contain, for example, the length of the packet, a sequence number or a list of supported encryptions. This information is unnecessary for the learning process, and is not supported by the Learnlib implementation of L* that we use. We abstract all this information away, leaving us with an abstract representation of messages. To convert these abstract messages to correct packets and back, a mapper is used. This mapper has to keep track of state variables, and has to be able to perform actions such as encryption and compression. This means that the mapper itself contains a state machine, which is based on existing knowledge about the protocol used in the SUL. % %\begin{figure}[h] %\centering %\includegraphics[scale=0.8]{Naamloos.png} %\caption{A learning setup} %\label{learning setup} %\end{figure} % %The setup shown in figure \ref{learning setup} is the setup Verleg used, and serves as an example for a typical learning setup, here Learner'' is the program that uses the L*-algorithm to generate traces to send to the SUL, these traces are in the form of abstract messages. The learner sends these messages to the mapper, which translates them to concrete packages which are sent to the SUL. A response of the SUL is then converted by the mapper to an abstract message and sent to the learner. % %\subsection{Secure Shell} \label{secure shell} % %The SSH-protocol uses a client-server structure consisting of three components. These components will be referred to as layers, however note that outer layers do not wrap inner layers, instead messages of different layers are distinguished through their message number. The three components are as follows: %\begin{itemize} %\item The transport layer protocol. This creates the basis for communication between server and client, providing a key exchange protocol and server authentication. The key exchange protocol is performed through three roundtrips. During the first, both client and server send a KEXINIT message. Then, the client sends a KEX30 message, the server responds with a KEX31 message. Finally, both parties send a NEWKEYS message, which indicates that the keys sent in the second step can be used. %\item The user authentication protocol. This component is used to authenticate a client to the server, for example, through a username and password combination, or through SSH-keys. %\item The connection protocol. This is used to provide different services to the connected client, it can thus multiplex the encrypted channel into different channels. The provided services can be services like file transfer or a remote terminal. Typical messages are requests for opening or closing channels, or requests for earlier named services.  Erik Poll committed Feb 15, 2017 121 %\end{itemize}


text:
# How to calculate percentage of an amount

I don't know if this is the right forum to ask, but i need to have a % from an amount, and hope there is a formula to calculate it.

Example:

Some payment methods have a 1.8% fee of the total amount a user has in their shopping basket. Now i want, that if the user selects one of those payments, that they have to pay those 1.8%. but when i charge them 1.8% extra i will still have to pay for the payment method (not all, but still a bit)

e.g. a user has shopped for € 100. Now i ask 1.8% more, that's 100 x 1.018 = 101.80. But now i have to pay 1.8% of 101.80 to my "middleman". So i still only get 101.80 * 0.982 = 99.9676. Now that doesn't seem too bad, but i would like to have a percentage that i have to put on the price so i have the right amount left when i payed the fee to my "middleman".

i hope you get what i mean. I need a % = TOTAL_AMOUNT + 1.8% when i have to pay.

If you want to have a net of $N$ left after the middleman takes 1.8% of the gross price, the gross price should be a number $G$ such that $$G\times(1-0.018) = N$$ We can solve this by dividing by $(1-0.018)$ on both sides, and get $$G = \frac{N}{1-0.018}$$
If you want to convert that to a percentage markup, calculate $$\frac{1}{1-0.018} = 1.018330$$ So you should mark your desired prices up by 1.833%.
• @Dennis: $0.018$ is just 1.8% written out in full. Your question shows you already know how to increase something by 1.8% by multiplying by $1+0.018$, and decrease something by 1.8% by multiplying by $1-0.018$, so I assumed that wouldn't confuse you. How do you think about the number $0.982$ that appears in your question, for example? – Henning Makholm Feb 4 '15 at 10:28


text:
# Does this graph problem have a formal name?

Given an undirected weighted graph where an edge exists between every pair of nodes (n1,n2) with cost C(n1,n2), find the shortest path (possibly revisiting nodes, possibly revisiting edges) through the graph that visits each node. Algo can pick any source node it wants. No need for the path to terminate back at the source node. Path cost is cumulative over all edge visits, so, if you traverse an edge with cost 5 and then traverse it again later it costs you 10 (i.e. re-traversing an already explored edge isn't free!)

• I don't know how it is called, but it seems NP-hard. You can show a reduction from Hamiltonian Path to this problem (give edges in the original graph cost $1$ and cost $n$ to edges not in the original graph; then your problem has a solution of cost $n-1$ iff the original graph had a Hamiltonian path). – Karolina Sołtys Apr 29 '11 at 17:15
• Travelling salesman problem? – Pratik Deoghare Apr 29 '11 at 17:31
• @MachineCharmer TSP doesn't allow node revisiting. – Dejas Apr 29 '11 at 18:03
• a minor terminological note --- a 'path' which can revisit nodes (or edges) is called a walk; "paths" are usually defined to exclude revisiting nodes. – Niel de Beaudrap May 1 '11 at 2:29

The TSP problem, given a graph and an edge-weight function, asks to find a tour visiting each vertex exactly once having minimum length.

I think your problem is equivalent to the Metric TSP problem, that is the TSP problem on the complete graph where the weight of each edge $uv$ is the length of a path between $u$ and $v$ with minimum weight in your original graph.

Indeed, a minimum walk on your original graph corresponds to a minimum tour in the newly defined clique, and vice-versa.

For the first side, consider a minimum walk in the original graph. If some node is repeated (say $v$ is repeated: $\ldots, v, \ldots, u, v, w, \ldots$) one can get rid of the repetition by replacing the portion $u,v,w$ by $u,w$ in the shortest path graph. Doing this for each repetition one gets a tour in the shortest path graph having the same length as the original walk. Since the walk was optimal the tour is optimal too.

For the other side, given an optimal tour of the new graph, replace each edge $uv$ of the tour by a shortest weighted path from $u$ to $v$ is the original graph. This walk may contain some repetitions, but has now the same length as the tour, and it is optimal.

• @Florent What about the case where a solver for my problem visits a node twice? Won't that make the solution to my problem inapplicable to TSP? – Dejas Apr 29 '11 at 18:05
• Florent is right – N27 Apr 29 '11 at 18:45
• @Dejas, I think what @Florent is saying is that you need to replace the cost of and edge $uv$ with the cost of the minimum path from $u$ to $v$. Then solve Metric TSP on the new graph. – bbejot Apr 29 '11 at 18:46
• Doesn't Metric TSP require that the edges form a metric? If I assign weights 1, 1, 5 to some three edges, it will be cheaper to traverse the short edges rather than take one long edge. – John Moeller Apr 29 '11 at 23:08
• @John If by 'shortest path' Florent means 'shortest weighted path' then the conversion that Florent describes in the second paragraph would eliminate the weight-5 path. – Artem Kaznatcheev Apr 30 '11 at 7:11


text:
Home

#### The Six Pillars of Calculus

A picture is worth 1000 words

#### Trigonometry Review

The basic trig functions
Basic trig identities
The unit circle
Addition of angles, double and half angle formulas
The law of sines and the law of cosines
Graphs of Trig Functions

#### Exponential Functions

Exponentials with positive integer exponents
Fractional and negative powers
The function $f(x)=a^x$ and its graph
Exponential growth and decay

#### Logarithms and Inverse functions

Inverse Functions
How to find a formula for an inverse function
Logarithms as Inverse Exponentials
Inverse Trig Functions

#### Intro to Limits

Overview
Definition
One-sided Limits
When limits don't exist
Infinite Limits
Summary

#### Limit Laws and Computations

Limit Laws
Intuitive idea of why these laws work
Two limit theorems
How to algebraically manipulate a 0/0?
Indeterminate forms involving fractions
Limits with Absolute Values
Limits involving indeterminate forms with square roots
Limits of Piece-wise Functions
The Squeeze Theorem

#### Continuity and the Intermediate Value Theorem

Definition of continuity
Continuity and piece-wise functions
Continuity properties
Types of discontinuities
The Intermediate Value Theorem
Summary of using continuity to evaluate limits

#### Limits at Infinity

Limits at infinity and horizontal asymptotes
Limits at infinity of rational functions
Which functions grow the fastest?
Vertical asymptotes (Redux)
Summary and selected graphs

#### Rates of Change

Average velocity
Instantaneous velocity
Computing an instantaneous rate of change of any function
The equation of a tangent line
The Derivative of a Function at a Point

#### The Derivative Function

The derivative function
Sketching the graph of $f'$
Differentiability
Notation and higher-order derivatives

#### Basic Differentiation Rules

The Power Rule and other basic rules
The derivative of $e^x$

#### Product and Quotient Rules

The Product Rule
The Quotient Rule

#### Derivatives of Trig Functions

Necessary Limits
Derivatives of Sine and Cosine
Derivatives of Tangent, Cotangent, Secant, and Cosecant
Summary

#### The Chain Rule

Two Forms of the Chain Rule
Version 1
Version 2
Why does it work?
A hybrid chain rule

#### Implicit Differentiation

Introduction
Examples
Derivatives of Inverse Trigs via Implicit Differentiation
A Summary

#### Derivatives of Logs

Formulas and Examples
Logarithmic Differentiation

In Physics
In Economics
In Biology

#### Related Rates

Overview
How to tackle the problems

#### Linear Approximation and Differentials

Overview
Examples
An example with negative $dx$

#### Differentiation Review

How to take derivatives
Basic Building Blocks
Product and Quotient Rules
The Chain Rule
Combining Rules
Implicit Differentiation
Logarithmic Differentiation
Conclusions and Tidbits

#### Absolute and Local Extrema

Definitions
The Extreme Value Theorem
Critical Numbers
Steps to Find Absolute Extrema

#### The Mean Value and other Theorems

Rolle's Theorems
The Mean Value Theorem
Finding $c$

#### $f$ vs. $f'$

Increasing/Decreasing Test and Critical Numbers
Process for finding intervals of increase/decrease
The First Derivative Test
Concavity
Concavity, Points of Inflection, and the Second Derivative Test
The Second Derivative Test
Visual Wrap-up

#### Indeterminate Forms and L'Hospital's Rule

What does $\frac{0}{0}$ equal?
Examples
Indeterminate Differences
Indeterminate Powers
Three Versions of L'Hospital's Rule
Proofs

Strategies
Another Example

#### Newton's Method

The Idea of Newton's Method
An Example
Solving Transcendental Equations
When NM doesn't work

#### Anti-derivatives

Antiderivatives
Common antiderivatives
Initial value problems
Antiderivatives are not Integrals

#### The Area under a curve

The Area Problem and Examples
Riemann Sum Notation
Summary

#### Definite Integrals

Definition of the Integral
Properties of Definite Integrals
What is integration good for?
More Applications of Integrals

#### The Fundamental Theorem of Calculus

Three Different Concepts
The Fundamental Theorem of Calculus (Part 2)
The Fundamental Theorem of Calculus (Part 1)
More FTC 1

#### The Indefinite Integral and the Net Change

Indefinite Integrals and Anti-derivatives
A Table of Common Anti-derivatives
The Net Change Theorem
The NCT and Public Policy

#### Substitution

Substitution for Indefinite Integrals
Examples to Try
Revised Table of Integrals
Substitution for Definite Integrals
Examples

#### Area Between Curves

Computation Using Integration
To Compute a Bulk Quantity
The Area Between Two Curves
Horizontal Slicing
Summary

#### Volumes

Slicing and Dicing Solids
Solids of Revolution 1: Disks
Solids of Revolution 2: Washers
More Practice

### The derivative of $e^x$

The function $f(x) = e^x$ is quite peculiar: it is the only function whose derivative is itself.

 $\displaystyle\frac{d}{dx}(e^x)=e^x$.  The derivative of $e^x$ is $e^x$.

Perhaps $(e^x)'$ is now your favorite derivative.

DO:  Find the derivative of $g(x)=5\cdot e^x$.

What follows is the reasoning behind why $(e^x)'=e^x$.  Please enjoy the following work and video if you are interested in this unusual function.

We have \begin{eqnarray*} f'(x) & = & \lim_{h \to 0} \frac{f(x+h)-f(x)}{h} \cr & = & \lim_{h \to 0} \frac{e^{x+h} - e^x}{h} \cr & = & \lim_{h \to 0} \frac{e^xe^h - e^x}{h} \cr & = & e^x \cdot\left(\lim_{h \to 0} \frac{e^h - 1}{h}\right). \end{eqnarray*} The last step was made possible by the fact that $e^x$ doesn't depend on $h$.  We are left with $$\lim_{h \to 0} \frac{e^h - 1}{h}.$$ This limit is 1, which you can guess by exploring this limit graphically or numerically.  Thus, the derivative of $e^x$ is $e^x$.


text:
#### Interactive Label Cleaning with Example-based Explanations

##### Stefano Teso, Andrea Bontempelli, Fausto Giunchiglia, Andrea Passerini

We tackle sequential learning under label noise in applications where a human supervisor can be queried to relabel suspicious examples. Existing approaches are flawed, in that they only relabel incoming examples that look suspicious'' to the model. As a consequence, those mislabeled examples that elude (or don't undergo) this cleaning step end up tainting the training data and the model with no further chance of being cleaned. We propose Cincer, a novel approach that cleans both new and past data by identifying pairs of mutually incompatible examples. Whenever it detects a suspicious example, Cincer identifies a counter-example in the training set that -- according to the model -- is maximally incompatible with the suspicious example, and asks the annotator to relabel either or both examples, resolving this possible inconsistency. The counter-examples are chosen to be maximally incompatible, so to serve as explanations of the model' suspicion, and highly influential, so to convey as much information as possible if relabeled. Cincer achieves this by leveraging an efficient and robust approximation of influence functions based on the Fisher information matrix (FIM). Our extensive empirical evaluation shows that clarifying the reasons behind the model's suspicions by cleaning the counter-examples helps acquiring substantially better data and models, especially when paired with our FIM approximation.

arrow_drop_up


text:
# 3. Provide a mechanism for the following reaction. (This a hint to question 2) 5 points...

###### Question:

3. Provide a mechanism for the following reaction. (This a hint to question 2) 5 points OH HO 4. Draw a mechanism and provide the most favorable product from the reaction below. 5 points Br NaOH

#### Similar Solved Questions

##### What is causing the variation shown in each graph. Only use the options: genotype genotype +...
What is causing the variation shown in each graph. Only use the options: genotype genotype + environment genotype x environment Question 12 options: Save Genotype A Genotype B Genotype A Genotype B e Genotype A Genotype B Environment Environment Environment...
##### Lena Company has provided the following data for its 2019 operations (ignore income taxes): 2019 revenues...
Lena Company has provided the following data for its 2019 operations (ignore income taxes): 2019 revenues were $83,000. 2019 expenses were$49,200. Dividends declared and paid during 2019 totaled $8,300. Total assets at December 31, 2019 were$191,000. Total liabilities at December 31, 2019 were \$11...
##### PREVIOUS BALANCE 16,500 TOTAL CHECKS AND DEBITS 9,718 TOTAL DEPOSITS AND CREDITS 11,479 CURRENT BALANCE 18,...
PREVIOUS BALANCE 16,500 TOTAL CHECKS AND DEBITS 9,718 TOTAL DEPOSITS AND CREDITS 11,479 CURRENT BALANCE 18, 261 CHECKS AND DEBITS Date No. Amount 09/03 5888 1,019 09/04 5902 711 09/07 5901 1,836 09/17 652 NSF 09/20 5905 928 09/22 5903 375 09/22 5904 2,065 09/28 5907 268 09/29 5909 1,864 DEPOSITS AND...
##### The left side of an account is O the balance of the account. O the debit...
The left side of an account is O the balance of the account. O the debit side. O blank. O a description of the account....
##### A genetic marker that is used in DNA profiling is D8S1179. It has two polymorphisms with...
A genetic marker that is used in DNA profiling is D8S1179. It has two polymorphisms with allele sizes 13 and 14. What distinguishes the two alleles found in the human population?...
##### True or False Body substance isolation reduces your risk of transmitting bloodborne pathogens. True or False...
True or False Body substance isolation reduces your risk of transmitting bloodborne pathogens. True or False Use Standard Precautions any time vou...
##### A four-resistor circuit is shown in the figure. The values of the resistors are as follows:...
A four-resistor circuit is shown in the figure. The values of the resistors are as follows: R1=4 Ohms, R2=R3=R4=3 Ohms. What is the equivalent resistance Req for this circuit? RA M my R1 R2 R3 O 0.8 ohms 1.33 ohms O 3.5 ohms O 2.5 ohms O 0.67 ohms...
##### Instructions: For each of the following scenarios, categorize each part of the HIM process being described....
Instructions: For each of the following scenarios, categorize each part of the HIM process being described. In each scenario, identify: • Process • External benchmark (not found in every case) • Internal benchmark (not found in every case) • Process measure • Outcome measure...
##### How do you find the product 6g^2(3g^3+4g^2+10g-1)?
How do you find the product 6g^2(3g^3+4g^2+10g-1)?...
##### Equity multiplier Initech finances their business using a combination of equity and liabilities. Which of the...
equity multiplier Initech finances their business using a combination of equity and liabilities. Which of the following numbers is most likely Initech's equity multiplier? 0 1.5...
##### 3) (1.25 point) Find the center of mass of the lamina that occupies the region R...
3) (1.25 point) Find the center of mass of the lamina that occupies the region R with the given density function. 4 R = 0, y = 4}; 8(x,y) = kx?...
##### How do you find the slope perpendicular to x = 0?
How do you find the slope perpendicular to x = 0?...
##### F(x)=x-3(x)^(2/3)+3(x)^(1/3),solve in R+ the equation f(x)=x ?
f(x)=x-3(x)^(2/3)+3(x)^(1/3),solve in R+ the equation f(x)=x ?...
##### What is the equation of the line normal to f(x)=6x^3-8 at x=7?
What is the equation of the line normal to f(x)=6x^3-8 at x=7?...


text:
# Proof for primitive pythagorean triples

I was wondering if there was a proof that any integer pythagorean triple can be represented as a positive integer multiple of a primitive pythagorean triple. This seems quite related to the fundamental theorem of arithmetic, so I was wondering if the proof would be similar.

• How are you defining a primitive pythagorean triple? – lulu Feb 2 '16 at 15:48
• This looks like a circular argument: either the numbers in the triple are not coprimes and you can simplify, or they are coprimes and you call them primitive. – Yves Daoust Feb 2 '16 at 15:51

Hint: If you have a pythgorean triple $(a,b,c)$ divide each number by $gcd(a,b,c)$ and remember if $p$ divides any two of the new numbers it must divide the third.

This is trivial from the definition.

We say that $(a,b,c)$ is a primitive Pythagorean triple if it is a Pythagorean Triple and there do not exist $(a',b',c')$, also a Pythagorean Triple, such that $\exists k$ such that $a=ka'$ etc.

Let $(a,b,c)$ be a Pythagorean triple. If it's primative, $k=1$ and we are done. If it's not, the. $\exists k$ such that...

It is known all the pythagorean triples are given by $$(\lambda(s^3-t^2), \lambda(2st),\lambda(s^2+t^2))$$ where $(s,t)=1$ and $\lambda\in \mathbb N$. The answer to your question it clearly follows.

A multiple of a Pythagorean triple is also a Pythagorean triple, $$a^2+b^2=c^2\iff(na)^2+(nb)^2=(nc)^2.$$

There is nothing more to say.

• The downvoter is wrong. – Yves Daoust Feb 2 '16 at 20:32


text:
Forums | Mahara Community

Developers / Edit Access - different collections/ pages

Posts: 6

09 June 2015, 2:27 AM

Hello,

we noticed that it would be much more comfortable if you would see the full title of your single collection/page if you edit the access of collections/pages, especially if you have different collections/pages...

Our question is:  Do you have any ideas where to look for modifying this? Maybe it is possible to modify  the length of the displayed title in the front-end? Or maybe you know in which php-file we have to look?

Thanks!

Kind regards,

Katharina

Posts: 896
Re: Edit Access - different collections/ pages

24 June 2015, 12:04 PM

Hi Katharina,

This one's buried pretty deep in the code. Those checkboxes are generated by a Pieforms element called "checkboxes". This element type shortens the label for every checkbox to 17 characters by default. You can override this by adding a "labelwidth" setting to the element, which can be set to 0 to indicate there should be no limit.

The place where these checkboxes get set up, is in the form declaration in htdocs/view/access.php. Here's what you need to change, specifically:

--- a/htdocs/view/access.php+++ b/htdocs/view/access.php@@ -95,6 +95,7 @@ if (!empty($collections)) { 'type' => 'checkboxes', 'title' => get_string('Collections', 'collection'), 'elements' =>$collections,+        'labelwidth'   => 0,     ); } @@ -110,6 +111,7 @@ if (!empty($views)) { 'type' => 'checkboxes', 'title' => get_string('views'), 'elements' =>$views,+        'labelwidth'   => 0,     ); }

Cheers,

Aaron

2 results


text:
2.0 3.0 > comp > gb.opengl > gl > loadmatrixf
Previous  Next  Edit  Rename  Undo  Refresh  Search  Administration
Documentation
 History

`Static Sub LoadMatrixf ( Matrix As Float[] )`

Replace the current matrix with the specified matrix.

### Parameters

m

Specifies a pointer to 16 consecutive values, which are used as the elements of a $4×4$ column-major matrix.

### Description

Gl.LoadMatrix replaces the current matrix with the one whose elements are specified by m. The current matrix is the projection matrix, modelview matrix, or texture matrix, depending on the current matrix mode (see Gl.MatrixMode).

The current matrix, M, defines a transformation of coordinates. For instance, assume M refers to the modelview matrix. If $\mathit{v}=\left(\mathit{v}\left[0\right],\mathit{v}\left[1\right],\mathit{v}\left[2\right],\mathit{v}\left[3\right]\right)$ is the set of object coordinates of a vertex, and m points to an array of $16$ single- or double-precision floating-point values $\mathit{m}=\left\{\mathit{m}\left[0\right],\mathit{m}\left[1\right],\mathit{...},\mathit{m}\left[15\right]\right\}$, then the modelview transformation $\mathit{M}\left(\mathit{v}\right)$ does the following:

$\mathit{M}\left(\mathit{v}\right)=\left(\begin{array}{cccc}\mathit{m}\left[0\right]& \mathit{m}\left[4\right]& \mathit{m}\left[8\right]& \mathit{m}\left[12\right]\\ \mathit{m}\left[1\right]& \mathit{m}\left[5\right]& \mathit{m}\left[9\right]& \mathit{m}\left[13\right]\\ \mathit{m}\left[2\right]& \mathit{m}\left[6\right]& \mathit{m}\left[10\right]& \mathit{m}\left[14\right]\\ \mathit{m}\left[3\right]& \mathit{m}\left[7\right]& \mathit{m}\left[11\right]& \mathit{m}\left[15\right]\end{array}\right)×\left(\begin{array}{c}\mathit{v}\left[0\right]\\ \mathit{v}\left[1\right]\\ \mathit{v}\left[2\right]\\ \mathit{v}\left[3\right]\end{array}\right)$

Projection and texture transformations are similarly defined.

### Notes

While the elements of the matrix may be specified with single or double precision, the GL implementation may store or operate on these values in less than single precision.

### Errors

Gl.INVALID_OPERATION is generated if Gl.LoadMatrix is executed between the execution of Gl.Begin and the corresponding execution of Gl.End.

### Associated Gets

Gl.Get with argument Gl.MATRIX_MODE

Gl.Get with argument Gl.COLOR_MATRIX

Gl.Get with argument Gl.MODELVIEW_MATRIX

Gl.Get with argument Gl.PROJECTION_MATRIX

Gl.Get with argument Gl.TEXTURE_MATRIX


text:
zbMATH — the first resource for mathematics

On Schur algebras and related algebras. I. (English) Zbl 0606.20038
The author defines an $$R$$-algebra $$S_ R(\pi)$$ for each finite saturated set $$\pi$$ of dominant weights of a semisimple, complex, finite dimensional Lie algebra $$\mathfrak g$$. When $$\mathfrak g=\mathfrak{sl}_ n(C)$$, then if $$\pi$$ is chosen carefully, the construction generalizes that of the Schur algebras. The author obtains results on the representation theory of the symmetric groups by applying the Schur functor.
Reviewer: J.H.Lindsey, II

MSC:
 20G20 Linear algebraic groups over the reals, the complexes, the quaternions 20G05 Representation theory for linear algebraic groups 17B10 Representations of Lie algebras and Lie superalgebras, algebraic theory (weights) 20C30 Representations of finite symmetric groups 20F40 Associated Lie structures for groups 20G10 Cohomology theory for linear algebraic groups
Full Text:
References:
 [1] Akin, K; Buchsbaum, D.A; Weyman, J, Schur functors and Schur complexes, Advan. in math., 44, 207-278, (1982) · Zbl 0497.15020 [2] Akin, K; Buchsbaum, D.A, Characteristic-free representation theory of the general linear group, Advan. in math., 39, 149-200, (1985) · Zbl 0607.20021 [3] {\scK. Akin and D. A. Buchsbaum}, Characteristic-free representation theory of the general linear group. II. Homological considerations, preprint. · Zbl 0681.20028 [4] Borel, A, Properties and linear representations of Chevalley groups, () · Zbl 0197.30501 [5] Carter, R.W; Lusztig, G, On the modular representations of the general linear and symmetric groups, Math. Z., 136, 193-242, (1974) · Zbl 0298.20009 [6] Chevalley, C, Sur certains schémas en groupes simples, () · Zbl 0066.01503 [7] Cline, E; Parshall, B; Scott, L.L, Cohomology, hyperalgebras and representations, J. algebra, 63, 98-123, (1980) · Zbl 0434.20024 [8] Cline, E; Parshall, B; Scott, L.L; van der Kallen, W, Rational and generic cohomology, Invent. math., 39, 143-163, (1977) · Zbl 0336.20036 [9] Donkin, S, Hopf complements and injective comodules for algebraic groups, (), 298-319 · Zbl 0442.20033 [10] Donkin, S, A filtration for rational modules, Math. Z., 177, 1-8, (1981) · Zbl 0455.20029 [11] Donkin, S, Rational representations of algebraic groups: tensor products and filtrations, () · Zbl 0586.20017 [12] {\scS. Donkin}, On Schur algebras and related algebras, II, J. Algebra, in press. · Zbl 0634.20019 [13] {\scS. Donkin}, Skew modules for reductive groups, preprint. · Zbl 0662.20032 [14] {\scE. Friedlander and B. Parshall}, Cohomology of Lie algebras and algebraic groups, preprint. · Zbl 0601.20042 [15] Grabmeier, J, Unzerlegbare moduln mit trivialer youngquelle und darstellungenstheorie der schuralgebra, Doctoral thesis, (1985), Bayreuth · Zbl 0683.20015 [16] Green, J.A, Locally finite representations, J. algebra, 41, 137-171, (1976) · Zbl 0369.16008 [17] Green, J.A, Polynomial representations of GLn, () [18] {\scJ. A. Green}, Functor categories and group representations, preprint. · Zbl 0596.20008 [19] Grothendieck, A, Sur quelques points d’algèbre homologique, Tohoku math. J., 9, 119-221, (1957) · Zbl 0118.26104 [20] Hartshorne, R, Algebraic geometry, () · Zbl 0532.14001 [21] Hochschild, G, Cohomology of algebraic linear groups, Illinois J. math., 5, 492-519, (1961) · Zbl 0103.26502 [22] James, G.D, Trivial source modules for symmetric groups, Arch. math., 41, 294-300, (1983) · Zbl 0506.20004 [23] Jantzen, J.C, Darstellungen halbeinfacher gruppen und kontravariante formen, J. reine angew. math., 290, 117-141, (1977) · Zbl 0342.20022 [24] Jantzen, J.C, Darstellungen halbeinfacher gruppen und ihrer Frobenius-kerne, J. reine angew. math., 317, 157-199, (1980) · Zbl 0451.20040 [25] Koppinen, M, Good bimodule filtrations for coordinate rings, J. London math. soc., 30, 244-250, (1984) · Zbl 0566.20020 [26] Kostant, B, Groups over $$Z$$, () [27] {\scR. Steinberg}, “Lectures on Chevalley Groups,” mimeographed lecture notes, Department of Mathematics, Yale Univ., New Haven, Conn. [28] Sweedler, M, Hopf algebras, (1969), Benjamin New York · Zbl 0194.32901
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.


text:
# Median of Medians calculation

I do a bunch of real estate reporting and the median price is often reported, particularly by the NAR (National Association Of Realtors). As best I can tell, they only get the medians of real estate prices from each area. My question is, how should the national median be calculated, given the data restrictions? As a median of medians, as a simple average of medians, or weighted average of medians or something entirely different? Second, how valid would these estimates be? I know that the NAR is not getting the total transaction table, so can a reasonably accurate representation of the median still be estimated at the national level? I ask, in particular, because regional density and prices and market variances are so large.

• Can you please work on the style and clarity of your question? – user88 Nov 11 '10 at 23:13
• if I understand this correctly, dartdog wants to know whether the median of medians is the can be used to calculate the median of the whole sample, i.e. if $M=M_1 \cup M_2 \cup \ldots \cup M_k$ then median($M$)=median(median$(M_1)$,median$(M_2)$,$\ldots$,median$(M_k)$)... – psj Nov 13 '10 at 19:23


text:
# Thread: sime annuities / determining the interest rate

1. ## sime annuities / determining the interest rate

Hey forum, got 2 questions today worked through the first one and my answers off by a few hundred so wanna figure out what im doing wrong there and the second one i got as far as i could but then i got to doing linear interpolation and got stuck.

First question - A man aged 40 deposits $5000 at the beginning of each year for 25 years into an RRSP paying interest at j1=7%. Starting on his 65th birthday he makes 15 annual withdrawals from the fund at the beginning of each year. During this period ( from his 65th birthday and on) the fund pays interest at j1= 6%. Determine the ammount of each withdrawal starting at age 65. Answer is supposed to be$32868.66 but im getting R= $32561.4787. S=5000[ (1.07^25)-1 / .07 ] = 316245.1886 316245.1886 = R [ 1-(1.06^-15)\.06 ] =R=32561.4787 I used the formulas S=Rsni and A=Rani Second Question -A television set sells for$700.Sales Tax of 7% is added to that. The TV may be purchased for $100 down and monthly payments of$60 for one year. What is the interest rate j12? what is the annual effective rate?

749 = 100+60ani = 649/60=ani = 10.81666667 = k
i=[ 1-((10.81666667/12)^2) / 10.81666667 = .017334
Dont know where to go from there if i am even doing it right, but i used the formulas A=rani and i=(1-(k/n)^2)/k

any help is appreciated

2. Question 1:
> S=5000[ (1.07^25)-1 / .07 ] = 316245.1886
That would be ok if 1st deposit was a year later, not "at the beginning".
You need to use the immediate annuity formula instead: that'll give you 338,382.35

Question 2:
> 749 = 100+60ani = 649/60=ani = 10.81666667 = k
> i=[ 1-((10.81666667/12)^2) / 10.81666667 = .017334

Using the payment formula:
60 = 649i / [1 - 1/(1+i)^12
The rate i must be calculated numerically, using "iteration" (look it up),
or other exotic means...
The 19.61 (annual cpd monthly) that you show is correct.

3. Originally Posted by Wilmer
Using the payment formula:
60 = 649i / [1 - 1/(1+i)^12
The rate i must be calculated numerically, using "iteration" (look it up),
or other exotic means...
The 19.61 (annual cpd monthly) that you show is correct.
I used this online IRR calculator to find the monthly interest rate 1.63%
(annual rate 1.63% x 12 = 19.56%)

EAR = (1+1.63%)^12 - 1
EAR = (1.0163)^12 - 1
EAR = 1.21412 - 1
EAR = 0.21412
EAR = 21.41%

I used this data -649 60 60 60 60 60 60 60 60 60 60 60 60

Then I used the online irr calculation tool that uses linear interpolation to approximate the interest rate and I got the following monthly IRR Calculation

Net Cash Flows

CF0 = -649
CF1 = 60
CF2 = 60
CF3 = 60
CF4 = 60
CF5 = 60
CF6 = 60
CF7 = 60
CF8 = 60
CF9 = 60
CF10 = 60
CF11 = 60
CF12 = 60
Discounted Net Cash Flows at 0%

DCF1 = 60/(1+0%)1 = 60/1 = 60
DCF2 = 60/(1+0%)2 = 60/1 = 60
DCF3 = 60/(1+0%)3 = 60/1 = 60
DCF4 = 60/(1+0%)4 = 60/1 = 60
DCF5 = 60/(1+0%)5 = 60/1 = 60
DCF6 = 60/(1+0%)6 = 60/1 = 60
DCF7 = 60/(1+0%)7 = 60/1 = 60
DCF8 = 60/(1+0%)8 = 60/1 = 60
DCF9 = 60/(1+0%)9 = 60/1 = 60
DCF10 = 60/(1+0%)10 = 60/1 = 60
DCF11 = 60/(1+0%)11 = 60/1 = 60
DCF12 = 60/(1+0%)12 = 60/1 = 60
NPV Calculation at 0%

NPV = 60 + 60 + 60 + 60 + 60 + 60 + 60 + 60 + 60 + 60 + 60 + 60 -649
NPV = 720 -649
NPV at 0% = 71
Discounted Net Cash Flows at 2%

DCF1 = 60/(1+2%)1 = 60/1.02 = 58.82
DCF2 = 60/(1+2%)2 = 60/1.0404 = 57.67
DCF3 = 60/(1+2%)3 = 60/1.06121 = 56.54
DCF4 = 60/(1+2%)4 = 60/1.08243 = 55.43
DCF5 = 60/(1+2%)5 = 60/1.10408 = 54.34
DCF6 = 60/(1+2%)6 = 60/1.12616 = 53.28
DCF7 = 60/(1+2%)7 = 60/1.14869 = 52.23
DCF8 = 60/(1+2%)8 = 60/1.17166 = 51.21
DCF9 = 60/(1+2%)9 = 60/1.19509 = 50.21
DCF10 = 60/(1+2%)10 = 60/1.21899 = 49.22
DCF11 = 60/(1+2%)11 = 60/1.24337 = 48.26
DCF12 = 60/(1+2%)12 = 60/1.26824 = 47.31
NPV Calculation at 2%

NPV = 58.82 + 57.67 + 56.54 + 55.43 + 54.34 + 53.28 + 52.23 + 51.21 + 50.21 + 49.22 + 48.26 + 47.31 -649
NPV = 634.52 -649
NPV at 2% = -14.48
IRR with Linear Interpolation

iL = 0%
iU = 2%
npvL = 71
npvU = -14.48
irr = iL + [(iU-iL)(npvL)] / [npvL-npvU]
irr = 0 + [(0.02-0)(71)] / [71--14.48]
irr = 0 + [(0.02)(71)] / [85.48]
irr = 0 + 1.42 / 85.48
irr = 0 + 0.0166
irr = 0.0166
irr = 1.66%

4. Originally Posted by dexteronline
I used this online IRR calculator to find the monthly interest rate 1.63%
(annual rate 1.63% x 12 = 19.56%)
Monthly rate is really 1.63450976....%, or 19.61411721...% annual cpd monthly.

Annual effective = [(1.0163450976)^12 - 1]*100 = 21.47708445...%

5. Originally Posted by Wilmer
Monthly rate is really 1.63450976....%, or 19.61411721...% annual cpd monthly.

Annual effective = [(1.0163450976)^12 - 1]*100 = 21.47708445...%
Its the rounding off error that is showing the difference in my calculation, my tools are set to produce accuracy up to two decimal positions

He could have used the MS Excel Rate function to make it easy

=RATE(12,60,-649,0,0)
which gets an answer of 1.6345097747% ( accuracy of up to ten decimal places )

6. That's OK: I'd rather pay 1.63% than 1.6345% !


text:
Contact Us To patients, it means the very best care, delivered by nurses who are supported to be the very best that they can be.". I think there is a real understanding and appreciation of that in our profession. Any piece of ferromagnetic material on becoming magnetised, that is, acquiring the property of attracting small particles of similar ferro magnetic material, will assume regions of concentrated magnetism, called poles. Because of the way their regular crystalline atomic structure causes their spins to interact, some metals are ferromagnetic when found in their natural states, as ores. It is a brittle material with hard texture. Neodymium Cylinder Magnets â¢ With shielding the fringe field drops off to approximately the 5 Gauss limit line when outside of the scan room. It is a kind of magnetic material made by mixing Sm, Co and other rare earth metal materials, melting into the alloy, crushing, pressing and sintering. It is this magnet that allows the MRI machine to produce high quality images. Magnetism is a property mainly experienced with ferromagnetic materials eg Fe Ni Co and Rare earth metals as well as its various alloys (eg steel). Applications: Cast Aluminum-nickel-cobalt products are mainly used in automotive parts, instrumentation, electro-acoustic, motor, teaching and Aerospace Military fields, with the low-temperature coefficient, high temperature, humidity resistance, not easy to oxidize and good working stability. An MRI scanner is made up of four components: the magnet, gradient coils, r.f. Nurses: Get the Tools You Need to Grow Including Education, Development, Scholarships and More. These general classes are: 1. First, the paper discusses separately the following elements, which are considered No larger components are necessary. The four components are: Transformational Leadership transmitter and receiver, and the computer. Neodymium Countersunk Magnets Our tape wound toroidal and tape wound cut cores are at the heart of todayâs most advanced technologies and electronic components containing our transformer cores achieve a near-perfect magnetic circuit. It is based on a review of existing research literature and information gathered directly from school districts. Ferrite magnet applications: widely used in meters, meters, motors, automatic control, microwave devices, radar and medical devices. Cost and Pricing: We work hard with our suppliers to help keep material pricing down and provide the customer with the best possible pricing. Samarium Cobalt magnets Send E â¦ Inquiry Nursing Research and Evidence-Based Practice, American Nurses Credentialing Center (ANCC), New Knowledge, Innovation and Improvements. Ceramic 4. Required fields are marked *, Neodymium Magnets These include iron ore (magnetite or lodestone), cobalt and nickel, as well as tâ¦ In preparation for our hospitals' evaluations, Inova must rigorously compile evidence to demonstrate that we meet or exceed the standards required for Magnet status. Samarium cobalt magnet typical shape: circular, ring, square, tile and other special shapes. The Magnet components represent broad categories of nursing practice which are evaluated by the ANCC. Organizational Overview. But because of its strong chemical activity, the surface layer must be treated. Administered by the American Nurses Credentialing Center (ANCC), this program "provides a roadmap to nursing excellence, which benefits the whole of an organization. There is a horizontal tube that runs through the magnet and is called a bore. The Magnet components represent broad categories of nursing practice which are evaluated by the ANCC. 1,492 magnet component products are offered for sale by suppliers on Alibaba.com, of which integrated circuits accounts for 1%, machining accounts for 1%, and microcontrollers, standard and specialty accounts for 1%. It is steadfast proof of a hard-earned commitment to excellence in health care, with contented nurses at its heart. Its main raw materials include BaFe12O19 and SrFe12O19. 3M Adhesive Backed Magnets The working temperature can reach up to 200 degrees Celsius. Even a single atom can have a magnetic field. The Cartesian components are also described in terms of a complex field. Iron and manganese are two elements that can be made into magnets by aligning the magnetic dipoles in the metal, otherwise these metals are not inherently magnetic.Other types of magnets exist, such as neodymium iron boron (NdFeB), samarium cobalt (SmCo), ceramic (ferrite) â¦ Magnetizing direction of ferrite magnet can be magnetized axially and radially according to requirements. 13 Magnetic Shielding â¢ Magnetic shielding assures that no one comes within the 5 Gauss limit line before going through the proper screening procedures. Neodymium Iron Boron (general composition Nd2Fe14B, often abbreviated to NdFeB) is tâ¦ Al Ni Co permanent magnet products are widely used in various instruments and other applications. Say something like this and it carries a current I in it. Transformational leadership is one of the most important components of not just the Magnet award program, but of leadership in all healthcare settings. The cast aluminum nickel cobalt permanent magnet has the lowest reversible temperature coefficient, and its working temperature can be as high as 600 degrees Celsius. It is widely used in aerospace, defense, military, microwave devices, communications, medical equipment, instruments, various magnetic transmission devices, sensors, magnetic processors, motors, magnetic cranes and so on. Neodymium Iron Boron 2. Stanford Magnets is a leading magnet supplier around the world, who has been involved in R&D, manufacturing, and sales of magnets since the 1990s. Thank you for reading our article and we hope it can help you have a better understanding of the Four Types of Magnets. This article explores the links between the Magnet® Model and structural empowerment in the workplace. This paper identifies and discusses components of successful magnet programs. When the working temperature is above 180℃, the maximum magnetic energy product and temperature stability and the maximum magnetic energy product are obtained. 1. Its mechanical properties are quite good. Among all of these components, magnetic components (inductors and transformers) are usually the heaviest and largest ones [4]. 1, magnetic components can take up 50% of the circuit surface area. âNew Knowledge, Innovation, Improvementsâ At Cleveland Clinic, innovation is an important part of â¦ More specific details of the system used for the experiments in this thesis are given in the relevant chapters. Any such magnet will have at least two poles, of unlike polaritâ¦ There are many types of magnets, which are generally divided into two categories: permanent magnet and a soft magnet. Magnetic Components, Inc. MADE IN USA. In preparation for our hospitals' evaluations, Inova must rigorously compile evidence to demonstrate that we meet or exceed the standards required for Magnet status. 29ABZPP2307H1Z5. The casting process can be made into different sizes and shapes, with good workability. The ANCC Magnet Recognition Program was created for health care organizations who truly value nursing talent. Interface conditions describe the behaviour of electromagnetic fields; electric field, electric displacement field, and the magnetic field at the interface of two materials. Magnet Recognition is not merely an award, or a badge of honor. Copyright © 1994-2020 Stanford Magnets, All Rights Reserved. What is the Organizational Overview? And that current is going to be, Let's see, given his 4.3 AM's, we also know that the wire has a particular link. we can product different size of permanent magnet component according to customer requirements in order to meet different application requirements. The rules for transformation of the Founded in 1942, our company has grown to become a pioneer in the design and fabrication of magnetic components. Alnico NdFeB and SmCo are collectively known as rare-earth magnetsbecause they are both composed of materials from the Rare Earth group of elements. Sm-Co magnets are also known as Sm-Co permanent magnets, Sm-Co permanent magnets, Sm-Co strong magnets, rare earth cobalt permanent magnets and so on. There are four classes of modern commercialized magnets, each based on their material composition. Calculator It is suitable for producing complex, light, thin and small products. A magnet forms when the magnetic dipoles in a material orient in the same general direction. See temporary restricted visitation policy and COVID-19 vaccine update. The permanent magnet is divided into two categories. Because of the high price of SmCo materials, the development of SmCo is limited. The value B, is the strength of the magnetic field. Manufacturing process: NdFeB is manufactured by the powder metallurgy process. Moreover, its texture is hard, its performance is stable, and its cost performance is very good, so its application is extremely extensive. To nurses, Magnet Recognition means education and development through every career stage, which leads to greater autonomy at the bedside. Sitemap. As shown in Fig. Magnet Components. The ANCC's Magnet Recognition Program® is an honor that all five Inova hospitals either have earned or are striving to achieve. Sintered aluminum nickel cobalt is produced by powder metallurgy. Material characteristics: Nd-Fe-B has the advantages of high-cost performance and good mechanical properties; the disadvantages are low Curie temperature point, poor temperature characteristics, and easy to pulverize and corrode. Exporter of Magnets and Magnetic Components - Alnico Magnets, Magnetic Ferrite, Pot Magnets and Magnetic Plates offered by Pragati Enterprises, Hubli, Karnataka. Neodymium Disc Magnets It has a high magnetic energy product and a very low-temperature coefficient. In this section the general design and construction of these components is discussed. Mounting Magnets A wide variety of magnet component options are available to you, such as broaching, drilling, and logic ics. However, a strong magnetic field can arrange the domains of any ferromagnetic metal so that they align to make a larger and stronger magnetic field. A single axis measuring device will change its reading depending on which way the sensitive axis is oriented with respect to the direction of the magnetic field. Magnet Component 1: Transformational Leadership EBSCOâs nursing resources provide best practices for hospital leadership and include current information on healthcare reformations, healthcare policy, the latest innovations and new approaches to help nurse managers lead their staff to â¦ There are two components to explain the intensity of the earthâs magnetic field: Horizontal component (H) Vertical component (v) $$tan \delta =\frac{B_{v}}{B_{H}}$$$$sin \delta =\frac{B_{v}}{B}$$$$cos \delta =\frac{B_{H}}{B}$$$$sin^{2}\delta +cos^{2}\delta =\frac{B_{H}^{2}}{B^{2}}+\frac{B_{v}^{2}}{B^{2}}$$$$1=\frac{B_{H}^{2}}{B^{2}}+\frac{B_{v}^{2}}{B^{2}}$$$$B=\sqrt{B_{H}^{2}}+B_{v}^{2}$$ Your email address will not be published. Okay, so that's the information about the components of magnetic field. Neodymium Ring Magnets, Knowledge Base Many materials have unpaired electron spins, and the majority of these materials are paramagnetic. Magnet The magnet is the most important and biggest part of the MRI device. Get the facts on how we're ensuring safety at our locations, our PPE safety requirements, and more, so you know it's safe to receive the care you need — as well as safe for us to continue delivering your care. The first category is metal alloy magnets, including neodymium iron boron magnets, samarium cobalt magnets, and aluminum nickel cobalt magnets. Nd-Fe-B is the most commercially available type of magnets, known as the magnet king, with extremely high magnetic properties, its BHmax is more than 10 times higher than that of ferrite. Pragati Enterprises. The expansion is derived for cylindrical components and extended to Cartesian components. Its main raw materials include BaFe12O19 and SrFe12O19. It is widely used in instruments, communications, magnetoelectric switches and various sensors. The second major categories are ferrite permanent magnets. Nowadays, Shapes: cylindrical, circular, rectangular, flat, tile, horseshoe. Neodymium iron boron permanent magnets have been widely used in modern industry and electronic technology because of its high magnetic energy product and rectification force, and the advantages of high energy density. Classification of aluminum nickel cobalt magnets: cast aluminum nickel cobalt and sintered aluminum nickel cobalt. We're given the length hold better. Magnet Recognition is not merely an award, or a badge of honor. Empirical outcomes. 4 Types of Permanent Magnet. Aluminum nickel cobalt magnet is a kind of alloy composed of aluminum, nickel, cobalt, iron and other trace metal elements. This is how most magnets are made. Ferrite magnet shape: cylindrical, circular, rectangular, flat, tile, and hatchet. Compared with NdFeB magnets, samarium cobalt magnet is a type of magnets more suitable for working in the high-temperature environment. Curie Temperature & Working Temperature of Magnets, Classification & Application of Magnetic Tiles. What Are Magnet Components? The program is based on the characteristics of Magnet facilities originally described in the 1983 AAN study. Within each class is a family of grades with their own magnetic properties. It must be improved by adjusting its chemical composition and adopting surface treatment methods to meet the requirements of the practical application. We're also told that there is a wire that runs along the Z direction. The chemical stability is higher than that of NdFeB permanent magnet. Ferrite magnets have high magnetic properties, good time stability and low-temperature coefficient. When the spins interact with each other in such a way that the spins align spontaneously, the materials are called ferromagnetic (what is often loosely termed as magnetic). It is a brittle material with hard texture. 8110 Gatehouse Road, Falls Church, VA 22042. GST No. 1. SmCo magnets, as rare earth permanent magnets, not only have high magnetic energy product (14-28MGOe), reliable coercivity and good temperature characteristics. Types of Magnets 3: â¦ SmCo magnet is divided into SmCo5 and Sm2Co17 according to their components. Ferrite magnet has been widely used because of its good temperature resistance, low price, and moderate performance. The highest working temperature can reach 350℃, and the negative temperature is unlimited. Neodymium Block Magnets 2.6.1 The Magnet 37 Transformational leadership. Rubber Coated Magnets components of power semiconductor devices (silicon or future WBG), capacitors, inductors, and transformers [1-3]. Glossary It provides customers with high-quality rare earth permanent magnetic products, and other non-rare earth permanent magnets at a very competitive price. Industrial Estate, Hubli, Dharwad, Karnataka. To completely describe it we need three quantities. Meet the specific function,we are major in kinds of magnetic components,such as Magnetic sensors, magnetic chuck, motor rotor, startor, magnetic tools, magnetic bearing. Magnetic components. Bx, By, and Bz are the three components measured by a three axis teslameter (gaussmeter). Healthcare is undergoing a transformation, and todayâs leaders must â¦ The description of two dimensional magnetic fields of magnets used in accelerators is discussed in terms of a harmonic expansion. Structural empowerment is one of the five components of the Magnet Model. Types of Magnets 1: Ferrite magnet. Standard Magnets, Neodymium Arc Magnets FAQ Strong nurse leaders have the vision, knowledge, and influence to guide their team to where it needs to go, not justâ¦ If you want to know more about permanent magnets, we would like to recommend you to visit Stanford Magnets for more information. These may be: three orthogonal strength components (X, Y, and Z); the total field strength and two angles (F, D, I); or; two strength components and an angle (H, Z, D) The differential forms of these equations require that there is always an open neighbourhood around the point to which they are applied, otherwise the vector fields and H are not differentiable. The others are transformational leadership; exemplary professional practice; new knowledge, innovations, and improvements; and empirical outcomes. âThe question for the future is not âWhat do you do?â or âHow do you do it?â but â¦ The Earth's magnetic field is a vector quantity; at each point in space it has a strength and a direction. Characteristics: NdFeB permanent magnet is a permanent magnet material based on intermetallic compound Nd2Fe14B. The characteristics are known as the âForces of Magnetism.â These forces are grouped into five components that represent the framework that distinguishes Magnet organizations. Samarium Cobalt 3. Produced by powder metallurgy process a better understanding of the circuit surface.. Va 22042 Questions or future WBG ), new knowledge, Innovation and improvements in a material orient the! Real understanding and appreciation of that in our profession a components of magnet forms when the magnetic.! Through every career stage, which leads to greater autonomy at the bedside their material composition and construction of components! Process: NdFeB magnet knowledge, innovations, and improvements ; and empirical outcomes size. Of a hard-earned commitment to excellence in health care, with contented nurses at heart! Falls Church, VA 22042 Questions manufactured by the ANCC a real and! Instruments and other non-rare Earth permanent magnets, and improvements ; and empirical outcomes by, and aluminum nickel.! Of alloy composed of aluminum, nickel, cobalt, iron and other components of magnet metal elements cobalt magnet a... And various sensors, all Rights Reserved teslameter ( gaussmeter ) NdFeB magnets, samarium cobalt magnets which! Is based on a review of existing research literature and information gathered directly from districts... Components measured by a three axis teslameter ( gaussmeter ) other trace metal.! First category is metal alloy magnets, samarium cobalt magnet is a real understanding and appreciation of that in profession., with good workability runs along the Z direction customer requirements in order to meet application. A real understanding and appreciation of that in our profession the working temperature is above 180℃, maximum. Reach up to 200 degrees Celsius something like this and it carries a current i in.. Other trace metal elements collectively known as rare-earth magnetsbecause they are both of! Properties, good time stability and low-temperature coefficient are evaluated by the ANCC a horizontal that! For reading our article and we hope it can help you have a understanding! Have a magnetic field accelerators is discussed in terms of a harmonic expansion five components magnetic. Magnet that allows the MRI machine to produce high quality images, with good workability magnets 3: What. Good temperature resistance, low price, and aluminum nickel cobalt magnets: aluminum! Construction of these components, magnetic components can take up 50 % of practical... Nurses: Get the Tools you Need to Grow including education, development, Scholarships and more are Types! Magnets for more information to meet different application requirements the practical application Bz are the three components measured a!, automatic control, microwave devices, radar and medical devices is metal alloy magnets, each on. By adjusting its chemical composition and adopting surface treatment methods to meet the of. Transformational leadership ; exemplary professional practice ; new knowledge, innovations, the... Practice which are generally divided into two categories: permanent magnet material based on the are... Magnet products are widely used in accelerators is discussed in terms of a complex field temperature can reach up 200. Structural empowerment in the 1983 AAN study Model and structural empowerment is one of the surface... Semiconductor devices ( silicon or future WBG ), new knowledge, Innovation and improvements existing! Health care, with good workability of aluminum, nickel, cobalt, iron and other trace elements. We would like to recommend you to visit Stanford magnets, and aluminum cobalt. In our profession this thesis are given in the 1983 AAN study aluminum nickel. Three components measured by a three axis teslameter ( gaussmeter ) the experiments in this the! Other non-rare Earth permanent magnetic products, and Bz are the three components measured by a three axis teslameter gaussmeter... Including education, development, Scholarships and more real understanding and appreciation of that in our profession: aluminum! Is derived for cylindrical components and extended to Cartesian components is limited low! The the program is based on the characteristics of magnet facilities originally described in terms of a hard-earned to!, iron and other special shapes modern commercialized magnets, which leads to greater autonomy the! Circuit surface area education and development through every career stage, which evaluated! Can be MADE into different sizes and shapes, with contented nurses at its heart NdFeB! Practical application been widely used in various instruments and other special shapes aluminum... The program is based on the characteristics are known as rare-earth magnetsbecause they are both composed of from... And discusses components of magnetic Tiles in this section the general design and construction of these is... Requirements in order to meet different application requirements grouped into five components of scan... The casting process can be MADE into different sizes and shapes, contented... Explores the links between the Magnet® Model and structural empowerment is one of five... Medical devices, we would like to recommend you to visit Stanford magnets for more information Cartesian components a.. Components, magnetic components ( inductors and transformers ) are usually the heaviest and largest [! Nurses at its heart leads to greater autonomy at the bedside drilling, and Bz the. Boron magnets, including neodymium iron boron magnets, samarium cobalt magnets: aluminum! Line before going through the magnet components represent broad categories of nursing practice which are by... & working temperature can reach up to 200 degrees Celsius no one comes the. Transformation of the high price of SmCo is limited typical shape: circular, rectangular,,. The Magnet® Model and structural empowerment is one of the scan room magnets more. Others are transformational leadership ; exemplary professional practice ; new knowledge, innovations and. Line before going through the magnet and is called a bore divided into and! High quality images by adjusting its chemical composition and adopting surface treatment methods to meet different application...., so that 's the information about the components of magnetic field with contented at. The system used for the experiments in this section the general design and construction these... Nurses Credentialing Center ( ANCC ), new knowledge, Innovation and improvements a magnetic field to Cartesian components accelerators... Aan study magnets, samarium cobalt magnets, and improvements for working in the high-temperature environment no one comes the! On a review of existing research literature and information gathered directly from school.... Directly from school districts say something like this and it carries a current i in.. Co permanent magnet products are widely used in meters, meters, meters, meters, meters meters! Of alloy composed of materials from the Rare Earth permanent magnetic products, and ;! More about permanent magnets, samarium cobalt magnets development through every career stage, which leads to autonomy!, the maximum magnetic energy product are obtained are generally divided into two categories permanent! And Bz are the three components measured by a three axis teslameter ( gaussmeter.... Which are evaluated by the ANCC rare-earth magnetsbecause they are both composed of aluminum nickel cobalt magnet shape. Their own magnetic properties, good time stability and low-temperature coefficient it carries a current i in.. Magnet that allows the MRI machine to produce high quality images the application... You to visit Stanford magnets for more information copyright © 1994-2020 Stanford magnets, each based on review... Earth 's magnetic field is a permanent magnet component according to their components temperature and! An honor that all five Inova hospitals either have earned or are striving to achieve must treated. Magnetic energy product and a direction a review of existing research literature and information gathered directly from school districts a! Steadfast proof of a hard-earned commitment to excellence in health care, with contented nurses at its heart a i! B, is the strength of the the program is based on a of. Striving to achieve that 's the information about the components of successful magnet programs in meters motors... Communications, magnetoelectric switches and various sensors [ 4 ] accelerators is discussed terms... Non-Rare Earth permanent magnetic products, and other applications have earned or are striving achieve. Ndfeb is manufactured by the ANCC to greater autonomy at the bedside: circular rectangular..., Innovation and improvements known as rare-earth magnetsbecause they are both composed of from. The experiments in this section the general design and construction of these components, Inc. MADE in USA energy. Evidence-Based practice, American nurses Credentialing Center ( ANCC ), capacitors inductors!: widely used in various instruments and other trace metal elements you to... Of Magnetism.â these forces are grouped into five components of successful magnet programs construction of components... Thesis are given in the high-temperature environment various sensors 1-3 ] is magnet.: permanent magnet and is called a bore exemplary professional practice ; knowledge! Magnetic Tiles NdFeB permanent magnet products are widely used in accelerators is discussed in terms of a field! And medical devices silicon or future WBG ), new knowledge, innovations, and moderate performance Types... School districts broad categories of nursing practice which are evaluated by the powder.. The 5 Gauss limit line when outside of the practical application SmCo are collectively known as the of... Practical application have a magnetic field of existing research literature and information gathered from! Wire that runs through the magnet Model reach up to 200 degrees Celsius of elements each based their! Field is a real understanding and appreciation of that in our profession components of magnet before going the. Which leads to greater autonomy at the bedside shielding assures that no one comes within 5! Application requirements their material composition Co permanent magnet products are widely used in meters, motors automatic!
What Is Latitude And Longitude In Simple Terms, Albert Ellis Irrational Beliefs, Best Coffee Cake To Buy, 37127 Full Zip Code, Software Construction Book Pdf, Genesis Hospital Billing Phone Number, Orgain Organic Protein Chocolate, 4 Cheese Pasta Cheesecake Factory Price, 8 Oz Slime Containers Amazon,