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## Energy of ejected electron from energy of light

$\lambda=\frac{h}{p}$

marcus lin 1E
Posts: 54
Joined: Fri Sep 28, 2018 12:28 am

### Energy of ejected electron from energy of light

Can someone explain how I'd go about calculating the freq/wavelength/energy of an electron when only given the energy of the light used to eject said electron?

Posts: 77
Joined: Fri Sep 28, 2018 12:28 am
Been upvoted: 1 time

### Re: Energy of ejected electron from energy of light

When you're given the amount of energy of light used to eject an electron it's important to know if there's any excess energy in the form of kinetic energy given off. If there isn't any excess kinetic energy given off then we know the energy of the incoming light is equivalent to the energy of that electron. We would then use c=(wavelength)(frequency) and E=(h)(frequency) (which you would have to combine together when solving for light) to calculate the frequency and wavelength.

Hope this helped! :)


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# R / Sweave formatting numbers with \Sexpr{} in scientific notation

I am just starting to write some documents with Sweave/R and I like the \sexpr{} command that lets one tow write numbers directly within text.

If I have a number like mus=0.0002433121, well I can say round it to a number of decimal places e.g.

\Sexpr{round(mus,7)}

How to write it in the scientific notation i.e. as LaTeX would be outputting

2.43 \times 10^{-4}

and can we control the number of significant digits to be outputted like 3 in this example?

I note that a number like sigma = 2000000 is written automatically to 2e + 06 if I specify

\Sexpr{round(sigma,2)}.

I would prefer that it would be written as

2 \times 10^6

same as we would get in LaTeX notation and perhaps giving us the possibility to control the number of significant digits as well.

How to achieve this?

I think this function should work:

sn <- function(x,digits)
{
if (x==0) return("0")
ord <- floor(log(abs(x),10))
x <- x / 10^ord
if (!missing(digits)) x <- format(x,digits=digits)
if (ord==0) return(as.character(x))
return(paste(x,"\\\\times 10^{",ord,"}",sep=""))
}

Some tests:

> sn(2000000)
[1] "2\\\\times 10^{6}"
> sn(0.001)
[1] "1\\\\times 10^{-3}"
> sn(0.00005)
[1] "5\\\\times 10^{-5}"
> sn(10.1203)
[1] "1.01203\\\\times 10^{1}"
> sn(-0.00013)
[1] "-1.3\\\\times 10^{-4}"
> sn(0)
[1] "0"

If you want the result in math mode you could enter $ signs in the paste() call. # Edit: Here is a Sweave example: \documentclass{article} \begin{document} <<echo=FALSE>>= sn <- function(x,digits) { if (x==0) return("0") ord <- floor(log(abs(x),10)) x <- x / 10^ord if (!missing(digits)) x <- format(x,digits=digits) if (ord==0) return(as.character(x)) return(paste(x,"\\\\times 10^{",ord,"}",sep="")) } @ Blablabla this is a pretty formatted number$\Sexpr{sn(0.00134,2)}$. \end{document}  • Changed the function to also include 0 and negative cases. – Sacha Epskamp Dec 3 '11 at 11:46 • @Sacha Epskamp I have decided to take away my "accept answer" as you didn't explicitly show how to use with \Sexpr{}. Yes you code does work fine in R but I couldn't make it work with \Sexpr{} in Sweave. Thanks. – yCalleecharan Dec 5 '11 at 6:36 • @Sacha Epskamp. Thanks for your patience. Now I am getting 1.34times10-3 (the power being well represented). How to insert$ $properly so that the number 1.34 also is between$ and $and the \times appears as x. Thanks. – yCalleecharan Dec 5 '11 at 10:44 • Ah I hadn't actually tested it in Sweave and apearently I had to use four backslashes instead of two:) This should work. – Sacha Epskamp Dec 5 '11 at 12:24 • @Sacha Epskamp Thanks. This is very helpful and it works. If you can consider adding a possibility to specify the number of significant digits as I mention in my post e,g. in your example I can just want to output 1.3 x 10 -3 instead of 1.34 x 10-3. Setting options(digits=1) or similar won't work. – yCalleecharan Dec 5 '11 at 12:43 An example using siunitx link to pdf. In the preamble you can define your default options which you may override later in the document. For numeric output : num <- function(x,round_precision=NULL) { if (is.null(round_precision)) { return(sprintf("\\num{%s}", x)) } else { return(sprintf("\\num[round-precision=%s]{%s}",round_precision, x)) } }  For scientific output : sci<- function(x,round_precision=NULL){ if (is.null(round_precision)) { return(sprintf("\\num[scientific-notation = true]{%s}", x)) } else { return(sprintf("\\num[round-precision=%s,scientific-notation = true]{%s}",round_precision, x)) } }  Here is a full reproducible .Rnw script (to be used with knitr ... for sweave use four antislashes in the functions instead of two see this SO post.) \documentclass[a4paper]{article} \usepackage{siunitx} %\usepackage{Sweave} \title{siunitx} \sisetup{ round-mode = figures, round-precision = 3, group-separator = \text{~} } \begin{document} \maketitle <<sanitize_number,echo=FALSE>>= num <- function(x,round_precision=NULL) { if (is.null(round_precision)) { return(sprintf("\\num{%s}", x)) } else { return(sprintf("\\num[round-precision=%s]{%s}",round_precision, x)) } } sci<- function(x,round_precision=NULL){ if (is.null(round_precision)) { return(sprintf("\\num[scientific-notation = true]{%s}", x)) } else { return(sprintf("\\num[round-precision=%s,scientific-notation = true]{%s}",round_precision, x)) } } @ Examples :\\$num$for number formatting : \begin{itemize} \item \textbf{num(pi, round\_precision=2)}$\Rightarrow$\num[round-precision=2]{3.14159265358979} \item \textbf{num(pi, round\_precision=4)}$\Rightarrow$\num[round-precision=4]{3.14159265358979} \item The default formatting (here round-precision=3) is taken from \textbf{\textbackslash sisetup} \textbf{num(pi)}$\Rightarrow$\num{3.14159265358979}\\ \end{itemize} \noindent$sci$for scientific notation : \begin{itemize} \item \textbf{sci(12.5687e4)}$\Rightarrow$\num[scientific-notation = true]{125687} \item \textbf{sci(125687.11111)}$\Rightarrow$\num[scientific-notation = true]{125687.11111} \item \textbf{sci(125687.11111, round\_precision=4)}$\Rightarrow\$
\Sexpr{sci(125687.11111, round_precision=4)}
\end{itemize}

\end{document}

format: plain


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# Monotonic bijections of rational numbers

How can one characterize monotonic bijections from $\mathbb{Q}$ to $\mathbb{Q}$? It is easy to see that piecewise linear functions which are strictly monotonic and surjective will do the trick, but are these functions already all monotonic bijections of the rationals, or are there also "curved" ones? Classical curved bijections of the reals like $x^3$ are of no help, because, for example, 2 has no preimage in $\mathbb{Q}$.

The context of this question is the transformation of random variables and the degrees of freedom associated with these transformations, or more specific, what kind of results valid for continuous random variables can be transferred to rational random variables, and which results are sensitive to the special structure of the reals.

-
There are loads of non-monotonic continuous bijections from $\mathbb{Q}$ to itself that don't extend to continuous functions on $\mathbb{R}$. But the way you phrase the question suggests that you may not be interested in these? –  Jeremy Rickard Jan 30 at 16:30
For example, swap the intervals $(\sqrt{2},\sqrt{2}+1)$ and $(\sqrt{2}+1,\sqrt{2}+2)$, leaving everything outside these intervals fixed. –  Jeremy Rickard Jan 30 at 16:48
Consider the zig-zag construction showing that any two countable dense linear orders are isomorphic, and apply it with $\mathbb Q$ on both sides. You have a lot of freedom in the construction: in particular, you can do it in such a way that the image or preimage being chosen is slightly off from the position it would have if the bijection were linear between the nearest points where the function has already been fixed (I hope this description is clear). You will end up with a monotone bijection which is not linear on any nonempty interval. –  Emil Jeřábek Jan 30 at 17:03
In fact, if $F$ is a countable set of functions, you can arrange that the the value of the bijection in the $n$-th chosen point disagrees with the values given by the first $n$ functions from $F$. You will get a monotone bijection which differs from any $f\in F$ in all but finitely many points. For example, $F$ can be the set of all rational functions with rational coefficients. –  Emil Jeřábek Jan 30 at 17:12
You can biject two given countable dense subsets of $\mathbb{R}$ by an entire diffeomorphism: mathoverflow.net/questions/42460/… –  Pietro Majer Feb 2 at 19:52
A simple example of a monotonic homeomorphism from $\mathbb{Q}$ to itself that is not piecewise linear is given by $f(x)=x$ if $x<0$, $f(x)=\frac{x}{1-x}$ if $x\in[0,\frac{1}{2}]$ and $f(x)=x+\frac{1}{2}$ if $x>\frac{1}{2}$.
You can even have $f$ be an entire function. See e.g. https://groups.google.com/d/msg/sci.math/UjRgb0y_iBE/iTFqP5GjRgwJ


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Tags
#m249 #mathematics #open-university #statistics #time-series
Question
if a [...] model is appropriate for the time series Xt , then an additive model is appropriate for the time series of logarithms, Yt =log Xt .
Answer
multiplicative

Tags
#m249 #mathematics #open-university #statistics #time-series
Question
if a [...] model is appropriate for the time series Xt , then an additive model is appropriate for the time series of logarithms, Yt =log Xt .
Answer
?

Tags
#m249 #mathematics #open-university #statistics #time-series
Question
if a [...] model is appropriate for the time series Xt , then an additive model is appropriate for the time series of logarithms, Yt =log Xt .
Answer
multiplicative
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Contents

• # For Solution

Before becoming a successful trader William got a university degree. During his education an interesting situation happened, after which William started to listen to homework assignments much more attentively. What follows is the correct formal description of the homework assignment:

You are given a string ss of length nn only consisting of characters “a“, “b” and “c“. There are qq queries of format (pos,cpos,c), meaning replacing the element of string ss at position pospos with character cc. After each query you must output the minimal number of characters in the string, which have to be replaced, so that the string doesn’t contain string “abc” as a substring. A valid replacement of a character is replacing it with “a“, “b” or “c“.

A string xx is a substring of a string yy if xx can be obtained from yy by deletion of several (possibly, zero or all) characters from the beginning and several (possibly, zero or all) characters from the end.

Input

The first line contains two integers nn and qq (1n,q105)(1≤n,q≤105), the length of the string and the number of queries, respectively.

The second line contains the string ss, consisting of characters “a“, “b” and “c“.

Each of the next qq lines contains an integer ii and character cc (1in)(1≤i≤n), index and the value of the new item in the string, respectively. It is guaranteed that character’s cc value is “a“, “b” or “c“.

Output

For each query output the minimal number of characters that would have to be replaced so that the string doesn’t contain “abc” as a substring.

Example
input

Copy
9 10
abcabcabc
1 a
1 b
2 c
3 a
4 b
5 c
8 a
9 b
1 c
4 a

output

Copy
3
2
2
2
1
2
1
1
1
0

Note

Let’s consider the state of the string after each query:

1. s=s= “abcabcabc“. In this case 33 replacements can be performed to get, for instance, string s=s= “bbcaccabb“. This string does not contain “abc” as a substring.
2. s=s= “bbcabcabc“. In this case 22 replacements can be performed to get, for instance, string s=s= “bbcbbcbbc“. This string does not contain “abc” as a substring.
3. s=s= “bccabcabc“. In this case 22 replacements can be performed to get, for instance, string s=s= “bccbbcbbc“. This string does not contain “abc” as a substring.
4. s=s= “bcaabcabc“. In this case 22 replacements can be performed to get, for instance, string s=s= “bcabbcbbc“. This string does not contain “abc” as a substring.
5. s=s= “bcabbcabc“. In this case 11 replacements can be performed to get, for instance, string s=s= “bcabbcabb“. This string does not contain “abc” as a substring.
6. s=s= “bcabccabc“. In this case 22 replacements can be performed to get, for instance, string s=s= “bcabbcabb“. This string does not contain “abc” as a substring.
7. s=s= “bcabccaac“. In this case 11 replacements can be performed to get, for instance, string s=s= “bcabbcaac“. This string does not contain “abc” as a substring.
8. s=s= “bcabccaab“. In this case 11 replacements can be performed to get, for instance, string s=s= “bcabbcaab“. This string does not contain “abc” as a substring.
9. s=s= “ccabccaab“. In this case 11 replacements can be performed to get, for instance, string s=s= “ccabbcaab“. This string does not contain “abc” as a substring.
10. s=s= “ccaaccaab“. In this case the string does not contain “abc” as a substring and no replacements are needed.


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## Minimal permutation representation of the simple group $$F_ 5$$.(English. Russian original)Zbl 0659.20015

Algebra Logic 26, No. 3, 167-178 (1987); translation from Algebra Logika 26, No. 3, 298-317 (1987).
Let $$F_ 5$$ be the finite simple group of Harada of order $$2^{14}\cdot 3^ 6\cdot 5^ 6\cdot 7\cdot 11\cdot 19$$. It contains a subgroup H which is isomorphic to the alternating group $$A_{12}$$. The authors prove that H is of minimal possible index in $$F_ 5$$ and study the permutation representation of $$F_ 5$$ on $$F_ 5/H$$. Namely, three theorems are proved:
Theorem 1. The minimal index of a proper subgroup in $$F_ 5$$ is $$1,140,000=| F_ 5:H|$$. Each subgroup of this index in $$F_ 5$$ is conjugate to H. - The permutation representation of $$F_ 5$$ on cosets $$F_ 5/H$$ has rank 12 (Theorem 2; it also gives subdegrees and stabilizers of two points). Theorem 3. $$F_ 5\neq HA$$ for every proper subgroup A in $$F_ 5$$. In particular, $$F_ 5$$ does not contain wide subgroups.
Reviewer: S.A.Syskin

### MSC:

 20D08 Simple groups: sporadic groups 20D60 Arithmetic and combinatorial problems involving abstract finite groups
Full Text:

### References:

 [1] V. D. Mazurov and A. N. Fomin, ”On finite simple nonabelian groups,” Mat. Zametki,34, No. 6, 821–824 (1983). · Zbl 0555.20010 [2] K. Harada, ”On the simple group F of order 214{$$\cdot$$}36{$$\cdot$$}56{$$\cdot$$}7{$$\cdot$$}11{$$\cdot$$}19,” in: Proc. Conf. Finite Groups (Utah, 1975). New York Academic Press (1976), pp. 119–276. [3] N. P. Mazurova, ”Subgroups of finite groups and the problem of linear optimization,” Algebra Logika,25, No. 4, 405–414 (1986). · Zbl 0631.20008 [4] J. Fischer and J. McKay, ”The nonabelian simple groups G, |G| < 106 – maximal subgroups,” Math. Comp.,32, No. 144, 1293–1302 (1978). · Zbl 0388.20010 [5] J. McKay, ”The non-Abelian simple groups G, |G| < 106 – character tables,” Commun. Algebra,7, No. 13, 1407–1445 (1979). · Zbl 0418.20009 [6] M. Hall, Jr., The Theory of Groups, Macmillan, New York (1959). [7] B. Huppert, Endliche Gruppen. I, Springer, Berlin (1979). [8] C. C. Sims, ”Computational methods in the study of permutation groups,” in: Computational Problems in Abstract Algebra (Proc. Conference, Oxford, 1967), Pergamon Press, Oxford (1970), pp. 169–183. [9] S. P. Norton and R. A. Wilson, ”Maximal subgroups of the Harada-Norton group,” J. Algebra,103, No. 1, 362–376 (1986). · Zbl 0595.20017 [10] J. H. Conway, R. T. Curtis, S. P. Norton, R. A. Parker, and R. A. Wilson, An Atlas of Finite Groups, Oxford Univ. Press, London (1985). · Zbl 0568.20001
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.


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# topology question: bounded linear operator

#### Poirot

##### Banned
Let T be a bounded linear operator on X and x be in X. Assume ||x||>||T||=1 It follows that ||Tx||>1. Can anyone explain why 0 is not in the closure of T(B), where B is the ball centered at x, with radius 1. Thanks

#### Ackbach

##### Indicium Physicus
Staff member
Re: topology question

Counterexample: let $X=\mathbb{R}^{2}$, and let $x= \langle 2,0 \rangle$, and let $T$ be the operator that rotates through an angle $\theta$. That is,
$$T= \begin{bmatrix} \cos( \theta) &-\sin( \theta) \\ \sin( \theta) & \cos( \theta) \end{bmatrix}.$$
Then $T$ is bounded and linear, with norm 1, and maps the ball $B_{1}(x)$ to another ball of radius 1, but it will not include zero. The rotated ball will always be a distance of 1 away from zero.

#### Poirot

##### Banned
Re: topology question

#### MarkFL

Staff member
Re: topology question

#### Klaas van Aarsen

##### MHB Seeker
Staff member
Re: topology question

Let T be a bounded linear operator on X and x be in X. Assume ||x||>||T||=1 It follows that ||Tx||>1. Can anyone explain why 0 is not in the closure of T(B), where B is the ball centered at x, with radius 1. Thanks
$||T||=1$ means that $||Tv|| \le ||v||$ for all $v$.

So with $||x||>||T||=1$, what follows, is that $||Tx|| \le ||x||$.
It does not follow that $||Tx|| > 1$.

In particular that means that it is possible for 0 to be in the closure of T(B).
Pick for instance $T: \mathbb R^2 \to \mathbb R^2, (x,y) \mapsto (x,0)$.
It is not necessarily the case that 0 is in the closure of T(B) as Ackbach has shown.

With the additional condition that $||Tx||>1$, it becomes a different matter.
For instance, the closest point to the origin is $$\displaystyle \frac{||x|| - 1}{||x||} x$$.
Its image is:
$$T\left(\frac{||x|| - 1}{||x||} x\right) = \frac{||x|| - 1}{||x||} Tx$$
This point is still on the same side of the origin.


text:
Just for interest, I point out that it is well-known that for every finite group $G,$ we can make the group algebra $\mathbb{C}G$ into a $G \times G$-module, by setting $a(g,h) = g^{-1}ah$ for all $a \in \mathbb{C}G,$ for all $(g,h) \in G \times G.$ It is easy to check that this representation is multiplicity-free. In fact, calculating the trace with respect to the natural group basis for $\mathbb{C}G,$ we see that ${\rm trace}(g,h) = |C_{G}(g)|$ if $g$ and $h$ are conjugate in $G,$ and $0$ if they are not conjugate. Hence the character of $G \times G$ afforded by this representation is $\sum_{i=1}^{k} \overline{\chi}_{i} \otimes \chi_{i}$, where $\{ \chi_{i} : 1 \leq i \leq k \}$ are the irreducible charactes of $G.$ It is also easy to see directly that the algebra of $G \times G$ endomorphisms of this module is isomorphic to the commutative ring $Z(\mathbb{C}G).$ This way of looking at the group algebra as a $G \times G$-module ( or as a bimodule for $G$) was profitably exploited in modular representation theory by J.A. Green. But in the context of this question, it illustrates that the existence of a faithful multiplicity free representation puts very little restriction on the structure of the group. Notice that if we take the group basis for $G,$ the module may be viewed as a transitive permutation module, and that the stabilizer of the identity element $1_{G}$ is the "diagonal" subgroup $\Delta(G) = \{ (g,g) : g \in G \}.$ It is easy to check that the module is faithful if and only if $G$ has trivial center. Another theme related to the question is that of "models". For example, R. Richardson constructed monomial representations of the symmetric group which contain every irreducible complex representation just once. These are not quite permutation representations: one takes a certain sign character for the centralizer of each involution (up to conjugacy) of the symmetric group $S_{n}$, and induces that up to $S_{n}$. This can be done in such a way that the sum of these is multiplicity free, and contains every irreducible character. For the general linear group, there are also Gelfand-Graev representations.
Just for interest, I point out that it is well-known that for every finite group $G,$ we can make the group algebra $\mathbb{C}G$ into a $G \times G$-module, by setting $a(g,h) = g^{-1}ah$ for all $a \in \mathbb{C}G,$ for all $(g,h) \in G \times G.$ It is easy to check that this representation is multiplicity-free. In fact, calculating the trace with respect to the natural group basis for $\mathbb{C}G,$ we see that ${\rm trace}(g,h) = |C_{G}(g)|$ if $g$ and $h$ are conjugate in $G,$ and $0$ if they are not conjugate. Hence the character of $G \times G$ afforded by this representation is $\sum_{i=1}^{k} \overline{\chi}_{i} \otimes \chi_{i}$, where $\{ \chi_{i} : 1 \leq i \leq k \}$ are the irreducible charactes of $G.$ It is also easy to see directly that the algebra of $G \times G$ endomorphisms of this module is isomorphic to the commutative ring $Z(\mathbb{C}G).$ This way of looking at the group algebra as a $G \times G$-module ( or as a bimodule for $G$) was profitably exploited in modular representation theory by J.A. Green. But in the context of this question, it illustrates that the existence of a faithful multiplicity free representation puts very little restriction on the structure of the group. Notice that if we take the group basis for $G,$ the module may be viewed as a transitive permutation module, and that the stabilizer of the identity element $1_{G}$ is the "diagonal" subgroup $\Delta(G) = \{ (g,g) : g \in G \}.$ It is easy to check that the module is faithful if and only if $G$ has trivial center.


text:
# ProjectiveHilbertPolynomial ZZ -- value of polynomial

## Description

P i -- the value of a projective Hilbert polynomial P at an integer i.

 i1 : P = projectiveHilbertPolynomial 2 o1 = P 2 o1 : ProjectiveHilbertPolynomial i2 : apply(0 .. 12, i -> P i) o2 = (1, 3, 6, 10, 15, 21, 28, 36, 45, 55, 66, 78, 91) o2 : Sequence


text:
# Linear Algebra/Complex Vector Spaces

Linear Algebra
 ← Introduction to Similarity Complex Vector Spaces Factoring and Complex Numbers: A Review →

This chapter requires that we factor polynomials. Of course, many polynomials do not factor over the real numbers; for instance, ${\displaystyle x^{2}+1}$ does not factor into the product of two linear polynomials with real coefficients. For that reason, we shall from now on take our scalars from the complex numbers.

That is, we are shifting from studying vector spaces over the real numbers to vector spaces over the complex numbers— in this chapter vector and matrix entries are complex.

Any real number is a complex number and a glance through this chapter shows that most of the examples use only real numbers. Nonetheless, the critical theorems require that the scalars be complex numbers, so the first section below is a quick review of complex numbers.

In this book we are moving to the more general context of taking scalars to be complex only for the pragmatic reason that we must do so in order to develop the representation. We will not go into using other sets of scalars in more detail because it could distract from our goal. However, the idea of taking scalars from a structure other than the real numbers is an interesting one. Delightful presentations taking this approach are in (Halmos 1958) and (Hoffman & Kunze 1971).

## References

• Halmos, Paul P. (1958), Finite Dimensional Vector Spaces (Second ed.), Van Nostrand .
• Hoffman, Kenneth; Kunzy, Ray (1971), Linear Algebra (Second ed.), Prentice-Hall .
Linear Algebra
 ← Introduction to Similarity Complex Vector Spaces Factoring and Complex Numbers: A Review →


text:
# Can y=3x^2-39x-90  be factored? If so what are the factors ?

Dec 16, 2015

#### Answer:

Factor y = 3x^2 - 39x - 90

Ans: y = (x +2)(x -15)

#### Explanation:

I use the new AC Method to factor trinomials (Socratic Search)
$y = 3 Y = 3 \left({x}^{2} - 13 x - 30\right)$. Factor Y = x^2 - 13x - 30).
Find 2 numbers knowing sum (b = -13) and product (c = -30)
Factor pairs of (-30) --> (-2, 15). This sum is 13 = -b. Then the opposite sum gives 2 numbers: 2 and - 15.

Factored form: $y = 3 \left(x + 2\right) \left(x - 15\right) =$


text:
# Homework Help - Area of Trapezium

• September 3rd 2008, 12:12 PM
ajnokia
Homework Help - Area of Trapezium
Struggling with this question and so are my mates.

Attached is the Graph.

The line PQ has equation y=2x+4

(a) Find, without using calculus, the area of the shaded trapezium shown in the diagram.

(b) Express the area of this trapezium as a definite integral.

(c) Evaluate this integral

I just dont know where to go about tackling this question at all, I feel its the english putting me off. If anyone can help me to find the solutions to this question it will help me and assure I know what I am doing when I face similar questions in future.

Many Thanks

PS. The Q should be right at the end of the line with P..... the digit close to the P is a 0 not a Q, sorry about that
• September 3rd 2008, 12:22 PM
Chop Suey
The area of a trapezoid is:
$A_{\text{trapezoid}} = \frac{h (b_1 +b_2)}{2}$

Where h is the height and $b_1$ and $b_2$ are the upper and lower bases.

You might need to change your perspective. Remember that the bases have to be parallel sides!

Now for the next question....you have to know that the integral is the area under the curve (or line). Here, you're trying to find the area under the curve between 2 and 5.

As for question (c): I guess you have to integrate $f(x) = 2x + 4$
• September 3rd 2008, 12:27 PM
Chop Suey
Quote:

Originally Posted by Chop Suey
The area of a trapezoid is:
$A_{\text{trapezoid}} = \frac{h (b_1 +b_2)}{2}$

Where h is the height and $b_1$ and $b_2$ are the upper and lower bases.

You might need to change your perspective. Remember that the bases have to be parallel sides!

Or, you can think of the trapezium as a combination of rectangle and a triangle.

Recall that:
$A_{Triangle} = \frac{1}{2}bh$

$A_{Rectangle} = b \cdot h$
• September 3rd 2008, 01:03 PM
ajnokia
Quote:

Originally Posted by Chop Suey
Or, you can think of the trapezium as a combination of rectangle and a triangle.

Recall that:
$A_{Triangle} = \frac{1}{2}bh$

$A_{Rectangle} = b \cdot h$

So the base is 3. But I dont know the height for the rectangle or the triangle. So unless I'm missing something (which I probably am due to tiredness) I cant manage to work out the area of the triangle nor the rectangle?

I know what Im doing with and (b) and (c) now though (Happy) Thanks!
• September 3rd 2008, 01:09 PM
Chop Suey
The height of the rectangle is given by $f(x) = 2x+4$. The height of the triangle is given by $f(5) - f(2)$. Do you understand why?
• September 3rd 2008, 01:11 PM
ajnokia
Quote:

Originally Posted by Chop Suey
The height of the rectangle is given by $f(x) = 2x+4$. The height of the triangle is given by $f(5) - f(2)$. Do you understand why?

y = 2x+4 is for the line PQ so I dont understand how that becomes the height of the rectangle.
• September 3rd 2008, 01:19 PM
Chop Suey
The endpoints of the two segments lie on the line y = 2x + 4. To find the length of these two lines, we find f(5) and f(2).
• September 3rd 2008, 01:19 PM
leftyguitarjoe
Quote:

Originally Posted by ajnokia
y = 2x+4 is for the line PQ so I dont understand how that becomes the height of the rectangle.

your base goes from 2 to 5

just find y then x=5
• September 3rd 2008, 01:20 PM
leftyguitarjoe
Quote:

Originally Posted by Chop Suey
The endpoint of the two lines lie on the line y = 2x + 4. To find the length of these two lines, we find f(5) and f(2).

gah, ya beat me(Giggle)


text:
Ngo's support theorem & the Jacobian over the versal deformation of a planar curve - MathOverflow most recent 30 from http://mathoverflow.net 2013-05-21T12:54:00Z http://mathoverflow.net/feeds/question/38549 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/38549/ngos-support-theorem-the-jacobian-over-the-versal-deformation-of-a-planar-curv Ngo's support theorem & the Jacobian over the versal deformation of a planar curve Vivek Shende 2010-09-13T09:23:21Z 2010-09-13T10:44:42Z
I am trying to understand what exactly Ngo's support theorem says about how the cohomology of compactified Jacobians varies over the versal deformation of a curve with plane singularities.

That is, let $C$ be a compact complex curve with singularities of embedding dimension 2. Let $\mathcal{C} \to \mathbf{V}$ be a versal deformation, i.e., there is a point $0 \in \mathbf{V}$ over which the fibre is $C$, and any deformation of $C$ pulls back from this family. Let $\pi: \overline{\mathcal{J}} \to \mathbf{V}$ be the relative compactified Jacobian of the family; I want to understand the family of cohomologies $\mathrm{R}\pi_* \mathbb{Q}$. As pointed out in the few references I've browsed, this is a situtation to which Ngo's support theorem applies: there is a relative action of the (uncompactified but generalized) Jacobian $\mathcal{J}$, and the locus $\mathcal{V}_h$ where the affine part of $\mathcal{J}$ is of dimension $h$ is of precisely codimension $h$ (by an old result of Teissier). However, said references go on to treat a more famous example instead -- the Hitchin fibration.

Unfortunately my competence with the machinery of perverse sheaves is so poor that I can barely understand the statement of the support theorem. So, I apologize if the following questions are overly trivial or confused.

0.) Is the application of the support theorem to the situation I describe above treated explicitly in the literature?

1.) Does the support theorem imply that $\mathrm{R} \pi_* \mathbb{Q}$ is the IC-sheaf on $\mathbf{V}$ determined by the local system of cohomologies of Jacobians on the locus $\mathbf{V}_{\mathrm{reg}} \subset \mathbf{V}$ of smooth curves? If not, does it specify what other local systems on what other loci must be added?

2.) Does the knowledge of the Picard-Lefshetz monodromy around nodal degenerations determine (explicitly) the local system of cohomologies of Jacobians on $\mathbf{V}_{\mathrm{reg}}$, and, if so, does it allow an in principle calculation of the cohomology of the central fibre? Can it be done in practice?

3.) Does it follow that the cohomology of the compactified Jacobian of the central fibre is determined by the topology of the singularity of the discriminant (the complement of the locus of smooth curves)? Or -- perhaps it is necessary to know how monodromies around different nodes interact -- the stratification of the discriminant by the (closures of the) loci with a fixed number of nodes?

4.) Any other comments on how this (or anything else) would be used to compute the cohomology of the compactified Jacobian are welcome.
 http://mathoverflow.net/questions/38549/ngos-support-theorem-the-jacobian-over-the-versal-deformation-of-a-planar-curv/38557#38557 Answer by Sam Gunningham for Ngo's support theorem & the Jacobian over the versal deformation of a planar curve Sam Gunningham 2010-09-13T10:37:10Z 2010-09-13T10:44:42Z
Very partial answer - I don't think I can comment yet...

I found it helpful to rephrase the statement of the support theorem like this:

Let $R\pi_\ast \mathbb Q = \bigoplus _i IC_{Z_i}(L_i)[d_i]$ be the decomposition of the pushforward sheaf. If Z is one of the $Z_i$ appearing in the sum above, then $Z$ is the support of a direct summand of $R^{2d}\pi_\ast \mathbb Q$ (i.e. there is extra stuff appearing in the top cohomology of the fibres). Such a summand only will occur when there are extra irreducible components in the fibres of $\pi$.

In particular, if the fibres of $\pi$ are irreducible, then the only possible support appearing in the direct sum is the whole of V. This means that the pushforward is the IC extension of the local system over the locus where $\pi$ is smooth.

I think this means that in your situation, the answer to (1) is yes (as compactified Jacobians are irreducible for such singularities). I would be interested to find out the answers to (2) and (3).


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Hardcover | $100.00 Short | £68.95 | ISBN: 9780262112789 | 648 pp. | 8.5 x 11 in | 128 illus.| September 2003 Ebook |$70.00 Short | ISBN: 9780262252423 | 648 pp. | 8.5 x 11 in | 128 illus.| September 2003

# The MIT Encyclopedia of Communication Disorders

Edited by Raymond D. Kent

## Overview

A massive reference work on the scale of MITECS (The MIT Encyclopedia of Cognitive Sciences), The MIT Encyclopedia of Communication Disorders will become the standard reference in this field for both research and clinical use. It offers almost 200 detailed entries, covering the entire range of communication and speech disorders in children and adults, from basic science to clinical diagnosis.MITECD is divided into four sections that reflect the standard categories within the field (also known as speech-language pathology and audiology): Voice, Speech, Language, and Hearing. Within each category, entries are organized into three subsections: Basic Science, Disorders, and Clinical Management. Basic Science includes relevant information on normal anatomy and physiology, physics, psychology and psychophysics, and linguistics; this provides a scientific foundation for entries in the other subsections. The entries that appear under Disorders offer information on the definition and characterization of specific disorders, and tools for their identification and assessment. The Clinical Management subsection describes appropriate interventions, including behavioral, pharmacological, surgical, and prosthetic.Because the approach to communication disorders can be quite different for children and adults, many topics include separate entries reflecting this. Although some disorders that are first diagnosed in childhood may persist in some form throughout adulthood, many disorders can have an onset in either childhood or adulthood, and the timing of onset can have many implications for both assessment and intervention.Topics covered in MITECD include cochlear implants for children and adults, pitch perception, tinnitus, alaryngeal voice and speech rehabilitation, neural mechanisms of vocalization, holistic voice therapy techniques, computer-based approaches to children?s speech and language disorders, neurogenic mutism, regional dialect, agrammatism, global aphasia, and psychosocial problems associated with communicative disorders.


text:
# Research articles for the 2020-08-24

A Blockchain Transaction Graph based Machine Learning Method for Bitcoin Price Prediction
Xiao Li,Weili Wu
arXiv

Bitcoin, as one of the most popular cryptocurrency, is recently attracting much attention of investors. Bitcoin price prediction task is consequently a rising academic topic for providing valuable insights and suggestions. Existing bitcoin prediction works mostly base on trivial feature engineering, that manually designs features or factors from multiple areas, including Bticoin Blockchain information, finance and social media sentiments. The feature engineering not only requires much human effort, but the effectiveness of the intuitively designed features can not be guaranteed. In this paper, we aim to mining the abundant patterns encoded in bitcoin transactions, and propose k-order transaction graph to reveal patterns under different scope. We propose the transaction graph based feature to automatically encode the patterns. A novel prediction method is proposed to accept the features and make price prediction, which can take advantage from particular patterns from different history period. The results of comparison experiments demonstrate that the proposed method outperforms the most recent state-of-art methods.

A Computational Approach to Hedging Credit Valuation Adjustment in a Jump-Diffusion Setting
T. van der Zwaard,L.A. Grzelak,C.W. Oosterlee
arXiv

This study contributes to understanding Valuation Adjustments (xVA) by focussing on the dynamic hedging of Credit Valuation Adjustment (CVA), corresponding Profit & Loss (P&L) and the P&L explain. This is done in a Monte Carlo simulation setting, based on a theoretical hedging framework discussed in existing literature. We look at hedging CVA market risk for a portfolio with European options on a stock, first in a Black-Scholes setting, then in a Merton jump-diffusion setting. Furthermore, we analyze the trading business at a bank after including xVAs in pricing. We provide insights into the hedging of derivatives and their xVAs by analyzing and visualizing the cash-flows of a portfolio from a desk structure perspective. The case study shows that not charging CVA at trade inception results in an expected loss. Furthermore, hedging CVA market risk is crucial to end up with a stable trading strategy. In the Black-Scholes setting this can be done using the underlying stock, whereas in the Merton jump-diffusion setting we need to add extra options to the hedge portfolio to properly hedge the jump risk. In addition to the simulation, we derive analytical results that explain our observations from the numerical experiments. Understanding the hedging of CVA helps to deal with xVAs in a practical setting.

Binary Funding Impacts in Derivative Valuation
Junbeom Lee,Chao Zhou
arXiv

We discuss the binary nature of funding impact in derivative valuation. Under some conditions, funding is either a cost or a benefit, i.e., one of the lending/borrowing rates does not play a role in pricing derivatives. When derivatives are priced, considering different lending/borrowing rates leads to semi-linear BSDEs and PDEs, and thus it is necessary to solve the equations numerically. However, once it can be guaranteed that only one of the rates affects pricing, linear equations can be recovered and analytical formulae can be derived. Moreover, as a byproduct, our results explain how debt value adjustment (DVA) and funding benefits are dissimilar. It is often believed that considering both DVA and funding benefits results in a double-counting issue but it will be shown that the two components are affected by different mathematical structures of derivative transactions. We find that funding benefit is related to the decreasing property of the payoff function, but this relationship decreases as the funding choices of underlying assets are transferred to repo markets.

Can Technical Analysis Generate Superior Returns in Securitized Property Markets? Evidence From East Asia Markets
SSRN
This study applies various popular technical trading rules to Asian property market indices from 1995 to 2015 to investigate the profitability of these rules. The results validate the predictive and profitability power of technical indicators in the markets of Indonesia, Malaysia, Taiwan, and Thailand but not in the markets of China, Hong Kong, Japan, the Philippines, and Singapore. For markets in which technical indicators are predictive, this ability is proven useful in generating returns in excess of buy-and-hold returns using four different trading strategies. The study's results hold even after including transaction costs, adjustments for risk, and data snooping. The results for the markets in which technical analysis is fruitful contradict weak-form market efficiency, whereas markets in which technical analysis is not successful are at least weakly efficient.

Competitive ride-sourcing market with a third-party integrator
Yaqian Zhou,Hai Yang,Jintao Ke,Hai Wang,Xinwei Li
arXiv

Recently, some transportation service providers attempt to integrate the ride services offered by multiple independent ride-sourcing platforms, and passengers are able to request ride through such third-party integrators or connectors and receive service from any one of the platforms. This novel business model, termed as third-party platform-integration in this paper, has potentials to alleviate the cost of market fragmentation due to the demand splitting among multiple platforms. While most existing studies focus on the operation strategies for one single monopolist platform, much less is known about the competition and platform-integration as well as the implications on operation strategy and system efficiency. In this paper, we propose mathematical models to describe the ride-sourcing market with multiple competing platforms and compare system performance metrics between two market scenarios, i.e., with and without platform-integration, at Nash equilibrium as well as social optimum. We find that platform-integration can increase total realized demand and social welfare at both Nash equilibrium and social optimum, but may not necessarily generate a greater profit when vehicle supply is sufficiently large or/and market is too fragmented. We show that the market with platform-integration generally achieves greater social welfare. On one hand, the integrator in platform-integration is able to generate a thicker market and reduce matching frictions; on the other hand, multiple platforms are still competing by independently setting their prices, which help to mitigate monopoly mark-up as in the monopoly market.

Discounted optimal stopping of a Brownian bridge, with application to American options under pinning
arXiv

Mathematically, the execution of an American-style financial derivative is commonly reduced to solving an optimal stopping problem. Breaking the general assumption that the knowledge of the holder is restricted to the price history of the underlying asset, we allow for the disclosure of future information about the terminal price of the asset by modeling it as a Brownian bridge. This model may be used under special market conditions, in particular we focus on what in the literature is known as the "pinning effect", that is, when the price of the asset approaches the strike price of a highly-traded option close to its expiration date. Our main mathematical contribution is in characterizing the solution to the optimal stopping problem when the gain function includes the discount factor. We show how to numerically compute the solution and we analyze the effect of the volatility estimation on the strategy by computing the confidence curves around the optimal stopping boundary. Finally, we compare our method with the optimal exercise time based on a geometric Brownian motion by using real data exhibiting pinning.

Does Global Outsourcing Affect Corporate Customersâ€™ Voluntary Disclosures?
Dai, Lili,Dai, Rui,Ng, Lilian K.,Peng, Zihang Ryan
SSRN
Global outsourcing has become an economic imperative for many major corporations worldwide, but at the same time, it has brought substantial risks and uncertainties to these firms. In this study, we examine whether global sourcing of goods or services shapes U.S. corporate disclosure policies. Our main results indicate a negative relation between global outsourcing and voluntary disclosure. Further, identification tests show that managers reduce voluntary disclosures when they first outsource their inputs globally and when suppliersâ€™ countries experience spikes in trade uncertainty and political risks. The global outsourcing effect on disclosure is stronger when corporate customers face a higher litigation risk, when there is less information transfer between customers and suppliers, and when customers are in more competitive product markets. Collectively, these results suggest that corporate customers decrease voluntary disclosures due to increased uncertainties along the global supply chain.

SSRN
This paper tests for the ability of a variety of technical indicators to generate excess returns at the individual stock level in the seven emerging and frontier markets of the Gulf region. While technical indicators show some early profitability promise, after controlling for the data snooping bias using the False Discovery Rate (FDR) methodology and non-synchronous trading, we fail to find any predictive ability or profitability for technical analysis. We arrive at a similar finding when assessing the risk-adjusted performance of a portfolio composed of stocks chosen based on technical indicators. The findings go to show the failure of technical analysis on the stock level. The findings are also evidence of the Gulf region markets being at least weak-form efficient and carry implications for investors choice of investment tools.

Geometric Arbitrage Theory and Market Dynamics Reloaded
Simone Farinelli
arXiv

We have embedded the classical theory of stochastic finance into a differential geometric framework called Geometric Arbitrage Theory and show that it is possible to:

--Write arbitrage as curvature of a principal fibre bundle.

--Parameterize arbitrage strategies by its holonomy.

--Give the Fundamental Theorem of Asset Pricing a differential homotopic characterization.

--Characterize Geometric Arbitrage Theory by five principles and show they they are consistent with the classical theory of stochastic finance.

--Derive for a closed market the equilibrium solution for market portfolio and dynamics in the cases where:

-->Arbitrage is allowed but minimized.

-->Arbitrage is not allowed.

--Prove that the no-free-lunch-with-vanishing-risk condition implies the zero curvature condition. The converse is in general not true and additionally requires the Novikov condition for the instantaneous Sharpe Ratio Dynamics to be satisfied.

Implications of the Tradeoff between Inside and Outside Social Status in Group Choice
arXiv

We investigate a group choice problem of agents pursuing social status. We assume heterogeneous agents want to signal their private information (ability, income, patience, altruism, etc.) to others, facing tradeoff between "outside status" (desire to be perceived in prestigious group from outside observers) and "inside status" (desire to be perceived talented from peers inside their group). To analyze the tradeoff, we develop two stage signaling model in which each agent firstly chooses her group and secondly chooses her action in the group she chose. They face binary choice problems both in group and action choices. Using cutoff strategy, we construct an partially separating equilibrium such that there are four populations: (i) choosing high group with strong incentive for action in the group, (ii) high group with weak incentive, (iii) low group with strong incentive, and (iv) low group with weak incentive. By comparative statics results, we find some spillover effects from a certain group to another, on how four populations change, when a policy is taken in each group. These results have rich implications for group choice problems like school, firm or residential preference.

Is culture related to strong science? An empirical investigation
Mahmood Khosrowjerdi,Lutz Bornmann
arXiv

National culture is among those societal factors which could influence research and innovation activities. In this study, we investigated the associations of two national culture models with citation impact of nations (measured by the proportion of papers belonging to the 10% and 1% most cited papers in the corresponding fields, PPtop 10% and PPtop 1%). Bivariate statistical analyses showed that of six Hofstede's national culture dimensions (HNCD), three dimensions of power distance, individualism, and uncertainty avoidance had statistically significant associations with citation impact of nations. The study also revealed that of two Inglehart-Welzel cultural values (IWCV), the value survival versus self-expression is statistically significantly related to citation impact indicators. We additionally calculated multiple regression analyses controlling for the possible effects of confounding factors including international migrant stock, investments in research and development, number of researchers, international co-authorships, and national self-citations. The results revealed that the statistically significant associations of HNCD with citation impact indicators disappeared. But the statistically significant relationship between survivals versus self-expression values and citation impact indicators remained stable even after controlling for the confounding variables. Thus, the freedom of expression and trust in society might contribute to better scholarly communication systems, higher level of international collaborations, and further quality research.

Optimizing tail risks using an importance sampling based extrapolation for heavy-tailed objectives
Anand Deo,Karthyek Murthy
arXiv

Motivated by the prominence of Conditional Value-at-Risk (CVaR) as a measure for tail risk in settings affected by uncertainty, we develop a new formula for approximating CVaR based optimization objectives and their gradients from limited samples. A key difficulty that limits the widespread practical use of these optimization formulations is the large amount of data required by the state-of-the-art sample average approximation schemes to approximate the CVaR objective with high fidelity. Unlike the state-of-the-art sample average approximations which require impractically large amounts of data in tail probability regions, the proposed approximation scheme exploits the self-similarity of heavy-tailed distributions to extrapolate data from suitable lower quantiles. The resulting approximations are shown to be statistically consistent and are amenable for optimization by means of conventional gradient descent. The approximation is guided by means of a systematic importance-sampling scheme whose asymptotic variance reduction properties are rigorously examined. Numerical experiments demonstrate the superiority of the proposed approximations and the ease of implementation points to the versatility of settings to which the approximation scheme can be applied.

Paying Too Much? Price Dispersion in the U.S. Mortgage Market
Bhutta, Neil,Fuster, Andreas,Hizmo, Aurel
SSRN
We document wide dispersion in the mortgage rates that households pay on identical loans, and show that borrowers' financial sophistication is an important determinant of the rates obtained. We estimate a gap between the 10th and 90th percentile mortgage rate that borrowers with the same characteristics obtain for identical loans, in the same market, on the same day, of 54 basis points|equivalent to about $6,500 in upfront costs (points) for the average loan. Time-invariant lender attributes explain little of this rate dispersion, and considerable dispersion remains even within loan officer. Comparing the rates borrowers obtain to the real-time distribution of rates that lenders could offer for the same loan and borrower type, we find that borrowers who are likely to be the least financially savvy tend to substantially overpay relative to the rates available in the market. In the time series, the average overpayment decreases when overall market interest rates rise, suggesting that a rising level of borrowing costs encourages more search and negotiation. Furthermore, new survey data provide direct evidence that financial knowledge and shopping both affect the mortgage rates borrowers get, and that shopping activity increases with the level of market rates. Portfolio Selection under Median and Quantile Maximization Xue Dong He,Zhaoli Jiang,Steven Kou arXiv In this paper, we study a portfolio selection problem in which an agent trades a risk-free asset and multiple risky assets with deterministic mean return rates and volatility and wants to maximize the$\alpha$-quantile of her wealth at some terminal time. Because of the time inconsistency caused by quantiles, we consider intra-personal equilibrium strategies. We find that among the class of time-varying, affine portfolio strategies, the intra-personal equilibrium does not exist when$\alpha>1/2$, leads to zero investment in the risky assets when$\alpha<1/2$, and is a portfolio insurance strategy when$\alpha=1/2$. We then compare the intra-personal equilibrium strategy in the case of$\alpha=1/2\$, namely under median maximization, to some other strategies and apply it to explain why more wealthy people invest more precentage of wealth in risky assets. Finally, we extend our model to account for multiple terminal time.

Power-type derivatives for rough volatility with jumps
Weixuan Xia
arXiv

In this paper we propose an efficient pricing-hedging framework for volatility derivatives which simultaneously takes into account path roughness and jumps. Instead of dealing with log-volatility, we directly model the instantaneous variance of a risky asset in terms of a fractional Ornstein-Uhlenbeck process driven by an infinite-activity L\'{e}vy subordinator, which is shown to exhibit roughness under suitable conditions and also eludes the need for an independent Brownian component. This structure renders the characteristic function of forward variance obtainable at least in semi-closed form, subject to a generic integrable kernel. To analyze financial derivatives, primarily swaps and European-style options, on average forward volatility, we introduce a general class of power-type derivatives on the average forward variance, which also provide a way of adjusting the option investor's risk exposure. Pricing formulae are based on numerical inverse Fourier transform and, as illustrated by an empirical study on VIX options, permit stable and efficient model calibration once specified.

Pricing Efficiency of Saudi Exchange Traded Funds (ETFs)
SSRN
This paper aims to investigate the pricing efficiency of Saudi Sharia-compliant (i.e. Islamic) exchange-traded funds (ETFs). The paper adheres to a positivist research philosophy with a deductive research approach where data is collected, analyzed and interpreted to examine a hypothesis. Ordinary least squares (OLS) regressions are applied to investigate pricing efficiency and persistence. The results show that Saudi ETFs do not currently offer proper diversification for investors, possibly due to their low trading volumes and the delays of market prices in reflecting net asset value (NAV). On average, ETFs trade at a premium to their NAVs. Moreover, the authors find that the deviations of ETF prices from their NAVs (i.e. premiums or discounts) do not disappear in one day. The results reveal a significant positive relationship between the trading volume of Saudi ETFs and volatility, a significant positive correlation between ETF returns and contemporaneous deviations and a significant negative relationship between returns and lagged deviations. These findings can be interpreted as evidence against the market efficiency of Saudi ETFs. Individual and institutional investors can use Saudi ETFs, especially as their efficiency improves with increased trading volume (liquidity). Saudi regulators must increase their efforts to educate market participants and expand the availability of information to enhance transparency and awareness of the benefits of investing in ETFs, which will positively affect liquidity and pricing efficiency in the future. This paper is the first to perform empirical tests on Saudi ETFs. Saudi Arabia deserves further attention because it is the most significant stock market in the Gulf Cooperation Council and only recently allowed foreigners to participate.

Stationarity of the detrended price return in stock markets
Karina Arias-Calluari,Morteza. N. Najafi,Michael S. Harré,Fernando Alonso-Marroquin
arXiv

This paper proposes a governing equation for stock market indexes that accounts for non-stationary effects. This is a linear Fokker-Planck equation (FPE) that describes the time evolution of the probability distribution function (PDF) of the price return. By applying Ito's lemma, this FPE is associated with a stochastic differential equation (SDE) that models the time evolution of the price return in a fashion different from the classical Black-Scholes equation. Both FPE and SDE equations account for a deterministic part or trend, and a stationary, stochastic part as a q-Gaussian noise. The model is validated using the S\&P500 index's data. After removing the trend from the index, we show that the detrended part is stationary by evaluating the Hurst exponent of the multifractal time series, its power spectrum, and its autocorrelation.

Statistical Learning for Probability-Constrained Stochastic Optimal Control
arXiv

We investigate Monte Carlo based algorithms for solving stochastic control problems with probabilistic constraints. Our motivation comes from microgrid management, where the controller tries to optimally dispatch a diesel generator while maintaining low probability of blackouts. The key question we investigate are empirical simulation procedures for learning the admissible control set that is specified implicitly through a probability constraint on the system state. We propose a variety of relevant statistical tools including logistic regression, Gaussian process regression, quantile regression and support vector machines, which we then incorporate into an overall Regression Monte Carlo (RMC) framework for approximate dynamic programming. Our results indicate that using logistic or Gaussian process regression to estimate the admissibility probability outperforms the other options. Our algorithms offer an efficient and reliable extension of RMC to probability-constrained control. We illustrate our findings with two case studies for the microgrid problem.

Stock Buybacks: Value Creation or Destruction?
Gordon, Michael
SSRN

Tail risk forecasting using Bayesian realized EGARCH models
Vica Tendenan,Richard Gerlach,Chao Wang
arXiv

This paper develops a Bayesian framework for the realized exponential generalized autoregressive conditional heteroskedasticity (realized EGARCH) model, which can incorporate multiple realized volatility measures for the modelling of a return series. The realized EGARCH model is extended by adopting a standardized Student-t and a standardized skewed Student-t distribution for the return equation. Different types of realized measures, such as sub-sampled realized variance, sub-sampled realized range, and realized kernel, are considered in the paper. The Bayesian Markov chain Monte Carlo (MCMC) estimation employs the robust adaptive Metropolis algorithm (RAM) in the burn in period and the standard random walk Metropolis in the sample period. The Bayesian estimators show more favourable results than maximum likelihood estimators in a simulation study. We test the proposed models with several indices to forecast one-step-ahead Value at Risk (VaR) and Expected Shortfall (ES) over a period of 1000 days. Rigorous tail risk forecast evaluations show that the realized EGARCH models employing the standardized skewed Student-t distribution and incorporating sub-sampled realized range are favored, compared to a range of models.

The Analysis of Benefits in Consumer Protection Regulation
Jackson, Howell E.,Rothstein, Paul
SSRN
Over the past decade, cost-benefit analysis in the field of financial regulation (â€œfinancial CBAâ€) has emerged as a topic of intense public interest. In reviewing rulemakings under the Administrative Procedure Act, courts have demanded greater rigor in the financial CBA that regulators provide in support of new regulations. Industry experts and other analysts have repeatedly questioned the adequacy of agency assessments of costs and benefits. And legal academics have engaged in a robust dialogue over the merits of financial CBA and the value of alternative institutional structures for overseeing financial CBA.This Article adds to the expanding literature on financial CBA by offering a detailed study of how regulatory agencies actually undertake benefit analysis in promulgating new regulations involving matters of consumer finance and other analogous areas of consumer protection. After a brief literature review, the Article proposes a taxonomy for categorizing benefit analysis in the area of consumer financial regulation. This taxonomy reflects traditional market failures, cognitive limitations of consumers, as well as several other beneficial outcomes commonly associated with regulations designed to protect consumers. Taking the taxonomy as a framework, the Article then reports on a detailed survey of seventy-two consumer protection regulations adopted in recent years, and presents an overview of the range and quality of benefit analysis that government officials actually undertook in the surveyed regulations. The Article next provides a more detailed discussion of twenty â€œexemplarsâ€ of benefit analysis drawn from regulations in the sample and focusing on the strengths and weaknesses of what might be considered state-of-the-art benefit analysis in consumer protection regulation in the years immediately following the enactment of the Dodd-Frank Act. The Article concludes with a discussion of potential lines of academic research and institutional reform that might assist financial regulators in conducting more complete benefit analysis for consumer protection regulation in the future.

The Economic Impacts of Direct Natural Disaster Exposure
Johar, Meliyanni,Johnston, David W.,Shields, Michael,Siminski, Peter,Stavrunova, Olena
SSRN
This paper studies how having your home damaged or destroyed by a natural disaster impacts on economic and financial outcomes. Our context is Australia, where disasters are frequent. Estimates of regression models with individual, area and time fixed-effects, applied to 10 waves of data (2009-2018), indicate that residential destruction has no average impact on employment and income, but increases financial hardship and financial risk aversion. These impacts are generally short-lived, larger for renters than home owners, and greater for smaller isolated disasters. Using a Group Fixed Effects estimator, we find that around 20% of the population have low resilience to financial shocks, and for these individuals we find a substantive increase in financial hardships. The most vulnerable are the young, single parents, those in poor health, those of lower socioeconomic status, and those with little social support. These results can help target government aid after future natural disasters to those with the greatest need.

The Economics of Soft Dollars: A Review of the Literature and New Evidence from the Implementation of MiFID II
Jackson, Howell E.,Zhang, Jeffery
SSRN
For nearly half a century, the bundling of research services into commissions that paid for the execution of securities trades has been the focus of both policy discussion and academic debate. The practice whereby asset management firms make use of investor funds to cover the costs of research, known as â€œsoft dollarâ€ payments in the United States, resembles a form of kickback or self-dealing. The payments allow asset managers to use investor funds to subsidize the cost of the asset managersâ€™ own research efforts even though those managers charge investors a separate and explicit management fee for advisory services. So why does this form of kickback continue to exist? Over the years, defenders of the practice have argued that soft dollars mitigate principal-agent problems between the investment manager and the broker, improve fund performance, and provide a public good in terms of the increased production of research on public companies. This article evaluates these theoretical arguments through the lens of academic work done in the past as well as an emerging new body of empirical studies exploring the impact of MiFID II, a European Union Directive that severely restricted the use of soft dollar payments in European capital markets as of January 2018. The weight of empirical evidence, including recent evidence coming out of Europe, suggests that the theoretical arguments in favor of soft dollars are not robust. In particular, MiFID IIâ€™s unbundling of commissions appears to have, on balance, improved European market efficiency by eliminating redundancy and producing information that is of greater value to investors.

The Nature of the Fintech Firm and its Implications for Financial Regulation
Jackson, Howell E.
SSRN
This chapter explores recent Fintech innovations through the lens of Ronald Coaseâ€™s classic article: The Nature of the Firm. Applying a transaction cost analysis, the chapter argues that developments in computer technology, data processing, and information networks are reshaping the manner in which financial services are produced, unsettling the boundaries separating regulated firms from outside vendors and open market transactions. These changes raise challenging questions as to the appropriate contours of regulatory perimeters as well as the structure of regulation and supervision in the many area of financial regulation. Fintech innovations also have the potential to be harnessed to serve public purposes, including expanding access to financial services and improving supervisory practices. At a minimum, Fintech innovations and most especially machine learning and artificial intelligence complicate the application of legal doctrines based on human intentionality. More broadly, the scale and scope of these technological developments may lead to a fundamentally rethinking of the appropriate goals of regulatory policy for financial firms and the economy more broadly, particularly with respect to privacy and the accumulation of personal information in private and public hands.

Voluntary Disclosure and Real Earnings Management in SEO Firms: Evidence from the 2005 Securities Offering Reform
Zhu, Yicheng
SSRN
The Securities Offering Reform (SOR) in 2005 eased restrictions on firm disclosure prior to equity offerings. Prior literature find a better information environment with increased disclosure and reduced information asymmetry in seasoned equity offering (SEO) firms after the regulation. Building on these findings, our paper documents that SEO firms reduce real earnings management activities after the SOR became effective. Our results indicate a substitution effect between voluntary disclosure and real earnings management in SEO firms. Specifically, when restrictions on disclosure prior to equity offerings are removed, SEO firms opt for increased disclosure and reduced level of real activities manipulation. Our paper adds to the literature on earnings management activities around SEOs. It contributes to the studies on the impact of voluntary disclosure on earnings management by employing a difference-in-differences design to address the endogeneity issue in this line of research. It also has important implications for regulations on securities offerings.


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# Recent questions tagged nonzero

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+
Shooting method for solving 1D quantum potentials

Developed by Andy Rundquist - Published October 3, 2016

Students will learn how to use the numeric shooting method to find the eigenenergies and eigenfunctions for 1D wells, including infinite square wells, finite square wells, and the Coulomb potential of the hydrogen atom.
Subject Areas Modern Physics and Quantum Mechanics Beyond the First Year Mathematica * Students will be able to describe the shooting method for solving for eigenenergies and eigenfunctions. * Students will be able to use appropriate units for distance, energy, and time so that analytical and numeric results can be compared. * Students will be able to model any shape potential well. * Students will be able to calculate other parameters from the eigenfunctions (like expectation values). 60 min

These exercises are not tied to a specific programming language. Example implementations are provided under the Code tab, but the Exercises can be implemented in whatever platform you wish to use (e.g., Excel, Python, MATLAB, etc.).

1. Use the shooting method to determine the eigenenergies for an electron in an infinite square well of width 2 nm. Compare with the expected analytical result (with appropriate units). 2. Use the shooting method to plot the lowest few eigenfunctions for an infinite well with a shape determined by you (anything other than V=0 is fine inside the well). 3. For one of the solutions you've found in (2), determine the expectation values of: * x * $x^2$ * p (momentum) * $p^2$ 4. Using the shooting method, determine the lowest 10 energies of the hydrogen atom. Don't forget to try different values of the angular momentum. 5. For one of the solutions in (4) determine the expectation values of: * r * $r^2$


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1

# The non-Markovian nature of turbulence 7: non-Markovian closures and the LET theory in particular.

The non-Markovian nature of turbulence 7: non-Markovian closures and the LET theory in particular.

We can sum up the situation regarding the failure of the pioneering closures as follows. Their form of the transfer spectrum $T(k)$, with its division into input and output parts, with the latter being proportional to the amount of energy in mode $k$, is only valid for Markov processes, so it is incompatible with the nature of turbulence which is non-Markovian. It is also incompatible with the phenomenology of turbulence, where the entire $T(k)$ acts as input (or output), depending on the value of $k$, as I pointed out in 1974 [1]. It is worth noting that the first measurement of $T(k)$ was made by Uberoi in 1963 [2], so turbulence phenomenology was in its infancy at the time the first closures were being developed. In later years, numerical experiments based on high-resolution direct numerical simulations, did not bear out the Markovian picture. In particular, we note the investigation by Kuczaj et al [3]. This is in fact the basic flaw in Kraichnan’s DIA and also the SCF theories of Edwards and Herring: the fault lies not in the covariance equations but in the relationship of the response function to them.

As mentioned in the first blog in the present series (posted on 13 October) a response to this problem took, and continues to take, the form of an extension of the DIA approach to Lagrangian coordinates. A consideration of these theories would take us too far away from our present objective although it should be mentioned that they are non-Markovian in that they are not expressible as Master equations. Instead we will concentrate on the LET theory which exposes the underlying physics of the turbulence energy transfer process.

The LET theory was introduced with the hypothesis that $\omega(k)$ is determined by the entire $T(k)$, not just part of it, and can be defined by a local energy balance [1]. It was extended to the two-time case [4] in 1978; and, less heuristically, in subsequent papers by McComb and co-workers: see [5] for a review. Essentially, the two-time LET theory comprises the DIA covariance equations plus the generalized fluctuation-response relation. It may be compared to Herring’s two-time SCF [6] which comprises the DIA response equation, single-time covariance equation and the generalized fluctuation-response equation. It may also be compared directly to DIA in terms of response equations. However, for our present purposes, we will go back to the simplest case, and show how LET arose in relation to the Edwards SCF.

It was argued by McComb [1], that a correct assignment of the system response in terms of $T(k)$ (i.e. `correct’ in the sense of agreeing with the turbulence phenomenology of energy transfer) could lead to a response function which was compatible with K41. This was found to be the case and, citing the form given in [1], we may write for the turbulence viscosity $\nu_T(k)$: $$\nu_T(k)= k^{-2}\int_{j\geq k}d^3 j \frac{L(\mathbf{k,j})C(|\mathbf{k j}|)[C(k)-C(j)]}{C(k)[\omega(k)+\omega(j)+\omega(|\mathbf{k-j}|)]},\label{let-visc}$$ where $\omega(k) = \nu_T(k) k^2$. The lower limit on the integral with respect to $j$ arises when we consider the flux through mode $k$. It was used in [1] to justify wavenumber expansions leading to differential forms but is not needed here and can be omitted. The interesting point here is made by rewriting this in terms of the Edwards dynamical friction $r(k)$. From equation (5) in the post on 3 November, rewritten as $\omega(k)=\nu k^2 + \nu_T(k)k^2 =\nu k^2 + r(k),$ we may rewrite (\ref{let-visc}) as: $$\nu_T(k)= r(k) – k^{-2}\int d^3 j \frac{L(\mathbf{k,j})C(|\mathbf{k,j}|)C(j)}{C(k)[\omega(k)+\omega(j)+ \omega(|\mathbf{k-j}|)]}. \label{let-visc-rk}$$

It was shown [1] that the second term in the LET  response equation  cancelled the divergence in $r(k)$ in the limit of infinite Reynolds number. Hence the term which destroys the Markovian nature of the renormalized perturbation theory is the term which makes the theory compatible with the Kolmogorov $-5/3$ spectrum.

In the next post we will consider the subject of almost-Markovian models, where the term refers to the integrals over time rather than to the energy transfer through wavenumber.

[1] W. D. McComb. A local energy transfer theory of isotropic turbulence. J. Phys. A, 7(5): 632, 1974.
[2] M. S. Uberoi. Energy transfer in isotropic turbulence. Phys. Fluids, 6:1048, 1963.
[3] Arkadiusz K. Kuczaj, Bernard J. Geurts, and W. David McComb. Nonlocal modulation of the energy cascade in broadband-forced turbulence. Phys. Rev. E, 74:16306-16313, 2006.
[4] W. D. McComb. A theory of time dependent, isotropic turbulence. J. Phys. A: Math. Gen., 11(3):613, 1978.
[5] W. D. McComb and S. R. Yoffe. A formal derivation of the local energy transfer (LET) theory of homogeneous turbulence. J. Phys. A: Math. Theor., 50:375501, 2017.
[6] J. R. Herring. Self-consistent field approach to nonstationary turbulence. Phys. Fluids, 9:2106, 1966.


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## LSU Doctoral Dissertations

#### Identifier

etd-07062016-114144

#### Degree

Doctor of Philosophy (PhD)

Mathematics

Dissertation

#### Abstract

The goal of this dissertation is to expand Berhard Koopman's operator theoretic global linearization approach to the study of nonautonomous flows. Given a system with states x in a set \Omega (the state space), a map t\to \gamma(t,s,x) (t\geq s \geq 0) is called a global flow if it describes the time evolution of a system with the initial state x \in \Omega at time t \geq s \geq 0. Koopman's approach to the study of flows is to look at the dynamics of the observables of the states instead of studying the dynamics of the states directly. To do so, one considers a vector space Z containing observables (measurements) and a vector space \mathcal{M}:=\mathcal{F}([0,\infty)\times \Omega,Z) of functions containing observations g: [0,\infty) \times \Omega \to Z. Then every global flow \gamma induces a family T(t)(t\geq 0) of linear maps on \mathcal{M}, where $$\label{abstract} T(t)g: (s,x) \mapsto g(t+s,\gamma(t+s,s,x)).$$ Since every global flow \gamma satisfies \gamma(s,s,x) = x and \gamma(t,r,\gamma(r,s,x)) = \gamma(t,s,x) for t\geq r \geq s \geq 0 and x \in \Omega, the linear maps T(t)(t\geq 0) define an operator semigroup on \mathcal{M}; that is, T(0) = I \text and T(t+s) = T(t)T(s) for t,s \geq 0. Following Koopman's approach, in pursuit of understanding the flow \gamma, we investigate the linear flow semigroup T(t)(t\geq 0) on \M given by \eqref{abstract}, and if \gamma(t,s,x) = U(t,s)x for some linear evolution family U(t,s), an associated special evolution semigroups on a subspace of \M given by S(t)f: s\mapsto f(t+s)U(t+s,s). Of primary concern are continuity properties of the associated linear evolution semigroups on different function spaces (Chapters 1-3). The Lie generator of the flow and a collection of open problems concerning general flow semigroups \eqref{abstract}, asymptotics and/or finite time blow-up, and Lie-Totter type approximations are described in Chapter 4.

2016

#### Document Availability at the Time of Submission

Release the entire work immediately for access worldwide.

#### Committee Chair

Neubrander, Frank

COinS


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# How to account for stretching in graph transformation of $y = \sqrt{x}$?

Hi guys, I got this question as an assignment from my university and have been trying to solve it for a whole day now but can't get the correct answer. Well, I know how to shift Graph towards left, right, upwards and downwards $C$ points. Well for example current Graph's equation is $y = x^2$ and if we have to move graph to left $C$ points. We will add $C$ to the function and it will become $y = (x+C)^2$ and if we have to shift Graph to right side $C$ points, then we will subtract $C$ from function, and the equation will become $y = (x-C)^2$, and similarly add $C$ to the $f(x)$ and subtract $C$ from $f(x)$ if we want to shift graph upwards $C$ points and downwards $C$ points respectively, and if we want to reflect it about $x$-axis, we will multiply $f(x)$ with minus "-" sign and if we want to reflect it about $y$-axis we will multiply the function with minus"-" sign thats makes it like $y = f(-x)$. Well now you now that I know these things. But I have been trying to solve this question and its been a day but I am not able to solve it correctly. Kindly help me, I want to submit my Assignment and I really do not want to lose even a single marks from Mathematics.

And yeah I forgot to tell that I have already figured out that the graph in second picture is reflected about $x$-axis and then moved $1$ unit/point toward left side and moved $1$ unit/point downwards, but if you look carefully the graph in second picture is little stretched as well and thats where I am getting problem because I am not figuring out that is it stretched towards x-direction or y-direction and also can't figure out that how many points/units it have stretched.

The original graph has been stretched by factor $3$ in the $y$ direction, reflected in the $x$ axis, and translated by $-1$ in both $x$ and $y$ directions.

Therefore your new graph is $$y=-1-3\sqrt{x+1}$$

Good work so far.

I am not figuring out that is it stretched towards x-direction or y-direction

Here's how to get started on that.

The transformations that stretch are the ones that replace $x$ by $ax$ or $y$ by $by$. To figure out which stretch you need and the value of the constant, look at a few easy interesting points on the graphs (after you've reflected and translated so that the new one starts at the origin and heads off into the first quadrant).

Hint: We know that $f(x) = \sqrt{x}$ grows from $0$ to $1$ as $x$ increases from $0$ to $1$ and that $f(x) = \sqrt{x}$ grows from $0$ to $2$ as $x$ increases from $0$ to $4$. If we call the second function $g$, then $g$ decreases by $3$ as $x$ increases from $-1$ to $0$ (one unit to the right of the endpoint) and decreases by $6$ as $x$ increases from $-1$ to $3$ (four units to the right of the endpoint). What factor do you need to multiply the square root by in order to account for this transformation?


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# Homework 4 Optional Problems

In the 2SAT problem, you are given a set of clauses, where each clause is the disjunction of two literals (a literal is a Boolean variable or the negation of a Boolean variable). You are looking for a way to assign a value “true” or “false” to each of the variables so that all clauses are satisfied — that is, there is at least one true literal in each clause. For this problem, design an algorithm that determines whether or not a given 2SAT instance has a satisfying assignment. (Your algorithm does not need to exhibit a satisfying assignment, just decide whether or not one exists.) Your algorithm should run in $O(m + n)$ time, where $m$ and $n$ are the number of clauses and variables, respectively. [Hint: strongly connected components.]

In computer science, 2-satisfiability, 2-SAT or just 2SAT is a computational problem of assigning values to variables, each of which has two possible values, in order to satisfy a system of constraints on pairs of variables. It is a special case of the general Boolean satisfiability problem, which can involve constraints on more than two variables, and of constraint satisfaction problems, which can allow more than two choices for the value of each variable. But in contrast to those more general problems, which are NP-complete, 2-satisfiability can be solved in polynomial time.

From the given formula, we construct a digraph $G$ where each vertex is a literal and each edge an implication between two literals. Each clause can be equivalently expressed as two implications: $(x \lor y) \equiv (\neg x \implies y) \equiv (\neg y \implies x)$

The above implications should be intuitive; Given $x$ is $false$ (not $true$), if the clause is to be $true$, $y$ must be $true$ (since the clause is a Boolean OR). Similarly, given $y$ is $false$, $x$ must be $true$. Of course, we can also deduce the same from a truth table.

So, whenever there is a clause $(x \lor y)$ is the formula, we add two edges to $G$ corresponding to the equivalent implications. Then we run Kosaraju’s algorithm for finding strongly connected components (SCC). The algorithms runs in $O(m + n)$ time. If a literal and its negation are in the same SCC, there is no possible assignment that can make both $true$, thus, the formula is not satisfiable. We need to do a linear scan of each SCC to see if a literal and its negation are both in it; this can be done in $O(m + n)$ time. Thus, the overall time complexity for determining satisfiability is $O(m + n)$.

Having determined satisfiability, we can go further and find assignments that satisfy the formula. In order to do that, we take advantage of a property of Kosaraju’s algorithm: the SCCs are returned in topological order. That is, if a node $u$ is in the ith SCC, $v$ is in the jth one, with $i < j$, $u$ appears before $v$.

That tells us that, if we take those $x$ and $\neg x$, the original formula can have the implication $x \implies \neg x$, but not in the other direction, otherwise, they wouldn’t be in separate SCCs.

Following the truth table for the implication operation, we want to avoid the case where $true \implies false$; the only case where the implication evaluates to $false$.

Therefore, if $x$ comes before $\neg x$, we choose $x = false$, thus making $(x \implies \neg x) = true$. On the other hand, if $\neg x$ comes before $x$, we choose $x = true$, thus making $(\neg x \implies x) = true$.


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Hellenica World

# .

In mathematics, the exponential integral is a special function defined on the complex plane given the symbol Ei.

Definitions

For real, nonzero values of x, the exponential integral Ei(x) can be defined as

$$\mbox{Ei}(x)=\int_{-\infty}^x\frac{e^t}t\,dt.\,$$

The function is given as a special function because $$\textstyle\int \frac{e^t}{t}\,dt$$is not an elementary function, a fact which can be proven using the Risch Algorithm. The definition above can be used for positive values of x, but the integral has to be understood in terms of the Cauchy principal value, due to the singularity in the integrand at zero. For complex values of the argument, the definition becomes ambiguous due to branch points at 0 and $$\infty$$.[1] In general, a branch cut is taken on the negative real axis and Ei can be defined by analytic continuation elsewhere on the complex plane.

The following notation is used,[2]

$$\mathrm{E}_1(z) = \int_z^\infty \frac{e^{-t}}{t} dt,\qquad|{\rm Arg}(z)|<\pi$$

For positive values of the real part of z, this can be written[3]

$$\mathrm{E}_1(z) = \int_1^\infty \frac{e^{-tz}}{t}\, dt = \int_0^1 \frac{e^{-z/u}}{u}\, du ,\qquad \Re(z) \ge 0.$$

The behaviour of E1 near the branch cut can be seen by the following relation[4]:

$$\lim_{\delta\to0\pm}\mathrm{E_1}(-x+i\delta) = -\mathrm{Ei}(x) \mp i\pi,\qquad x>0,$$

Properties

Several properties of the exponential integral below, in certain cases, allow to avoid its explicit evaluation through the definition above.
Convergent series

Integrating the Taylor series for $$e^{-t}/t$$, and extracting the logarithmic singularity, we can derive the following series representation for $$\mathrm{E_1}(x)$$ for real x[5]:

$$\mathrm{Ei}(x) = \gamma+\ln |x| + \sum_{k=1}^{\infty} \frac{x^k}{k\; k!} \qquad x \neq 0$$

For complex arguments off the negative real axis, this generalises to[6]

$$\mathrm{E_1}(z) =-\gamma-\ln z+\sum_{k=1}^{\infty}\frac{(-1)^{k+1} z^k}{k\; k!} \qquad (|\mathrm{Arg}(z)| < \pi)$$

where $$\gamma is the Euler–Mascheroni constant. The sum converges for all complex z, and we take the usual value of the complex logarithm having a branch cut along the negative real axis. This formula can be used to compute \( \mathrm{E_1}(x)$$ with floating point operations for real x between 0 and 2.5. For x > 2.5, the result is inaccurate due to cancellation.
Asymptotic (divergent) series
Relative error of the asymptotic approximation for different number ~N~ of term in the truncated sum

Unfortunately, the convergence of the series above is slow for arguments of larger modulus. For example, for x=10 more than 40 terms are required to get an answer correct to three significant figures.[7] However, there is a divergent series approximation that can be obtained by integrating $$ze^z\mathrm{E_1}(z)$$by parts[8]:

$$\mathrm{E_1}(z)=\frac{\exp(-z)}{z}\sum_{n=0}^{N-1} \frac{n!}{(-z)^n}$$

which has error of order O(N!z^{-N}) and is valid for large values of \mathrm{Re}(z). The relative error of the approximation above is plotted on the figure to the right for various values of N (N=1 in red, N=5 in pink).
Exponential and logarithmic behavior: bracketing
Bracketing of $$\mathrm{E_1}$$ by elementary functions

From the two series suggested in previous subsections, it follows that $$\mathrm{E_1}$$ behaves like a negative exponential for large values of the argument and like a logarithm for small values. For positive real values of the argument, $$\mathrm{E_1}$$ can be bracketed by elementary functions as follows[9]:

$$\frac{1}{2}e^{-x}\,\ln\!\left( 1+\frac{2}{x} \right) < \mathrm{E_1}(x) < e^{-x}\,\ln\!\left( 1+\frac{1}{x} \right) \qquad x>0$$

The left-hand side of this inequality is shown in the graph to the left in blue; the central part $$\mathrm{E_1}(x) is shown in black and the right-hand side is shown in red. Definition by \mathrm{Ein} Both \( \mathrm{Ei}$$ and $$\mathrm{E_1}$$ can be written more simply using the entire function $$\mathrm{Ein}$$[10] defined as

$$\mathrm{Ein}(z) = \int_0^z (1-e^{-t})\frac{dt}{t} = \sum_{k=1}^\infty \frac{(-1)^{k+1}z^k}{k\; k!}$$

(note that this is just the alternating series in the above definition of \mathrm{E_1}). Then we have

$$\mathrm{E_1}(z) \,=\, -\gamma-\ln z + {\rm Ein}(z) \qquad |\mathrm{Arg}(z)| < \pi$$
$$\mathrm{Ei}(x) \,=\, \gamma+\ln x - \mathrm{Ein}(-x) \qquad x>0$$

Relation with other functions

The exponential integral is closely related to the logarithmic integral function li(x) by the formula

$$\mathrm{li}(x) = \mathrm{Ei}(\ln x)\,$$

for positive real values of x

The exponential integral may also be generalized to

$${\rm E}_n(x) = \int_1^\infty \frac{e^{-xt}}{t^n}\, dt,$$

which can be written as a special case of the incomplete gamma function[11]:

{\rm E}_n(x) =x^{n-1}\Gamma(1-n,x).\, \)

The generalized form is sometimes called the Misra function[12] $$\varphi_m(x)$$, defined as

$$\varphi_m(x)={\rm E}_{-m}(x).\,$$

Including a logarithm defines the generalized integro-exponential function[13]

$$E_s^j(z)= \frac{1}{\Gamma(j+1)}\int_1^\infty (\log t)^j \frac{e^{-zt}}{t^s}\,dt.$$

Derivatives

The derivatives of the generalised functions \mathrm{E_n} can be calculated by means of the formula [14]

$$\mathrm{E_n}'(z) = -\mathrm{E_{n-1}}(z) \qquad (n=1,2,3,\ldots)$$

Note that the function $$\mathrm{E_0} is easy to evaluate (making this recursion useful), since it is just \( e^{-z}/z.[15]$$
Exponential integral of imaginary argument
\mathrm{E_1}(ix) against x; real part black, imaginary part red.

If z is imaginary, it has a nonnegative real part, so we can use the formula

$$\mathrm{E_1}(z) = \int_1^\infty \frac{e^{-tz}}{t} dt$$

to get a relation with the trigonometric integrals $$\mathrm{Si}$$ and $$\mathrm{Ci}$$:

$$\mathrm{E_1}(ix) = i\left(-\tfrac{1}{2}\pi + \mathrm{Si}(x)\right) - \mathrm{Ci}(x) \qquad (x>0)$$

The real and imaginary parts of $$\mathrm{E_1}(x)$$are plotted in the figure to the right with black and red curves.
Applications

Time-dependent heat transfer
Nonequilibrium groundwater flow in the Theis solution (called a well function)
Radial Diffusivity Equation for transient or unsteady state flow with line sources and sinks
Solutions to the neutron transport equation in simplified 1-D geometries.[16]

Notes

^ Abramowitz and Stegun, p.228
^ Abramowitz and Stegun, p.228, 5.1.1
^ Abramowitz and Stegun, p.228, 5.1.4 with n = 1
^ Abramowitz and Stegun, p.228, 5.1.7
^ For a derivation, see Bender and Orszag, p253
^ Abramowitz and Stegun, p.229, 5.1.11
^ Bleistein and Handelsman, p.2
^ Bleistein and Handelsman, p.3
^ Abramowitz and Stegun, p.229, 5.1.20
^ Abramowitz and Stegun, p.228, see footnote 3.
^ Abramowitz and Stegun, p.230, 5.1.45
^ After Misra (1940), p.178
^ Milgram (1985)
^ Abramowitz and Stegun, p.230, 5.1.26
^ Abramowitz and Stegun, p.229, 5.1.24
^ George I. Bell; Samuel Glasstone (1970). Nuclear Reactor Theory. Van Nostrand Reinhold Company.

References

Abramovitz, Milton; Irene Stegun (1964). Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables. Abramowitz and Stegun. New York: Dover. ISBN 0-486-61272-4.
Bender, Carl M.; Steven A. Orszag (1978). Advanced mathematical methods for scientists and engineers. McGraw-Hill. ISBN 0-07-004452-X.
Bleistein, Norman; Richard A. Handelsman (1986). Asymptotic Expansions of Integrals. Dover. ISBN 0486650820.
Busbridge, Ida W. (1950). "On the integro-exponential function and the evaluation of some integrals involving it". Quart. J. Math. (Oxford) 1 (1): 176–184. Bibcode 1950QJMat...1..176B. doi:10.1093/qmath/1.1.176.
Stankiewicz, A. (1968). "Tables of the integro-exponetial functions". Acta Astronomica 18: 289. Bibcode 1968AcA....18..289S.
Sharma, R. R.; Zohuri, Bahman (1977). "A general method for an accurate evaluation of exponential integrals E1(x), x>0". J. Comput. Phys. 25 (2): 199—204. Bibcode 1977JCoPh..25..199S. doi:10.1016/0021-9991(77)90022-5.
Kölbig, K. S. (1983). "On the integral exp(-μt)tν-1logmt dt". Math. Comput 41 (163): 171—182. doi:10.1090/S0025-5718-1983-0701632-1.
Milgram, M. S. (1985). "The generalized integro-exponential function". Mathematics of Computation 44 (170): 443–458. doi:10.1090/S0025-5718-1985-0777276-4. JSTOR 2007964. MR0777276.
Misra, Rama Dhar; Born, M. (1940). "On the Stability of Crystal Lattices. II". Mathematical Proceedings of the Cambridge Philosophical Society 36 (2): 173. doi:10.1017/S030500410001714X.
Chiccoli, C.; Lorenzutta, S.; Maino, G. (1988). "On the evaluation of generalized exponential integrals Eν(x)". J. Comput. Phys. 78: 278—287. Bibcode 1988JCoPh..78..278C. doi:10.1016/0021-9991(88)90050-2.
Chiccoli, C.; Lorenzutta, S.; Maino, G. (1990). "Recent results for generalized exponential integrals". Computer Math. Applic. 19 (5): 21—29. doi:10.1016/0898-1221(90)90098-5.
MacLeod, Allan J. (2002). "The efficient computation of some generalised exponential integrals". J. Comput. Appl. Math. 148 (2): 363—374. Bibcode 2002JCoAm.138..363M. doi:10.1016/S0377-0427(02)00556-3.
Press, WH; Teukolsky, SA; Vetterling, WT; Flannery, BP (2007), "Section 6.3. Exponential Integrals", Numerical Recipes: The Art of Scientific Computing (3rd ed.), New York: Cambridge University Press, ISBN 978-0-521-88068-8

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text:
Bernoulli's Equation (Water Outflow Speed of Tank)

1. Jan 18, 2013

Von Neumann

Problem:

Suppose the top of a tank is sealed and pressurized to twice atmospheric pressure. What is the speed of the liquid emerging from a small hole at the base of the tank?

My Solution:

Using Bernoulli's Equation, and assuming v_top ≈ 0,

P_top + 1/2*ρ*v_top^2 + ρgh = P_hole + 1/2*ρ*v_hole^2 + ρg*0

2*P_a + ρgh = P_a + 1/2*ρ*v_hole^2

Solving for v_hole I get,

$\sqrt{\frac{2(P_{a}+ρgh)}{ρ}}$

While my book has,

$\sqrt{\frac{2P_{a}}{ρ+gh}}$

Anyone know where I went wrong? Or, less likely, if my text is incorrect? Thank you in advance.

2. Jan 19, 2013

voko

The text is incorrect. The sum in the denominator is impossible, the units of the addends do not match.

3. Jan 19, 2013

Von Neumann

Oh wow, I'm foolish for not thinking of that myself. My units come out to be the correct units of velocity.