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In recent years there has been growing attention to the relative differences in access, quality, and costs of long-term services and supports (LTSS), especially community LTSS, for persons with different types of disabilities. These differences are attributed to various factors, including differences in the amounts of needed support and in the accepted approaches to support as well as relative effectiveness of advocacy on behalf of different constituencies and others. The interactions of such factors have led to actual and/or perceived differences among “disability groups” in access to and quality of LTSS that have been questioned in terms of fairness and appropriateness. In a discussion of these issues and their implications, Kane, Priester, and Newman (2004) observed that, “From a social justice perspective, it is not immediately apparent why different groups of persons with disabilities should be favored” (p. 16). Although perspectives may differ on whether the perceived differences in access, quality, and costs of LTSS represent social injustice, there is a general impression, not entirely without supporting evidence, that among the various “groups” of persons with disabilities who may need LTSS, persons identified as having ID/DD fare relatively well. There are two principal aspects to the perceived discrepancies in LTSS between persons with ID/DD and other Medicaid recipients with disabilities. One is that people with ID/DD have “favored” or at least disproportionate access to Medicaid LTSS funding in general; the other, that people with ID/DD are particularly favored in their access to community-based LTSS. Statistics presented below relate to such perceptions. As Kane et al. (2004) stated outright, “The large discrepancy in public support for community-based care between MR/DD and aging can be directly traced to persistent efforts of MR/DD advocates (pp. 13–14).

Figure 1 shows patterns of growth between FY 1993 and FY 2003 in Medicaid LTSS expenditures for persons with ID/DD, specifically for Medicaid Home and Community-Based Services (HCBS) and Medicaid Intermediate Care Facility for the Mentally Retarded (ICF/MR) services, along with patterns of growth for all Medicaid expenditures for institutional and community LTSS. Between 1993 and 2003, LTSS expenditures for persons with ID/ DD increased from $11.4 billion to$25.6 billion (125%), whereas all Medicaid LTSS expenditures increased from $42.1 billion to$84.5 billion (101%). In 1993, LTSS for persons with ID/DD made up 27.0% of all Medicaid LTSS expenditures; by 2003, they had encreased modestly to 30.3% of all LTSS expenditures. These differences are notable, but seem hardly the stuff of social injustice.

Figure 1.

Total Medicaid and Medicaid ID/DD long-term service and support (LTSS) expenditures (in billions of dollars) 1993–2003

Figure 1.

Total Medicaid and Medicaid ID/DD long-term service and support (LTSS) expenditures (in billions of dollars) 1993–2003

What seems more notable in comparing LTSS expenditures for persons with ID/DD and the total Medicaid LTSS expenditures are the substantial differences in the proportion of LTSS expenditures that have been committed to HCBS, as opposed to institutional services. As shown in Figure 2, in 2003, only 22.3% of the $84.5 billion spent on Medicaid LTSS was spent in HCBS programs. As a point of contrast, among LTSS programs for persons with ID/DD, 55.2% of expenditures were for HCBS; among all other LTSS programs, 8.0% of expenditures were for HCBS. Figure 2. Medicaid long-term services and supports expenditures for persons with ID/DD and all others Figure 2. Medicaid long-term services and supports expenditures for persons with ID/DD and all others While Medicaid LTSS expenditures for persons with ID/DD increased between 1993 and 2003 at a rate not appreciably greater than that of overall LTSS expenditures, the growth for the HCBS portion of those expenditures was very much greater. Still, the relative distribution of HCBS expenditures for persons with ID/DD among all HCBS expenditures remained quite stable during the 1993– 2003 decade. In FY 1993, 78% of all HCBS expenditures were for persons with ID/DD; between FY 1993 and FY 2003, 74.5% of the increase in total HCBS expenditures (from$2.8 billion to \$18.8 billion) was for persons with ID/DD; and in FY 2003, 75% of all HCBS expenditures were for persons with ID/DD. More notable than changing patterns was the maintenance of patterns. Similarly, the 9.8% of all Medicaid expenditures that went for LTSS for persons with ID/DD in FY 2003 remained in the range of 9% to 12% as it has for more than 20 years.

Table 1 shows levels of expenditures for various categories of Medicaid LTSS by state and for the US as a whole. It shows that nationally in FY 2003, 30.3% of all Medicaid LTSS went to support persons with ID/DD, including 75.0% of all HCBS expenditures, but only 17.5% of institutional expenditures (i.e., for nursing facilities and ICFs-MR). Patterns of LTSS expenditures among states tended to fall around the national averages, but with some notable variations. States varied in their proportions of all Medicaid LTSS expenditures that went to supporting persons with ID/DD, with 36 states falling between 25.0% and 39.0%, but with less than 25% in 8 states and more than 40% in 7 states. In all but 5 states most HCBS expenditures went to support persons with ID/DD, with 30 states falling between 60% and 90% and with more than 90% in 6 states. In all but 8 states, the total amount of Medicaid spending allocated to LTSS for persons with ID/DD was between 5.5% and 17.5%, with 4 above and 4 below. (Sources: Statistics for LTSS expenditures for persons with ID/DD are from R. Prouty, G. Smith, and K.C. Lakin (2004, June). Residential services for persons with developmental disabilities: Status and trends through 2003. Minneapolis: University of Minnesota, RTC on Community Living; Statistics on all Medicaid and Medicaid LTSS expenditures are from B. Burwell, K. Sredl, and S. Eiken (2004, May). Medicaid long-term care expenditures in FY 2003. Cambridge, MA: MEDSTAT. Modest amendments have been made to statistics appearing in cited reports based on subsequent updates from reporting states. The authors greatly appreciate the assistance of Brian Burwell of MEDSTAT in the preparation and review of the Medicaid LTSS expenditure statistics presented.

Table 1

Expenditures for Medicaid Long-Term Services and Supports (LTSS) for Persons With ID/DD, All Medicaid LTSS Recipients, and All Medicaid Recipients in Fiscal Year 2003

Table 1

Continued

## Reference

Reference
Kane
,
R. L.
,
R.
Priester
, and
D.
Neumann
.
2004, October 15
.
Disability across the age spectrum: Examining the disparities.
Unpublished paper prepared for The “Balkanization” of Disability: Examining the Disparities Conference, Minneapolis. (available from the Center on Aging, University of Minnesota)
.


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## 8.49 [ENDORSED]

$\Delta U=q+w$

Hyein Cha 2I
Posts: 103
Joined: Fri Sep 29, 2017 7:05 am
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### 8.49

Oxygen diflouride is a colorless, very poisonous gas that reacts rapidly with water vapor to produce O2, HF and heat: OF2 +H20 yields O2 +2HF and delta H is =- 318 kJ.

I just don't get why T remains constant when calculating for -PdeltaV = -deltanRT.

Also, why is T 298K and not other temperature? The question does not give any info about T, but solutions manual says that T=298K.

Chem_Mod
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Has upvoted: 716 times

### Re: 8.49  [ENDORSED]

We will usually take chemical reactions as occurring in isothermal environments. This is the basis for standard molar enthalpies of formation and standard reaction enthalpies we have discussed and can be found tabulated in the back of the book. All these values are predicated on the fact that temperature and pressure remain constant.

It will usually be pretty obvious from the problem description when temperature changes occur.

Hyein Cha 2I
Posts: 103
Joined: Fri Sep 29, 2017 7:05 am
Been upvoted: 1 time

### Re: 8.49

Chem_Mod wrote:We will usually take chemical reactions as occurring in isothermal environments. This is the basis for standard molar enthalpies of formation and standard reaction enthalpies we have discussed and can be found tabulated in the back of the book. All these values are predicated on the fact that temperature and pressure remain constant.

It will usually be pretty obvious from the problem description when temperature changes occur.

So will T always be 298K? If no info about T is given in the problem, do we just assume it's 298K?

mhuang 1E
Posts: 22
Joined: Fri Sep 29, 2017 7:05 am

### Re: 8.49

From what I read on another post, it's said that if the temperature is not explicitly statedg, we can assume that the reaction is occurring at standard temperature, which is 25C (room temp) or 298K.

Ramya Lakkaraju 1B
Posts: 67
Joined: Fri Sep 29, 2017 7:03 am

### Re: 8.49

Yes, if no other temperature is given in the problem, it is safe to assume that the reaction occurred at room temperature which is 298 K.

format: plain


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# XML - Document Type (Definition|Declaration) (DTD)

Document Type Definition (DTD) is a grammar (language) that defines the schema (ie data structure) of an XML document.

It

## Syntax

The syntax is a set of markup declarations that define a document type for SGML-family markup languages (SGML, XML, HTML).

DTDs were a precursor to XML schemas and have a similar function, although different capabilities.

A DTD can be declared:

• inline inside an XML document,
• or as an external reference.

The document type declaration can point to an external subset (a special kind of external entity) containing:

• markup declarations,
• or can contain the markup declarations directly in an internal subset,
• or can do both.

The DTD for a document consists of both subsets taken together.

DTD are context-free grammars


text:
A. Declined Finalists
time limit per test
2 seconds
memory limit per test
256 megabytes
input
standard input
output
standard output

This year, as in previous years, MemSQL is inviting the top 25 competitors from the Start[c]up qualification round to compete onsite for the final round. Not everyone who is eligible to compete onsite can afford to travel to the office, though. Initially the top 25 contestants are invited to come onsite. Each eligible contestant must either accept or decline the invitation. Whenever a contestant declines, the highest ranked contestant not yet invited is invited to take the place of the one that declined. This continues until 25 contestants have accepted invitations.

After the qualifying round completes, you know K of the onsite finalists, as well as their qualifying ranks (which start at 1, there are no ties). Determine the minimum possible number of contestants that declined the invitation to compete onsite in the final round.

Input

The first line of input contains K (1 ≤ K ≤ 25), the number of onsite finalists you know. The second line of input contains r1, r2, ..., rK (1 ≤ ri ≤ 106), the qualifying ranks of the finalists you know. All these ranks are distinct.

Output

Print the minimum possible number of contestants that declined the invitation to compete onsite.

Examples
Input
252 3 4 5 6 7 8 9 10 11 12 14 15 16 17 18 19 20 21 22 23 24 25 26 28
Output
3
Input
516 23 8 15 4
Output
0
Input
314 15 92
Output
67
Note

In the first example, you know all 25 onsite finalists. The contestants who ranked 1-st, 13-th, and 27-th must have declined, so the answer is 3.


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# Calculate [H+]?

1. Nov 5, 2004

### parwana

Calculate [H+]?????

Calculate [ H + ] of a 0.220 M solution of Aniline C6H5NH2 Kb = 7.4e-10

2. Nov 5, 2004

### Sirus

Consider that for a reaction
$$HA_{(aq)}\rightleftharpoons~H^{+}_{(aq)}~+~A^{-}_{(aq)}$$
we can use
$$K_{a}=\frac{[H^{+}_{(aq)}][A^{-}_{(aq)}]}{[HA_{(aq)}]}$$
How can you find $K_{a}$ when given $K_{b}$?

3. Nov 5, 2004

### chem_tr

Sirus is right; an alternative point of view may be using dissociative approach. In this, you start with 0.220 M of aniline, but only $\displaystyle x$ of it is ionized to give some $\displaystyle H^+$. We know the equilibrium constant of this reaction, i.e., $$\displaystyle \frac {10^{-14}}{K_b}$$.

$$\displaystyle \underbrace{C_6H_5NH_2_{(aq)}}_{0.220-x} \leftrightharpoons \underbrace{C_6H_5NH^-_{(aq)}}_{x}+\underbrace{H^+_{(aq)}}_{x}$$

Now it is better for you to consider the magnitude of $$\displaystyle \frac {10^{-14}}{K_b}$$; for the sake of simplification, you may omit the $\displaystyle x$ in $\displaystyle 0.220-x$, as we may omit values generally less than 5%. Of course, you may also want to solve the quadratic equation to find the exact answer, but believe me this is not very necessary.

What you'll do next is to find the $\displaystyle -\log x$.

4. Nov 6, 2004

### ShawnD

Aniline is a base, not an acid. Hydrogen will stick to the lone pair on the nitrogen so the reaction is like this:

$$C_6H_5NH_2 + H_2O \rightarrow C_6H_5NH_3^+ + OH^-$$

5. Nov 6, 2004

### Sirus

In that case, for a reaction
$$B_{(aq)}~+~H_{2}O_{(l)}\rightleftharpoons~BH^{+}_{(aq)}~+~OH^{-}_{(aq)}$$
we can use
$$K_{b}=\frac{[BH^{+}_{(aq)}][OH^{-}_{(aq)}]}{[B_{(aq)}]}$$

6. Nov 6, 2004

### chem_tr

Dear ShawnD, you are right about aqueous reactions, but don't forget that there are very powerful bases for using in non-aqueous phases like sodium hydride, lithium diisopropylamide, etc, which can take a proton to give anilinide anion.

In most cases, your reaction is sufficient, and same things may be said for Sirus' last post.


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## Finding Q

$E_{cell} = E_{cell}^{\circ}-\frac{RT}{nF}\ln Q$

TimVintsDis4L
Posts: 104
Joined: Sat Aug 17, 2019 12:17 am

### Finding Q

When we are given a Cell Diagram and concentrations and asked to find the Q what do we use as the Products and Reactants. Would we not use the right side (cathode, reduced) side as the Product and the (anode,oxidized) side as the Reactant?

Renee Grange 1I
Posts: 56
Joined: Fri Aug 30, 2019 12:16 am
Been upvoted: 1 time

### Re: Finding Q

You can write the total reaction to see what the products and reactions are. But, I believe the anode is usually the product and the cathode usually the reactant.

Brooke Yasuda 2J
Posts: 102
Joined: Sat Jul 20, 2019 12:17 am

### Re: Finding Q

Yes, the anode is the product and the cathode is the reactant. This makes sense because in the anode the concentrations of reactant are decreasing as it forms product in the cathode.

Mandeep Garcha 2H
Posts: 100
Joined: Sat Aug 24, 2019 12:17 am

### Re: Finding Q

Q=anode/cathode. So you can plug in all the values you know for the equation and solve for the unknown.

Anthony Hatashita 4H
Posts: 103
Joined: Wed Sep 18, 2019 12:21 am

### Re: Finding Q

Q is anode/cathode usually. You can write out the full reaction equation to verify on a case to case basis.

Sanjana Borle 2K
Posts: 111
Joined: Fri Aug 09, 2019 12:15 am

### Re: Finding Q

Since you flip the equation for the anode when writing out the total equation for a cell, it makes sense that Q would be anode/cathode because this way the anode becomes the product.

Juana Abana 1G
Posts: 100
Joined: Wed Sep 18, 2019 12:15 am

### Re: Finding Q

Brooke Yasuda 2J wrote:Yes, the anode is the product and the cathode is the reactant. This makes sense because in the anode the concentrations of reactant are decreasing as it forms product in the cathode.

So then the anode is always the product and the cathode is always the reactant?

Jasmine Kim 1L
Posts: 71
Joined: Fri Aug 02, 2019 12:16 am

### Re: Finding Q

Juana Abana 1G wrote:So then the anode is always the product and the cathode is always the reactant?

In most cases, yes, but I would always write out the reaction to be sure.

Return to “Appications of the Nernst Equation (e.g., Concentration Cells, Non-Standard Cell Potentials, Calculating Equilibrium Constants and pH)”

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text:
# Can you please help me find the Moment generating function of the negative binomial when it looks a bit different?

f(x) = $\binom{(r+x)-1}{x-1}p^{r}(1-p)^{x}$ noting 0 < p<1 and r>0 $$\sum_{X}^\infty \binom{(r+x)-1}{r-1}p^{r}(1-p)^{x} =1.$$ x=0,1,2,3......... I think it looks like this: $$\sum_{X}^\infty \binom{(r+x)-1}{r-1}p^{r}(1-p)^{x} =$$ I think i put an e^tx inside of the sumation. But do i try to turn all the x's to x+r or the r's to r+x?

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## 1 Answer

Your expressions are not equivalent.

Perhaps it would be helpful for you to consider the following identity.

$${x+r-1 \choose x-1}={x+r-1 \choose r}$$

In terms of x=failures and r=successes in the first we are counting the number of ways we can choose the x-1 failures (since the xth failure ends the trial and is determined), and in the other, the ways we can choose r successes. They both exist in the same basket that we are counting from and we know that the total number of events in the basket is x+r-1.

To count r successes before x-1 failures doesn't require a change to the pmf. Changing the r to r+x, or x to r+x in the exponents is equivalent to changing the number or success or failures needed to satisfy the problem and would require changes to the combinatoric element of the pmf as well. It would be a whole different probability.

Your understanding of the definition of the moment generation function is correct though.

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i got it thank you! – user43126 Oct 1 '12 at 22:31


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# Number of ways to partition a rectangle into n sub-rectangles

How many ways can a rectangle be partitioned by either vertical or horizontal lines into n sub-rectangles? At first I thought it would be:

      f(n) = 4f(n-1) - 2f(n-2)
where f(0) = 1
and f(1) = 1

but the recurrence relation only counts the cases in which at least one side (either top, bottom, left or right of the original rectangle) is not split into sub-rectangles. There are many other partitions that don't belong to those simple cases like

[EDIT ImageShack has removed the picture. One of the cases is the sixth partition when n = 4 in the picture in the accepted answer below.]

Any other related problem suggestions are welcome. Also it is nice to know how to traverse this partition efficiently.

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I'm not sure exactly how you're counting different "ways" to be the same, but depending on that definition this could be a very hard problem. (As in, open.) –  Qiaochu Yuan Jul 29 '10 at 8:28
Even counting the number of partitions into dominoes (rectangles 2x1) is quite difficult -- see "Concrete Mathematics", ex. 7.51 for the answer and some hints; there exists (e.g. mccme.ru/free-books/matpros/ia129142.pdf.zip -- but it's in Russian) a proof that uses methods from arxiv.org/abs/math/9810091 and arxiv.org/abs/math/0108150. –  Grigory M Jul 29 '10 at 15:45
(Offtopic) solution(s) for the domino problem etc: mathematik.uni-bielefeld.de/~sillke/PUZZLES/domino –  Grigory M Jul 30 '10 at 10:03
At what coordinates may the dividing lines lie? If you allow real coordinates, there are infinitely many way to split any rectangle into two sub-rectangles. Is this rectangle made of nxn tiles, e.g.? –  Jens Aug 4 '10 at 13:38
@Jens It doesn't matter where exactly the lines are. What matters is the overall form. –  puri Aug 5 '10 at 4:27

## 4 Answers

I had my student, Tim Michaels, work on this. We came up with a complicated recurrence relation, indicated below. The first few answers are 1, 2, 6, 25, 128, 758, 5014, 36194, 280433, 2303918, 19885534, 179028087, 1671644720, 16114138846, 159761516110, 1623972412726, 16880442523007, 179026930243822, 1933537655138482, 21231023519199575, 236674460790503286, 2675162663681345170, 30625903703241927542, 354767977792683552908, 4154708768196322925749, 49152046198035152483150, 587011110939295781585102, 7072674305834582713614923. Note that we are counting rotations and reflections as distinct tilings. An interesting pattern is that mod 2, there is an 8-fold periodicity which we don't understand and can't prove in general.

Here's a picture of the cases n=1,2,3,4, with 1,2,6,25 tilings in each case.The way to systematically generate these is to "push in" a vertical line from the right to all previously constructed tilings in all possible ways. That's how the recurrence relation is defined.

Okay, here is the recurrence: Let $a_{\ell,j,m}$ be the number of distinct tilings with $\ell$ tiles, $j$ edges that meet the right-hand side of the square and $m$ 4-valent vertices. $$a_{\ell,j,m}=\sum_{p=1}^\ell(-1)^{p+1}\sum_{i=0}^m\sum_{n=1}^{\ell-1}\sum_{k=0}^{n-1}\alpha_{n,k,i,\ell,j,m,p} a_{n,k,i}$$ where, letting $t=m-i, s=\ell-n-p-t$ and $r=k+s+t+p-j$, $$\alpha_{n,k,i,l,j,m,p}=\binom{r-1}{p-1}\binom{k-r+2}{p}\binom{s+r-1}{r-1}\binom{r-p}{t}.$$

Edit: I have posted a preprint describing the recurrence relation here. Comments are welcome. Is this sort of thing publishable anywhere to anyone's knowledge?

Edit 2: Nathan Reading has just posted a relevant preprint. He finds a bijection between generic tilings (no 4-valent vertices) and a set of permutations that avoid a certain pattern.

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By the way, the recurrence involved counting tilings with a specific number of edges hitting the right side of the square and a specific number of places where 4 rectangles meet in a vertex. –  Grumpy Parsnip Oct 31 '10 at 14:56
1, 2, 6, 15 are exactly the numbers I have when I count by hands (including all rotations and reflections.) I will appreciate if you can share the recurrence relation or point me to a publication. –  puri Nov 2 '10 at 0:10
I've added a picture to show there are indeed 25 tilings with 4 rectangles. –  Grumpy Parsnip Nov 2 '10 at 9:18
Okay now I've added the recurrence relation! –  Grumpy Parsnip Nov 2 '10 at 9:29
@Jim Conant: On the third iteration, aren't the four last figures redundant since they are just rotations of the first two? By the way, your picture is so annoying to look at, I'm prey of the grid illusion! :D –  Raskolnikov Feb 18 '11 at 15:52

This problem is tantalizing. It may be a "toy"-version of some substantial ongoing research in topology, so it's valuable and interesting. It has immediate analogs in areas of classical topology, knot theory, universal algebra, and topological field theory. By toy-version, I don't mean it's necessarily easy to solve, but that it can be used to present some advanced ideas in a non-technical way. Forgive me because what follows is not an answer to your question, but a rambling list of related problems and possible applications. I only hope it inspires someone here to investigate the connections in algebraic topology. (Thus, community wiki.)

You are asking to count equivalence classes of parametrized rectangle framings (where a parametrized framing is a framing with precise line positions). In some sense, this type of enumeration problem comes up often in algebraic topology when one wants to understand a cellular decomposition of a complicated space. Take for instance, the space of n-planes in R^d. These spaces are called Grassmannians and their topology can be analyzed by breaking them down into subsets of various dimensions. These subsets are called Schubert cells. Counting these Schubert cells and understanding how they fit together was an important advancement. See the wikipedia entries on Grassmannian and Enumerative Geometry.

Your parametrized rectangle framing spaces (denoted perhaps Fi^{n} where i indexes the equivalence classes of framings having n sub-rectangles) are analogous to the cells of the Grassmannian, but the resulting stratified space is some universal space of parametrized rectangular framings which includes framings with an arbitrarily large but finite number of sub-rectangles. That is, each of your framing spaces F (the equivalence classes you want to count) represents a "cell" of some fixed dimension which degenerates into lower degree framings at the boundaries. By carefully studying the simple shape of each cell and how they degenerate into other cells, we can create a universal space $F^{\infty}$ = $\bigcup$ $F^{n}_{i}$ (modulo the boundary case equivalence relations).

Sometimes one can approach the topology of this universal space by other means, and that global knowledge can be used to count the number of cells needed in each stratum. Finally, as you point out, the spaces have certain symmetries which may pass up to the full space in an interesting way.

This stratification trick comes up all the time If you want to understand some large space, find a way to break it down into strata which fit together in an analyzable way. Or, find a related space which has a natural stratification, analyze this stratified space, and make inferences about the original space. A great, non-trivial use of this in the 90's was when Vassiliev realized that knots could be studied by analyzing the space of non-knots which admitted a very clear stratification (essentially stratify the non-knots by the number of points where the circle map is non-injective) Vassiliev found clear topological structure in this space and this allowed him to make claims about the structure of the set of knots. This led others like Kontsevich and Bar Natan to provide real computational tools using tricks for counting and integrating over cells. For instance, Mathworld has two good introductory articles on Vassiliev Invariants and Kontsevich Integrals.

Thirdly, your rectangle framing space is evocative of the little-disks operad which encodes an algebraic structure. See the wikipedia page:
http://en.wikipedia.org/wiki/Operad_theory#.22Little_something.22_operads

Whenever we have a set of topological spaces such that elements of the spaces can be geometrically combined to get higher degree spaces, it hints that the geometry can be studied using techniques from algebra. If you imagine putting variables in each of your frame's sub-rectangles, you've sort of described an algebraic operation. Imagine that every one of your parametrized n-framings defined a map from n inputs to 1 output, an "n-ary operation." Further assume that the geometry forces some consistency condition that demands that when you place the result of a p-ary operation into one of the sub-rectangles in a q-ary frame, you get just what you'd get from the corresponding (p+q-1)-ary frame operation. This means your algebra must satisfy certain standard relations (maybe up to homotopy) and then these operations may descend to operations on an appropriately defined cohomology theory. That is, the topology of your universal framing space may index various operations which satisfy certain relations exactly. See for instance the ncatlab.org entries on operad, L-infinity-algebra, and A-infinity-algebra.

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I don't have enough rep to comment yet, so I'll make this CW since it's not a complete answer, just a remark.

Would it overcome some of the difficulty by using some sort of graph representation of the subdivided rectangle? Each rectangle is a vertex and an edge represents sharing a side, i.e. the first rectangle above would be represented by the edges ${(1,2),(1,3),(2,4),(3,4)}$ and the second by ${(1,2),(1,3),(1,4),(2,4),(2,5),(3,4),(3,5)}$, numbering the rectangles from top-left to bottom-right. The tricky part here would then be to implement the operations for subdividing by a vertical or horizontal line segment. Starting instead with lines instead of segments could be promising.

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Interesting but I wonder if your graph representation will be sufficient. From the second example above, sliding the partitions the other way around from "clockwise" to "counter-clockwise" will result in another way to partition with the exactly same graph. –  puri Aug 8 '10 at 18:22

Here is a non-answer which may prove interesting and tractable.

Given a configuration of rectangles that properly tile a surrounding rectangle, one can represent the tiling in a number of ways. Two that come to mind I will call C and LL. C represents each rectangle by giving its dimensions (x-span and y-span) and the coordinates for the center of the rectangle, and LL does the same, except it specifies the coordinate for the lower left vertex (instead of the center) of each rectangle.

With plenty of rectangles in a tiling, there is some redundancy in the LL form of a tiling. For example, it should be possible to deduce the dimensions of the rectangle in the extreme lower left corner of the tiling by noting stored coordinates of the rectangle above and the rectangle to the right of this corner rectangle. No such boundaries exist for the upper rightmost rectangle in a tiling, but perhaps an "end point" could be added to indicate the maximum x and maximum y values in the tiled rectangle. Suppose we add such an end point, and let LL' be a representation like LL that contains the lower left coordinates and the end point, and does NOT contain the dimension information.

Statement 1: LL' is an equivalent representation to LL. That is, given a representation in LL form, one can construct a representation in LL' form and use it to recreate the tiling; conversely, one can build an LL form from an LL' form to represent the same tiling.

Statement 1 appears to be true; it is clear how to make an LL' form from an LL form, while for the converse, it should be possible to check all the lower left points to see which would help determine the dimensions for a given rectangle.

Is statement 1 true?

Now consider C and make C' from C by again forgetting the dimensions, except that you can provide two endpoints representing the upper right and lower left corners of the tiled rectangle.

Statement 2: C and C' are equivalent in the sense used above for LL and LL' .

It is clear how to get a C' form from a C form. It is NOT clear that it is possible to get a C form from a C' form where you have no dimension information at all, except for the entire rectangle as implied by the two endpoints.

Is Statement 2 true?

Statements 1 and 2 have a bearing on the problem above: it may be possible to enumerate geometrically inequivalent tilings with n rectangles by looking at normalized forms of either C, C', LL, or LL'. Also, there may be advantages to switching between the forms of the tiling in determining equivalence.

There are interesting variations that can occur. For example, C'' might have the center of the tiled rectangle instead of the two endpoints, or even give just the dimensions of the tiled rectangle instead of the coordinates for its upper right and lower left corners. What information is lost by such representations?

Meta-question: Is there a geometric property at work here that might explain why LL' uses less information but is still equivalent to C (assuming Statement 1 is true), while C' does not (assuming Statement 2 is false)? Can something be said in general about what properties of a form for tiling are needed in order to represent the tiling both minimally and faithfully?

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text:
# php_mailparse.dll for PHP 5.3.x

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I tried a long time to get a working binary version of the PECL extension (PECL: „PHP Extension Community Library“) mailparse for PHP 5.3.x in the Internet. As that failed, I tried to compile the searched *.dll file by myself. Unfortunately, the attempt took more time than planned. Therefore thanks to Иван Гущин, who compiled the php_mailparse.dll for PHP 5.3.x and wrote a documentation for the compilation in my order. The compiled php_mailparse.dll is published as public download.

## Download and Installation of php_mailparse.dll for PHP 5.3.x

The resulting php_mailparse.dll, compiled in VC6 and against the actual PHP 5.3.5 source is publicly downloadable here. php_mailparse.dll can easily been setup as extension in PHP 5.3.x: Simply copy php_mailparse.dll to the subdirectory /ext of the PHP installation (e.g. C:\Program Files (x86)\PHP\ext) and activate the extension in the PHP config file (php.ini, usually found in the PHP installation directory, e.g. C:\Program Files (x86)\PHP). Up next there is an exemplary extract of a php.ini with activated mailparse extension:

;;;;;;;;;;;;;;;;;;;;;;
; Dynamic Extensions ;
;;;;;;;;;;;;;;;;;;;;;;

; If you wish to have an extension loaded automatically, use the following
; syntax:
;
;   extension=modulename.extension
;
; For example, on Windows:
;
;   extension=msql.dll
;
; ... or under UNIX:
;
;   extension=msql.so
;
; ... or with a path:
;
;   extension=/path/to/extension/msql.so
;
; If you only provide the name of the extension, PHP will look for it in its
; default extension directory.
;
; Windows Extensions
; Note that ODBC support is built in, so no dll is needed for it.
; Note that many DLL files are located in the extensions/ (PHP 4) ext/ (PHP 5)
; extension folders as well as the separate PECL DLL download (PHP 5).
; Be sure to appropriately set the extension_dir directive.
;

extension=php_mbstring.dll
extension=php_bz2.dll
extension=php_curl.dll
;extension=php_dba.dll
extension=php_exif.dll
extension=php_fileinfo.dll
extension=php_gd2.dll
extension=php_gettext.dll
;extension=php_gmp.dll
;extension=php_intl.dll
extension=php_imap.dll
;extension=php_interbase.dll
;extension=php_ldap.dll
;extension=php_ming.dll
;extension=php_mssql.dll
extension=php_mysql.dll
extension=php_mysqli.dll
;extension=php_oci8.dll      ; Use with Oracle 10gR2 Instant Client
;extension=php_oci8_11g.dll  ; Use with Oracle 11g Instant Client
extension=php_openssl.dll
;extension=php_pdo_firebird.dll
;extension=php_pdo_mssql.dll
extension=php_pdo_mysql.dll
;extension=php_pdo_oci.dll
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;extension=php_soap.dll
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;extension=php_sybase_ct.dll
extension=php_tidy.dll
extension=php_xmlrpc.dll
;extension=php_xsl.dll
;extension=php_zip.dll
extension=php_mailparse.dll
;extension=php_mcrypt.dll
;extension=php_mhash.dll
;extension=php_mime_magic.dll

The extension is being activated in line 73.

 mailparse requires the PECL extension mbstring (php_mbstring.dll) to be loaded before loading of php_mailparse.dll (see line 32)!

## Background Story

I run on several Windows servers the e-mail ticket system Cerberus Helpdesk. This system uses mailparse to parse in- and outgoing e-mails. Until PHP version 5.2.x the neccessary php_mailparse.dll as binary for Windows systems could be found at the build page of the PHP developer Pierre Joye at this page.

Some time ago, the Windows builds have been discontinued; a long term planned buildbox for the PECL extensions for Windows systems hasn't been implemented until today. Therefore every single user has to care by his own to compile the corresponding extensions since PHP 5.3.x.

Unfortunately, even the manufacturer of the ticket system haven't been able to remedy until today. Since the missing binary of php_mailparse.dll has been a major blocker of an update to PHP 5.3.x a solution was needed.


text:
# The definition of natural log and e

I'm asking this question from the point of view of an introductory non-rigorous calculus instructor. Calculus textbooks have different approaches about how to define $e$ and $\ln$. For example, my current textbook defines $e$ as the number such that

a) $\int_1^e \frac1x \, \mathrm{d}x= 1$.

I've also seen $e$ defined as the number such that

b) $\frac{\mathrm{d}}{\mathrm{d}x} e^x |_{x = 0} = 1$, or

c) $e := \lim_{n \to \infty} (1 + \frac1n)^n$, or

d) $e := \sum_{n=0}^\infty \frac{1}{n!}$

Of course no matter where you start, you can derive the others, but the distinction between which facts are definitions and which are derived feels very artificial, especially to the students! The questions:

• Do you prefer one of these approaches, and if so, why?
• Is it a good idea to explain the (arbitrarily) chosen definition and the subsequent derivations to students? I often wonder: why not just say $e$ is an important number that is about 2.718, and talk about its properties? Maybe the appropriate place for these worries is an analysis course.
• I have always hated saying <foo> is important and go on about their properties instead of showing why it is important. As a student I don't care if it's above my level, as long as you attempt to show me its importance. – Mark Fantini Jun 30 '14 at 22:27
• Please don't remove content from a question when editing it especially if you have enough rep to edit without peer review... the fact that I am teaching non-rigorous freshman calculus is important to the question. – Chris Cunningham Jul 1 '14 at 10:29
• Students are exposed to e and ln in high school, well before their first calculus class. Do you feel you need to 'undo' some harm or approach it as if for the first time? – JoeTaxpayer Jul 1 '14 at 13:08
• @JoeTaxpayer That's part of the question. Our calculus books certainly seem to think they should start from scratch, always defining either e or ln and never referring to prerequisite knowledge. I think that's odd and worth discussing! – Chris Cunningham Jul 1 '14 at 19:57

I think it is a good thing to talk about how there are some concepts where there are choices for where you start when definining them. It happens in linear algebra too, with the definition of linear dependence. You need to talk about how there is this web of connected properties, and it depends on what you're trying to achieve as to where you start. I do recommend prefacing any discussion of it with a general discussion on logs and exponentials using familar numbers, so they get a feel for how they ought to work first. This can even be done without graphs.

As to which definition of e I prefer...

Firstly, it's the functions ln(x) and $e^x$ that are the important thing, rather than the number $e$ itself per se. The number $e$ just crops up as a useful number to unify things.

I quite like the one where ln(x) is defined to be $\int_1^x \frac{1}{t} dt$, and e is the number for x that makes this area come out to 1. I have several reasons:

1. There is a real need for this function to have an integral because no power of $x$ differentiates to give $\frac{1}{x}$.

2. You can approximate e by using upper and lower sums of an easy function like 1/x.

3. The area truly exists because you can draw it there on the page, whereas it's a bit dicey as to whether it's reasonable to differentiate a function like $a^x$ when you don't know what it even means when $x$ is irrational!

4. Defining e using the area of a shape we can draw makes it feel a bit like $\pi$, which can be defined as the area of a circle and approximated by polygonal areas, and is one of the other famous irrationals we are familar with.

EDIT: For reference, here is the full sequence of definitions as I understand it (since it has been hotly debated in the comments):

As far as I understood it, it goes like this: for positive real $a$ you still define $a^1:=a$, $a^n:=a^{n−1}\times a$ for $n\in \mathbb{N}$, $a^0:=1$, $a^{−n}:=\frac{1}{a^n}$ for $n\in\mathbb{N}$, $a^\frac{1}{n}:=\sqrt[n]{a}$ for $n\in\mathbb{N}$ and $a^\frac{m}{n}:=(a^\frac{1}{n})^m$ for $n\in\mathbb{N}$ and $m\in\mathbb{Z}$.

Then you can still prove all the familiar properties of powers from these definitions, and that $\frac{d}{dx}x^r=rx^{r−1}$ for any rational $r$ using methods of first principles, without using logs at all. The issue is how you define $a^x$ for irrational $x$, because it needs a definition before you can differentiate $a^x$ or indeed define an inverse for the function.

If we want, can implicitly assume it does have a consistent definition which works for all the other properties we observe about powers, and investigate the behaviour of this function and its inverse and their derivatives, calling the inverse $\log_a (x)$, but realising that there is a hole in our definition of it and a hole in our arguments about derivatives until we know what $a^x$ means for irrational $x$.

Now we define $\ln(x) := \int_1^x \frac{1}{t}dt$ which immediately gives us that $\frac{d}{dx} \ln(x) = \frac{1}{x}$. From this definition we are able to prove all the familiar log laws for $\ln$. For example, to prove that $\ln(x^r) = r\ln(x)$ we can do the following:

By the chain rule, $\frac{d}{dx} \ln(x^r) = \frac{1}{x^r}\cdot rx^{r-1} = \frac{r}{x}$. However $\frac{d}{dx} r\ln(x) = \frac{r}{x}$. Since these two functions have the same derivative, they differ by a constant and so $\ln(x^r) = r\ln(x) +C$ for some $C$. Substituting $x=1$ gives $\ln(1) = r\ln(1) + C$, so $C=0$ and $\ln(x^r) = r\ln(x)$. Note that this only works for $r$ rational, because we still don't know what $x^r$ means for $r$ irrational.

We define $\exp$ to be the inverse of $\ln$, and using the properties of $\ln$ we are able to prove that it behaves the same as the powers laws we know. For example, we can prove that $\exp(a+b) = \exp(a)\times\exp(b)$ and $\exp(rx) = \exp(x)^r$. However this last one is only currently proved for rational $r$.

We define the particular number $e$ as the number such that $ln(e)=1$ or equivalently $e := exp(1)$, because we notice that $\exp(x)$ is the same as $e^x$ for all rational $x$ (and we know what $e^x$ is for rational $x$ because we defined what rational powers meant already without reference to $\exp$ or $\ln$). So now we define $e^x := \exp(x)$ for irrational $x$ and we define $a^x := \exp(\ln(a)\times x)$ for irrational $x$. We don't need to define $a^x$ for rational $x$ because it already has a definition, but we notice that even for rational $x$ this new calculation gives the correct result. We also notice that now $\exp(x) = e^x$ for all real $x$.

Finally, we can prove how derivatives of power functions work, and we can define $\log_a$ as the inverse of the function sending $x$ to $a^x$. And finally we observe that $\log_e(x)$ is the same as $\ln(x)$ for all real $x$.

This seems like a roundabout way of doing things, but if we define $a^x$ for $x$ irrational as the limit of a sequence, then all our proofs have to involve how limits of sequences interact with these functions and their derivatives and they are therefore actually harder.

• If one is after rigor, it still remains to be shown that this definition of $\ln(x)$ is an actual logarithm. This involves about as many steps as my answer to this question. Here is one proof filling in the missing steps math.stackexchange.com/a/486759/92427 – Matt Jul 1 '14 at 20:39
• @Matt:@Matt: This definition of $a^x$ is perfectly valid and used in several textbooks. The usual approach is not to assume any previous definition of powers and provide the intuitive $a^n=\prod_{k=1}^na$ for natural $n$ as a theorem. Your second sentence makes no sense; you must use the definition of $a^x$ for any property involving $a^x$. – Dennis Jul 2 '14 at 0:20
• @Dennis I see your point. I generally prefer to think of $a^2=a\times a$ rather than its "true" definition being $a^2=\exp(2\ln(a))$ but I don't disagree with the self-consistency of it. Cheers. – Matt Jul 2 '14 at 0:42
• +1 @DavidButlerUofA Since you're on the logarithm path, it is worth including the identity: $\ln(xy)=\int_1^{xy}\frac{1}{t}dt=\int_1^x\frac{1}{t}dt+\int_x^{xy}\frac{1}{t}dt=\ln(x)+\int_1^{y}\frac{1}{xu}xdu=\ln(x)+\ln(y)$ where the 2nd to last equality uses the substitution $t=xu$. It is from this that one can prove $\ln(a^x)=x\ln(a)$ for $x\in\mathbb{N},\mathbb{Z}$,and $\mathbb{Q}$. Lastly, I would disagree with "then all our proofs have to involve how limits of sequences interact..." You need not resort back to the limit definition; just stand upon whatever theorem is most convenient. – Matt Jul 3 '14 at 22:44
• @DavidButlerUofA Nice one! Maybe this logarithm path is faster than the exponential path proposed in my answer. Regarding the issue of using limits, one reason I like it is because it is a simple introduction to what a real number "really" is at least from the perspective of many real analysis courses. That is, Cauchy sequences of rational numbers. But I am not going to split hairs on this point. :) – Matt Jul 3 '14 at 23:23

I think a lot depends on context -- what course you are teaching and what the characteristics of that course are. If by "Freshman college calculus" you mean what I think you mean -- namely, a non-rigorous course that treats a lot of things using intuitive and heuristic arguments -- then I wonder if "define" is even the right verb to use. Students come into such a class having already encountered $e$; they may not feel like they really understand it well, but they should know it is a number about equal to 2.718, and they know it is used to describe exponential growth and decay, and that it has something to do with "continuous compounding". They have some familiarity with $f(x)=\ln x$, although the motivation for using $e$ as the base of a logarithm may seem far from "natural".

If that is the background of the students coming in, then I would not define $e$. Rather, I would do (have done) something roughly like this:

We want to find the derivative of functions of the form $y=a^x$, where $a$ is some positive real number. So we compute $$\lim_{h \to 0} \frac{a^{x+h}-a^x}{h} = a^x \lim_{h \to 0} \frac{a^h-1}{h}$$ and we notice that the limit does not depend on $x$ at all; it's just some constant that depends only on the choice of $a$. Let's denote that constant (temporarily) by the symbol $K_a$. Then we have $$y=a^x \implies y'=K_a a^x$$

This is a really interesting (and surprising) formula. When we take the derivative of a power function, the exponent decreases; but when we take the derivative of an exponential function, we end up with the same exponential function, times some constant that depends only on the base of the exponent.

So what is that constant? Well, students can do some numerical approximation with a graphing calculator or a spreadsheet: Choose a single specific value for $a$ and make a table of values of $\frac{a^h-1}{h}$ for smaller and smaller values of $h$ to estimate the corresponding value of $K_a$. Compare results with other students to compile a table of values of $K_a$ for different values of $a$. Graph them. Any ideas what the relationship is?

At this point I reveal the "amazing fact":

For any choice of $a$, the constant $K_a$ turns out to be exactly $\ln a$.

Students can, at the very least, convince themselves that this claim is consistent with what they found above.

Now as a consequence of this amazing fact, we have that if a happens to be equal to e, then the exponential function is its own derivative.

Now here is where my teaching sequence may differ from others: I make no attempt to prove this amazing fact -- to do so you would have to already have a rigorous definition of $\ln a$ available, and in this type of course we are dealing with students who have no such definition to work with. Nor do I take the amazing fact as a definition of $\ln a$; one could certainly do that, but then you would have to prove that this definition syncs up in some way with what they already know about $\ln$.

Instead, I take the position that $\ln$ and $e$ are basically already "out there" in their background knowledge, not really defined but available to them to use. They know some properties about them, but none of them are "primitive", nor are any of them "derived". And I am adding one more property to that constellation, making it seem plausible through numerical examples but not trying to do anything more rigorous than that.

All of that seems to me to make sense in a 1st-year, non-rigorous college Calculus course.

Is this were a Precalc course? Then we'd be having a completely different conversation. We'd be talking about compound interest, and what it means to compound interest twice per year, or three times per year, or $N$ times per year. We'd see the effective growth rate $(1 + 1/N)^N$ appear naturally in that context, and we'd explore what happens when $N$ gets very large. Eventually we'd be convinced (but would not prove) that as $N$ gets very large this value seems to converge to some number about equal to $2.718$, and we'd call that number $e$, and then talk about how whenever something is growing (or decaying) continuously (rather than at discrete intervals) it makes sense to express that growth (or decay) in terms of $e$.

If this were a rigorous (i.e. proof-based) Analysis course? The sequence of topics in such a course likely has convergence of sequences ("limit as $N \to \infty$") before either limit at a point (which necessarily precedes derivatives) and also before convergent series (which necessarily precedes integrals), so the first opportunity to define $e$ would be as $\lim_{N \to \infty} (1 + \frac{1}{N})^N$. I would start there (making sure to motivate it by connecting it to what students remember about compound interest and other contexts of continuous growth). Then later, when we are covering other types of limits, use that definition to prove both that $\lim_{h \to 0} \frac{e^h-1}{h} = 1$. Then go on from there, revisiting the topic when we cover derivatives, series and integrals, each time proving the "new" property in terms of the previously established ones.

When I teach calculus I, my initial working definition for $e$ is that it is the real number implicitly defined by: $$\lim_{h \rightarrow 0} \frac{e^h-1}{h}=1$$ Naturally, this allows me to derive that $\frac{d}{dx} e^x = e^x$ as $$\lim_{h \rightarrow 0} \frac{e^{x}e^h-e^x}{h}=e^x\lim_{h \rightarrow 0} \frac{e^{h}-1}{h}=e^x.$$ Of course, we know this. At this point, I usually say something like, you know, if we wanted, we could have defined $e$ to be the number for which $\frac{d}{dx}e^x = e^x$. That is an interesting equation, it says the slope of the tangent line to $y=e^x$ is the same as the value of the function at the point of tangency. In the world of all possible exponential functions the base $e$ is very special.

In what follows below, this is the sequence I make the comments, but, you could rearrange them depending on what sequence of topics you follow.

• Later, when I discuss the second derivative test and give an introduction to Taylor polynomials I show $e^x = 1+x+ \frac{1}{2}x^2 + \cdots$. This allows me to explicitly calculate $e$ with just a little arithmetic. Often at this point, I admit that this is our third definition and I know of at least 5 more or less distinct definitions. That said, the power series definition is near and dear to my heart as it is the method I use to define exponentials in supermath and other formal systems... (this I usually abstain from saying)
• Later, when I treat indeterminant powers I show $e = \lim_{n \rightarrow \infty} \left( 1+\frac{1}{n}\right)^n$. At this point, if I have my wits about me, I will foolishly attempt to explain how this ties into continuous compounding of interest. Indeed, continuous multiplication is how the exponential is thought of in physics and a few other places. So, it's worthwhile to mention this.
• Later, after we've covered FTC part I, I point out how $\frac{d}{dx} \int_1^x \frac{dt}{t} = \frac{1}{x}$ suggests that $\int_1^x \frac{dt}{t} = \ln (x)$ as we have already shown that $\frac{d}{dx} \ln (x) = \frac{1}{x}$ via implicit differentiation. Then, I'll show how to derive the various laws of logs as consequences of differential and integral calculus. I then confess that there are other instructors who take the integral as the definition of the log and so the exponential is introduced as the inverse function of the log. However, I tell them that was not our approach since I thought they all could use a whole semester to practice logs and exponential functions. Of course, my approach also has holes ( I don't prove laws of exponents, I just assume we know them and $a^x$ is somehow reasonable for $a>0$, admittedly, there is more substructure here, but I haven't found a class with sufficient curiosity to push the issue)

Naturally, there are other ideas to plan. Intuition for exponential functions is worth cultivating as it is one of those constructions which appears in a multitude of abstract fields of mathematics. I also like this topic because it gives us one topic which spans the entire course of calculus both integral and differential.

A textbook I have used first defines $$\ln x =\int_1^x \frac 1 t \, \mathrm{d}t,$$ and then defines the number $e$ as in your Definition a).

Next, the exponential function $e^x$ is defined as the inverse of the function $\ln x$. (They show that this definition of $e^x$ coincides with the usual one from repeated multiplication when $x$ is an integer.) Finally, for $a>0$, they define exponentiation with real exponent as $$a^x=e^{(\ln a)x}$$

This seems a bit backwards, but I can't find anything wrong with it. One advantage could be that formal properties like $$a^x a^y=a^{x+y}$$ can be proven more directly than the way it's done in the Real Analysis textbooks I have seen.

• This would only work if transcendentals are done after both differentiation and integration have been introduced. – Ben Crowell Jul 1 '14 at 13:07

My personal oppinion is that the number e itself has virtually no significance and one should rather focus on the exponential function. Depending on which course you are teaching, it can show up in various ways.

If you are giving an introduction to analysis, defining limits, series, etc. than c) and d) are very nice, but of course the exponential series $$\exp(x)= \sum_{n=0}^\infty \frac{x^n}{n!}$$ is much more important than only its value at $x=1$. It can be highlighted that this is a very important if not the stereotype of a power series, one can discuss radius of convergence, etc.

If the focus is on functions and derivatives or differential equations, one may of course ask for a function that solves $$f'=f, f(0)=1$$ to again find $\exp$. (Notice that your version of b) is not a definition, e.g. $f(x)=x$ also fulfils $f'(0)=1$.)

My favourite version for lower grades is to introduce exponentiation by integers in the obvious way: multiple addition -> multiplication with integers and similarly multiple multiplication -> exponentiation. Then one can introduce $\log_b$ as the inverse of $b^x$, explain the calculus of exponentials and logarithms, in particular $$b^x =c^{xlog_c(b)}$$ which highlights the insignificance of the basis and mention the preferred choices of $c=10$ for convenience and $c=e$. This leads then naturally to the question why $c=e$ is actually preferred for e.g. implementation in calculators, which could bring you back to the exponential series. (If pupils are actually interested: https://math.stackexchange.com/questions/18445/fastest-way-to-calculate-ex-upto-arbitrary-number-of-decimals )

• Among all numbers a, consider the derivatives of y = a^x at x = 0. There is a unique base such that the derivative is 1. – Chris Cunningham Jul 1 '14 at 10:30
• @Chris: Sure. But what's the point? ;) In my opinion f'=f is more interesting. – Echsecutor Jul 1 '14 at 10:35
• I'm clarifying that in fact (b) from the OP is a perfectly valid definition of e if you get the quantifiers right. Obviously f'=f is more interesting. – Chris Cunningham Jul 1 '14 at 19:53

Consider the question: What is the form of the function $f:\mathbb{R}\rightarrow\mathbb{R}$ that has the following two properties?

1. $f'(x)=f(x)$
2. $f(0)=1$

The unique answer is of course $f(x)=e^x$ however the proof is longer than people generally expect if they haven't considered this before, (for instance, we don't assume a priori that the solution is in the form of an exponential) but it consists of a series of steps that are mostly within the grasp of freshman calculus students. The only step that may perhaps have to be put off until later in the course is proof of absolute convergence. Here it is in outline form:

• Derive the Maclaurin series $f(x)=\sum_{n=0}^\infty\frac{x^n}{n!}$.
• Prove absolute convergence (can be put off until later.)
• Derive $f(x+y)=f(x)f(y)$ for all $x,y\in\mathbb{R}$ by multiplying out $\sum_{n=0}^\infty\frac{(x+y)^n}{n!}$ and invoking the binomial theorem.
• Prove $f(xy)=f(y)^x$ for all $y\in\mathbb{R}$, as $x$ varies over successively larger sets $\mathbb{N}\subset\mathbb{Z}\subset\mathbb{Q}\subset\mathbb{R}$.
• Define $e\equiv f(1)$. Thus $f(x)=e^x$.

This may seem like a lot of overhead for a seemingly elementary result, but I have asked this to a number of math grads and asked it here. There doesn't seem to be a substantially shorter proof that doesn't take for granted a known result about $e$ or the natural logarithm.

Do you prefer one of these approaches, and if so, why?

Not speaking as a professional educator, but as an occasional tutor, I prefer the above $f'(x)=f(x)$ approach as I feel it is both the most "natural" motivation, as well as one which is most directly applicable to real-life applications (e.g. Newton's law of cooling, etc.). The other results as listed in the question follow fairly readily from this approach. It is also a good introduction to more formalized/advanced proofs.

Is it a good idea to explain the (arbitrarily) chosen definition and the subsequent derivations to students? I often wonder: why not just say e is an important number that is about 2.718, and talk about its properties? Maybe the appropriate place for these worries is an analysis course.

One can begin to compile a list of all the places where $e$ shows up. E.g. question to college freshmen: Did you know that in order to create mp3 files, one must use $e$ in the fourier transform to encode the sound as frequencies? (get as in-depth on this point as desired.) Another: It is one of the most important numbers in mathematics. The one equation which relates the 5 most important constants is: $$e^{i\pi} + 1 = 0$$

This list can go on and on...

• If you assume existence and uniqueness of solutions to ODE, then one can prove $f(ax) = [f(x)]^a$ for the solution of $f'(x)=f(x)$, $f(0)=1$,by verifying that both $g_1(x) = f(ax)$ and $g_2(x)=[f(x)]^a$ both satisfy $g'(x) = ag(x)$ with $g(0) = 1$, so they must coincide. No need to follow the long and cumbersome sequence you outline above. – Steven Gubkin Oct 28 '14 at 23:44
• Yes certain assumptions allow one to skip over steps that might be considered "cumbersome". Also your method says nothing about the actual value of $e\equiv f(1)=2.718...$. In any case I appreciate your point, as in a first introduction this may be an appropriate and expedient way to present the material without sweeping too much under the rug like is all too often done. +1 to your answer. – Matt Oct 29 '14 at 17:10
• if you look at my answer, you can see that starting with the differential equation, Euler's method gives $e = \displaystyle\lim_{n \to \infty} (1+\frac{1}{n})^n$. I hope you did not interpret "cumbersome" as a "diss": I very much like your answer! – Steven Gubkin Oct 29 '14 at 17:14
• @StevenGubkin Yes I was referring to your comment in which you are suggesting that by assuming existence/uniqueness of ODE solutions, then there is "No need to follow the long and cumbersome sequence [I] outline above." This tells you nothing about the value of $e$ - that requires a separate step, as you are describing, which involves about as much work as the first two steps I am proposing, in my opinion. – Matt Oct 29 '14 at 17:26

I have few remarks which may be enlightening. I have not seen most of these spelled out anywhere for some reason.

I do not prefer one definition over the others: in fact I think this is a very nice place to try to get students to compare and contrast various definitions, and to derive each definition from the other.

Here are my observations:

1. $e^x = [e^{\frac{x}{n}}]^n$. For very large values of $n$, $e^{\frac{x}{n}} \approx 1+\frac{x}{n}$ by the tangent line approximation to $y=e^t$ at $t=0$. So we obtain $e^x \approx (1+\frac{x}{n})^n$. This approximation should get better and better as $n$ increases without bound, motivating the possibility that $e^x = \displaystyle\lim_{n \to \infty} (1+\frac{x}{n})^n$. Proving this is a nice challenge.

2. Another way to $(1+\frac{x}{n})^n$ is Euler's Method. Apply Euler's method to the diff EQ $f'(x)=f(x)$ with $f(0)=1$ to the interval $[0,x]$, with $n$ subintervals. We obtain:

\begin{align*} f\left(\frac{x}{n}\right) &\approx 1+\frac{x}{n}\\ f\left(\frac{2}{n}\right) &\approx f\left(\frac{x}{n}\right)+\frac{x}{n}f\left(\frac{x}{n}\right) = \left(1+\frac{x}{n}\right)f\left(\frac{x}{n}\right) \approx \left(1+\frac{x}{n}\right)^2\\ &\vdots\\ f(x) &\approx \left(1+\frac{x}{n}\right)^n \end{align*}

1. Assuming existence and uniqueness of ODE's is all that is needed to derive $e^{a+b} = e^ae^b$ and $e^{ax} = (e^x)^a$. For the first, observe that $f_1(x) = e^{a+x}$ and $f_2(x) = e^ae^x$ both satisfy the DE $y'=y$ with initial condition $y'(0)=e^a$. For the second note that $f_1(x) = e^{ax}$ and $f_2(x) = (e^x)^a$ both satisfy the DE $y'=ay$ with initial condition $y'(0)=1$.

2. You may also enjoy this post where I spell out how solving equations like $2^x=3$ is connected to finding areas under $\frac{1}{t}$, and how this is all linked with the number $e$. This story can also be turned into a very nice activity, where the students discover all of these connections themselves.

I prefer d), which fits with the all-important definition $$\exp(x):=\sum_{n=0}^\infty \frac{x^n}{n!}$$

It is a straightforward definition. It makes obvious sense. The proof that the sum converges is about as easy as it can be. By contrast:

Definition a), via the integral of $1/x$ is indirect.

Definition b), via the derivative of $e^x$, only makes sense if you've already defined both differentiation and exponentiation over arbitrary real bases.

Definition c), the limit of $(1+1/n)^n$, is both a more difficult limit than the sum, and not as central a property as the others.

So I prefer d).

• Totally straightforward, from the sky. When I took calculus I it was presented with (a) and I liked it very much. Indirect? Yes. Proving it's a series depends on commenting about series and how to use them, which is not typically calculus I, at least it wasn't until calculus III that I dealt with power series. – Mark Fantini Jul 1 '14 at 5:55
• I learned d) in sixth grade, and it made sense then. An occasional gift from the sky is ok by me. – user173 Jul 1 '14 at 11:59
• I've reached my threshold for gifts from the sky, it happened way too often. – Mark Fantini Jul 1 '14 at 14:44

When I taught high school Calculus, I talked about how e is a transcendental number and explained that it went on forever, never repeating, and that any finite length number sequence you search for could be found inside it. I had a student give me their date of birth and then looked it up in the pi and e search engine and found it in both pi and e (with 2 billion digits the odds of a 6 digit number being found it in is rather high): http://www.subidiom.com/pi/

which rather impressed them. I then had them consider whether e is found inside pi or pi inside of e - and that rather boggled their minds. I also presented the difference between an algebraic and trascendental irrational number.

I also had them consider e ^ pi and pi ^ e and whether those algebraic and transcendental. Also that e ^ pi is transcendental, but we don't know about pi ^ e. (Which is strange, because it feels like it should be, but no one has been able to prove it according to most sources I've looked at...) Here's some other fun transcendental numbers: http://sprott.physics.wisc.edu/pickover/trans.html

That seemed to be "good enough" for high school calculus - they were convinced that it was a number similar to pi and that it was important enough to have its own button on their calculator. And they could see that it worked similar to other logarithm bases to solve exponential functions - like, you could use ln instead of log and still get the same answers, and in fact you could use any base to solve the types of things we were doing. I didn't really rigorously prove it's derivative, but using the graphing calculator they could see rather readily that e^x and ln x are inverse functions and that the derivative of ln x was 1/x. (Most graphing calculators let you graph, say, the 1st derivative of a function, using a numerical approach).

I think if I had gone on to teach second semester calculus I might have used the area under the curve definition, or perhaps the limit of the compound interest formula. But the big thing, I think, is what you hope to get from the definitions - like, are there particular results you're hoping to prove that need a certain definition?

If not, maybe it's okay to have a bit of mystery around e and to merely pique students interest in learning more about it. Like, I'd rather have students going "oh wow, e is really fascinating - let's go read a book on it" (say, Eli Maor's excellent Story of E) then "I know a formal definition of e, but it put me to sleep."

• The claim that "any finite length number sequence you search for could be found inside it" (where by "it" I presume you mean "its representation as a decimal in base 10") is commonly believed to be true, but is not known to be so. It is not even known whether every digit appears infinitely many times. – mweiss Jul 3 '14 at 17:53
• That's part of the fascination of pi and e for me - the fact that so many things that seem obvious are not necessarily proveable. Like, why can't we prove pi^e is transcendental? – James S. Jul 4 '14 at 18:50
• But yes, I agree, qualifying that with "it is thought this is true" makes sense from a math perspective. – James S. Jul 4 '14 at 18:51
• Do we know if pi and e are evenly distributed - like, if we go far enough will the digits start to even out in number? – James S. Jul 4 '14 at 18:51


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# Find the cube of the number 16.

Last updated date: 25th Mar 2023
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${16^3} = 16 \times 16 \times 16 = 4096$
Note- In order to obtain the cube of any number a, we will write ${a^3} = a \times a \times a$ and then the final value of the RHS of this equal will give the cube of number a. For the cube root of any number a, we find the value of ${a^{\dfrac{1}{3}}}$.


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# Memory Forensics & Tor (part two)

In my previous post I used Volatility to examine a memory image from a hypothetical Tor user accessing webmail, the internet, and a Tor hidden service. From that analysis I could ascertain with good confidence a user of the operating system connected to the Tor network from a USB on drive E:.

In this post, I will continue with the same memory image but see what additional information can be revealed from data carving tools.

## Bulk Extractor

Bulk Extractor uses ‘scanners’ to carve data of interest from either memory or disk images without relying on the file system. This can include URLs, emails, credit card numbers, IP addresses, network traffic and much more.

There is an extensive range of options available to tune the processing to your needs by enabling or disabling the various scanners. This might be necessary if you are limited by what you are allowed to search for, or have a large disk image. However, running Bulk Extractor with default options is straightforward against a small memory image such as ours: bulk_extractor -o <output directory> <source file>.

[email protected]:/mnt/c/BoH$bulk_extractor -o output Win10_14393_Tor_Closed.vmem bulk_extractor version: 1.5.5 Hostname: LAPTOP Input file: Win10_14393_Tor_Closed.vmem Output directory: output Disk Size: 4294967296 Threads: 8 Attempt to open Win10_14393_Tor_Closed.vmem 15:43:18 Offset 67MB (1.56%) Done in 0:00:17 at 15:43:35 15:43:30 Offset 150MB (3.52%) Done in 0:05:30 at 15:49:00 15:43:32 Offset 234MB (5.47%) Done in 0:04:08 at 15:47:40 ***snipped*** 15:51:38 Offset 4177MB (97.27%) Done in 0:00:14 at 15:51:52 15:51:45 Offset 4261MB (99.22%) Done in 0:00:03 at 15:51:48 All data are read; waiting for threads to finish... Time elapsed waiting for 8 threads to finish: (timeout in 60 min.) All Threads Finished! Producer time spent waiting: 470.968 sec. Average consumer time spent waiting: 0.481216 sec. ******************************************* ** bulk_extractor is probably CPU bound. ** ** Run on a computer with more cores ** ** to get better performance. ** ******************************************* MD5 of Disk Image: 01b135f50fefa0de0a58704b6649e174 Phase 2. Shutting down scanners Phase 3. Creating Histograms Elapsed time: 516.226 sec. Total MB processed: 4294 Overall performance: 8.31994 MBytes/sec (1.03999 MBytes/sec/thread) Total email features found: 1103  The beauty (or fallacy!) of Bulk Extractor is it will use ~95% CPU to get the job done. This is great, but don’t expect to multi-task, especially over a big image. The output is placed into the specified directory and split into various text files. [email protected]:/mnt/c/BoH/output$ ls -sSh
total 153M
60M windirs.txt            56K url_services.txt               0 jpeg_carved.txt              0 gps.txt
59M url.txt                28K ip.txt                         0 telephone_histogram.txt      0 httplogs.txt
25M domain.txt             24K ether.txt                      0 ether_histogram.txt          0 kml.txt
7.2M winpe.txt              16K email_histogram.txt            0 ip_histogram.txt             0 pii_teamviewer.txt
1.1M json.txt              8.0K sqlite_carved.txt              0 ccn_histogram.txt            0 rar.txt
952K zip.txt               8.0K aes_keys.txt                   0 url_searches.txt             0 unrar_carved.txt
188K email.txt             4.0K exif.txt                       0 ccn_track2_histogram.txt     0 url_microsoft-live.txt
148K rfc822.txt            1.0K ccn.txt                        0 ccn_track2.txt               0 vcard.txt
80K winlnk.txt               0 jpeg_carved                    0 elf.txt                      0 winprefetch.txt
80K domain_histogram.txt     0 sqlite_carved                  0 find_histogram.txt
64K report.xml               0 pii.txt                        0 find.txt

From the memory image, Bulk Extractor has identified 153 MB of data; however, most of this data is from a select few files. Because there is so much data the problem is actually going to be looking through it and ascertaining its meaning within the memory image (if that is at all possible).

This brings the analysis to an important point. In this contrived scenario, the web browsing is solely from one known session; however, this cannot be assumed in a real investigation. Any artefact from Bulk Extractor should simply be treated as an indicator. Just because a URL or email address is in the output doesn’t mean that our hypothetical user accessed it, or it’s relevant in any way.

## Searching for Tor hidden services

Given the background to this investigation, we know that a user accessed the Tor network, so a logical start would be to grep in url.txt for ‘onion’ to determine if Tor hidden services were recorded in the memory image.

[email protected]:/mnt/c/BoH/output$cat url.txt | grep "onion" 339718862 https://www.nytimes3xbfgragh.onion/ \x00\x00M\x00\x00\x00\x08\x00\x00\x00\x01%\x00\x00\x00 https://www.nytimes3xbfgragh.onion/\x00\x00\x00\x00\x00\x00\x00\x00\x00\x00\x00\x00\x00\x00\x00\x00 462190668 https://www.nytimes3xbfgragh.onion/ \xBB\x01\x00\x00\x00\x00\x00\x00#\x00\x00\x00\x00\x00\x00\x00https://www.nytimes3xbfgragh.onion/\xBF\x00\x00\x00\x00\x00\x00\x00\x00\x05\x00\x00\x00\x08\x00\x00 458557510-ZIP-1599 https://3g2upl4pq6kufc4m.onion POST" template="https://3g2upl4pq6kufc4m.onion">\x0A Param name 458557510-ZIP-1693 https://3g2upl4pq6kufc4m.onion rl\x0ASearchForm https://3g2upl4pq6kufc4m.onion /SearchForm>\x0A/ 583230264 https://www.nytimes3xbfgragh.onion/2018/03/17/world/asia/us-technology-smuggling-foreign-weapons.html?action=click&module=In%20Other%20News&pgtype=Homepage&action=click&module=Latest&pgtype=Homepage \xBB\x01\x00\x00\x00\x00\x00\x00\xC6\x00\x00\x00\x00\x00\x00\x00https://www.nytimes3xbfgragh.onion/2018/03/17/world/asia/us-technology-smuggling-foreign-weapons.html?action=click&module=In%20Other%20News&pgtype=Homepage&action=click&module=Latest&pgtype=Homepage\xBF\xBF\x00\x00\x00\x00\x00\x00\x00\x00\x05\x00\x00\x00\x08\x00 739017720 https://www.nytimes3xbfgragh.onion 6\x00\x00\x00\x03\x00\xFF\xFF"\x00\x00\x80\x04\x00\xFF\xFFhttps://www.nytimes3xbfgragh.onion\x00\x00\x00\x00\x00\x00\x00\x00\x00\x00\x13\x00\xFF\xFF\x0C\x00 ***snipped and edited for display*** It’s here that we start to get an idea that there is a lot of data to cover. In fact there are 92 records containing ‘onion’. We can clean up this output and get an idea of the unique entries. [email protected]:/mnt/c/BoH/output$ cat url.txt | grep "onion" | awk '{print $2}' | sort | uniq -c | sort -nr 50 https://www.nytimes3xbfgragh.onion 14 https://www.nytimes3xbfgragh.onion/ 10 https://www.nytimes3xbfgragh.onion/2018/03/17/world/asia/us-technology-smuggling-foreign-weapons.html?action=click&module=In%20Other%20News&pgtype=Homepage&action=click&module=Latest&pgtype=Homepage 6 https:/01.graylady3jvrrxbe.onion 6 https://et.nytimes3xbfgragh.onion 2 https://pnytimes.chartbeat.net/ping?h=nytimes.com&p=%2F2018%2F03%2F17%2Fworld%2Fasia%2Fus-technology-smuggling-foreign-weapons.html%3Faction%3Dclick%26module%3DIn%2520Other%2520News%26pgtype%3DHomepage%26action%3Dclick%26module%3DLatest%26pgtype%3DHomepage&u=BbmfdHBIgFSHhtUPz&d=nytimes3xbfgragh.onion&g=16698&g0=World%2CAsia%20Pacific%2Cinternational_desk&g1=RON%20NIXON&n=1&f=00001&c=1.75&x 2 https://3g2upl4pq6kufc4m.onion 1 https://et.nytimes.com/?subject=page&url=https%3A%2F%2Fwww.nytimes3xbfgragh.onion%2F&assetUrl=https%3A%2F%2Fwww.nytimes3xbfgragh.onion%2F&referrer=&clientTimeStamp=1521372293400&totalTime=272800&evtUUID=35f71ccd-53d5-4880-8284-0ce4a04aaa7f&page_view_id=0307636b-421f-4866-9b06-6e73d188ba1a&performance=%7B%7D&presentation=%7B%22pageType%22%3A%22Homepage%22%2C%22program%22%3A%7B%22breakpoint% 1 https://djdkduep62kz4nzx.onion.to/ Even though we only accessed one hidden service, https://www.nytimes3xbfgragh.onion, there are a number of other URLs and we need ascertain their relevance. A quick check of the URL prefix indicates that https://3g2upl4pq6kufc4m.onion is the DuckDuckGo search engine. The URL https://djdkduep62kz4nzx.onion.to is definitely weird1. According to Malware Traffic Analysis (who incidentally have great PCAP challenges!) the URL was linked in 2015 to the TeslaCrypt ransomware.2 The server might have been temporarily compromised in 2015. However, now the site is inaccessible from either Tor2Web or Tor at the time of writing. (As an aside the .to URL suffix with an hidden service indicates a Tor2Web address, which is a way for a user to access a Tor hidden service without running Tor. Dropping the .to will access the same site over the Tor network.) The remaining onion URLs (https:/01.graylady3jvrrxbe.onion and https://www.nytimes3xbfgragh.onion) both relate to our tested browsing of the New York Times hidden service. However the large amount of URL records from simply browsing two pages reinforces the caveats on using Bulk Extractor for anything but lead generation. ## Email Addresses The amount of noise in Bulk Extractor extends to other identifiers such as email. Even with only basic browsing, the file email_domain_histogram.txt has 218 different email domains. [email protected]:/mnt/c/BoH/output$ cat email_domain_histogram.txt
# BANNER FILE NOT PROVIDED (-b option)
# BULK_EXTRACTOR-Version: 1.5.5 ($Rev: 10844$)
# Feature-Recorder: email
# Filename: Win10_14393_Tor_Closed.vmem
# Histogram-File-Version: 1.1
n=212   @eff.org
n=40    @mozilla.org
n=34    @torproject.org
n=29    @gmail.com
n=20    @ime.sy
n=15    @mozilla.com
n=15    @suscerte.gob.ve
n=14    @chambersign.org
n=12    @trustcentre.co.za
n=10    @emas.me
n=10    @globaltrust.info
n=9     @diginotar.nl
***snipped***

One way to identify email domains of interest is to look a the frequency of the domain in the full URLs for the major webmail providers. The output file url_histogram.txt provides a frequency of URLs which can then be grepped against major webmail providers.

For Gmail we have two entries with low scores on the histogram (which may actually be mangled URLs concatenated together):

[email protected]:/mnt/c/BoH/output$cat url_histogram.txt | grep gmail n=1 http://by137w.bay137.mail.live.com/mail/hiplight.aspx?n=http://bl103w.blu103.mail.live.com/mail/inboxlight.aspx?n=http://by142w.bay142.mail.live.com/mail/inboxlight.aspx?n=http://by137w.bay137.mail.live.com/mail/inboxlight.aspx?.mail.live.com/mail/[email protected] n=1 http://[email protected] (utf16=1) There are similar low results for Hotmail and Yahoo, and some specialist providers like Tuta Nota or ProtonMail have no results. Searching for ‘live.com’ produced a number of interesting entries, but not all appear to be related to email. [email protected]:/mnt/c/BoH/output$ cat url_histogram.txt | grep live.com
n=24    https://xsts.auth.xboxlive.com  (utf16=21)
n=11    https://xsts.auth.xboxlive.com/ (utf16=11)
n=8     https://onedrive.live.com/windows/      (utf16=5)
n=7     http://auth.xboxlive.com
n=7     http://xboxlive.com
n=7     https://account.live.com/       (utf16=5)
***snipped***

When we search for GMX we get a much larger and richer number of URLs, with much high scores in the histogram.

[email protected]:/mnt/c/BoH/output$cat url_histogram.txt | grep gmx n=56 https://navigator-bs.gmx.com n=46 https://3c-bs.gmx.com n=38 https://dl.gmx.com n=36 https://home.navigator-bs.gmx.com n=36 https://trackbar.navigator-bs.gmx.com n=27 https://mailderef.gmx.com n=26 https://dl.gmx.com/uim/container/generic_dfp_gmxcom_1.html#id=0#url=https%3A//mailderef.gmx.com/dl.gmx.com/uim/container/generic_dfp_gmxcom_1.html#lcid=1521372556900233436.54056410247 n=21 https://3c-bs.gmx.com/mail/client/mail/sentconfirm;jsessionid=1E8359D02DBEE35F86F0FC0BDB82D44E-n1.bs08a?editorAction=NEW&mailId=tmai151d0045aebe9761&uc=SHOW_CONFIRMPAGE n=20 https://3c-bs.gmx.com/mail/client/mail/detail;jsessionid=1E8359D02DBEE35F86F0FC0BDB82D44E-n1.bs08a?5&mailId=tmai151cfee574b629f3&page=0&sort=sort-DATE-desc&folderId=tfol151997642113d811 n=19 https://dl.gmx.com/uim/container/generic_dfp_gmxcom_1.html n=16 https://3c-bs.gmx.com/mail/client/mail/compose/html?editorAction=NEW (utf16=1) n=14 https://navigator-bs.gmx.com/?sid=8f7b6d9796a1d87e18679305cba3fda03a827dbee2aa77e48455ce496cb3d5d7a81e4f34c4788802ecc6616ea4b28f69 ***snipped*** The results also show a variety of URL parameters including words like ‘compose’, ‘session’ etc. which makes GMX mail more interesting. There are also references to parameter ‘sid’, which is likely a unique session identification hash. Testing on a throwaway GMX account indicates there likely is a new ‘sid’ for every login. Now we can go back to the email.txt file and look for GMX emails. [email protected]:/mnt/c/BoH/output$ cat email.txt | grep gmx
[email protected]:/mnt/c/BoH/output$ Nothing! In the tradition of memory forensics there is no record for any GMX email account. Despite our hypothetical user logging into GMX mail and reading and writing messages, sometimes data simply isn’t in memory at the time of capture! This would be one to refer to the disk image or other sources of intelligence. ## Network Traffic Bulk extractor also carves network traffic data from memory, which can be interesting. This is saved as a .pcap file which can be read in tcpdump or Wireshark. Although for Tor traffic it is less of interest due to its encrypted nature and IP obfuscation. In our scenario, there is no additional information gleamed from the Volatility netscan results with the same IP address, 54.37.17.235, recorded in the (mostly malformed) pcap traffic. The IP 40.77.228.92 resolves back to Microsoft. ## ‘Traditional’ data carving One hindrance to memory forensics is that ‘traditional’ file carving usually doesn’t provide good results. This kind of file carving looks for hexadecimal headers and footers to extract data and is commonly used in disk forensics. However, unlike disk forensics, memory is fragmented and not all of the data may be in memory at the times of capture. The ‘bible’ of memory forensics explains this succinctly: Occasionally, people still attempt to reconstruct a file from a memory sample using traditional file carving tools, such as Scalpel…Unfortunately, most of these tools assume the file data is contiguous and that the media being analyzed contains a whole copy of the file…As a result except for files smaller than one page of memory, you are probably not going to extract the data you expect. The Art of Memory Forensics: Detecting Malware and Threats in Windows, Linux and Mac in Memory, p.494 So, if we tried, what would we get? Foremost is data carving utility which will carve picture, document, executables, ZIP/RAR, HTML or audio files. It is simple and quick to run and places each file type in output folders. Here I will attempt to carve for picture and HTML data. [email protected]:/mnt/c/BoH$ foremost -t jpg,htm,gif,png -v -i Win10_14393_Tor_Closed.vmem -o foremost
Foremost version 1.5.7 by Jesse Kornblum, Kris Kendall, and Nick Mikus
Audit File

Foremost started at Tue May  1 09:31:44 2018
Invocation: foremost -t jpg,htm,gif,png -v -i Win10_14393_Tor_Closed.vmem -o foremost
Output directory: /mnt/c/BoH/foremost
Configuration file: /etc/foremost.conf
Processing: Win10_14393_Tor_Closed.vmem
|------------------------------------------------------------------
File: Win10_14393_Tor_Closed.vmem
Start: Tue May  1 09:32:12 2018
Length: 4 GB (4294967296 bytes)

Num      Name (bs=512)         Size      File Offset     Comment

0:      00009856.png           1 KB         5046368       (36 x 36)
1:      00009859.png           1 KB         5048104       (48 x 48)
2:      00092377.png          198 B        47297024       (50 x 50)
3:      00092378.png          224 B        47298032       (44 x 44)
4:      00092381.png          224 B        47299120       (44 x 44)
5:      00092383.png          136 B        47300160       (16 x 16)
6:      00104370.png          452 B        53437746       (16 x 16)
7:      00104371.png         1005 B        53438301       (32 x 32)
8:      00104373.png          485 B        53439397       (16 x 16)
9:      00144140.png          209 B        73799772       (9 x 5)
10:     00144141.png          215 B        73800560       (9 x 5)
***snipped***
560:    08269196_1.htm        236 B      4233828859
561:    08269197.htm          269 B      4233829195
562:    08269198.htm          247 B      4233829563
563:    08269199.htm          250 B      4233829915
564:    08269199_1.htm         3 KB      4233830267
565:    08269206.htm          281 B      4233833931
566:    08214113.gif           6 KB      4205626344       (1168 x 769)
567:    08212180.png          353 B      4204636576       (310 x 150)
568:    08212182.png          502 B      4204637616       (310 x 150)

Finish: Tue May  1 09:32:33 2018

569 FILES EXTRACTED

jpg:= 1
htm:= 41
gif:= 23
png:= 504

Foremost finished at Tue May  1 09:32:33 2018

Data was ‘found’ but as expected, the results are limited. No useful HTML or picture files were located. However, it did carve a number of small icons which could potentially be useful to either identify a hidden service of interest (e.g. a logo or a user avatar). For cases of child exploitation Foremost could be of greater interest, but ultimately it’s not the solution.

If we want to ‘carve’ files in memory we can return to Volatility and use the dumpfiles plugin. The syntax is vol.py -f <filename> --profile=<profile> dumpfiles -n -D ./<output_path> -r <regex_value> Here the -n appends the name of the original file and -r is using a regular expression to limit the output.

Results may still be limited. My attempts to locate HTML, GIF, JPG, and sqlite files all fail to locate any data.

However turning to Windows system files, using dumpfiles against Event Logs produced a good output to examine.

[email protected]:/mnt/c/BoH$vol.py -f Win10_14393_Tor_Closed.vmem --profile=Win10x64_14393 dumpfiles -n -D ./dump -r .evtx Volatility Foundation Volatility Framework 2.6 DataSectionObject 0xffff80814be1c460 908 \Device\HarddiskVolume1\Windows\System32\winevt\Logs\System.evtx SharedCacheMap 0xffff80814be1c460 908 \Device\HarddiskVolume1\Windows\System32\winevt\Logs\System.evtx DataSectionObject 0xffff80814be23080 908 \Device\HarddiskVolume1\Windows\System32\winevt\Logs\Application.evtx SharedCacheMap 0xffff80814be23080 908 \Device\HarddiskVolume1\Windows\System32\winevt\Logs\Application.evtx DataSectionObject 0xffff80814be24c80 908 \Device\HarddiskVolume1\Windows\System32\winevt\Logs\Security.evtx SharedCacheMap 0xffff80814be24c80 908 \Device\HarddiskVolume1\Windows\System32\winevt\Logs\Security.evtx ***snipped*** Located in the Security Event log was a reference to torbrowser-install-7.5.2_en-US.exe. As the log was only partially complete, as is common, and Windows Event Log Viewer failed load the log to view it natively. By reviewing baseline data (being logs from my Windows installation) and handy research previously done here, the key fields (for this log) can be broken down as follows: Offset (decimal) Data Type Length (bytes) Hex Data Converted Data / Comment 0 Event Record Signature 4 2A2A0000 Hex Signature 4 Event Record Size 4 60020000 608 bytes (LE) 8 Event Record Identifier 8 D801000000000000 472 (LE) 16 Windows FILETIME written 8 177D4A5BAABED301 18 March 2018, 11:15:00 UTC 24 Binary Data Header 4 0F010100 Hex Signature 118 Event ID 2 BE 12 4798 130 Windows FILETIME event created 8 177D4A5BAABED301 18 March 2018, 11:15:00 UTC 171 Event Log Provider Name 70 UTF-16 N/A 257 Event Log Channel 16 UTF-16 N/A 365 Target SID 28 010500000000000515000000 EB45C98E2A11D64863E30076E8030000 S-1-05-5-21-2395555307- 705812040-1221988650-1979769699-1000 393 Subject User SID 28 010500000000000515000000 EB45C98E2A11D64863E30076E8030000 S-1-05-5-21-2395555307- 705812040-1221988650-1979769699-1000 421 Subject User Name 8 UTF-16 N/A 431 Subject User Domain 30 UTF-16 N/A 479 Caller process name 116 UTF-16 N/A 604 Event Record Size (Footer) 4 60020000 608 bytes (LE) Which corresponds to the event log fragment as follows: The actual event (in this circumstance) is not particularly interesting ‘Event 4798 - A user’s local group membership was enumerated’. The presence of the log and the FILETIME stamp provides another interesting data point for a timeline. It also provides additional information that the Tor Browser Bundle installer was located on HarddiskVolume2, which may or may not be the same USB as the installed files. The date stamp, 18 March 2018, 11:15:00 UTC, indicates that the timing is definitely consistent to the other timestamps identified in this analysis (e.g. from the$MFT).

Having taken the time to parse out the hex in the event log fragment, I thought it might be useful to carve similar Event Log fragments via a script. The result is a simple Python script, event_carver.py, which searches a binary file (usually a .evtx file that is corrupted or zero padded) and locates the event record signature x\2Ax\2Ax\00x\00. It will then parse and extract some key fields including the record offset within the file, event log ID, and timestamp. The event fragment is then dumped into an ‘output’ folder for manual inspection.

(It certainly doesn’t appear to be as comprehensive as either EVTXract or python-evtx neither of which I have tested at this time.)

The output looks as follows:

c:\Scripts\event_carver>python event_carver.py test.bin
**snipped**

Located 139 Windows Event Record fragments
Event carver completed at 10:45:39, 08-May-2018

The output folder will appear as:

c:\Scripts\event_carver\output>dir /d /b
events_summary.csv
offset_0x1200_2018-03-08.bin
offset_0x1cf0_2018-03-08.bin
offset_0x31200_2018-03-07.bin
offset_0x31b48_2018-03-07.bin
offset_0x31d78_2018-03-07.bin
**snipped**

The script should only be used as a first triage, as it’s blindly looking at values that may be empty or corrupted. At this stage there is significant variability in the remaining binary content of the EVTX record so I haven’t yet attempted to parse this data out. (Otherwise, I’d never get this post out!)

## Summary

There is almost endless analysis that could be done with the Bulk Extractor output. Most likely (and useful) is to put the output into a word list which you can then run over the physical image or against password protected files. This could either be a selection of data, or through enabling the word list option on initial processing via bulk extractor -e wordlist -o <output directory> <source file>.

Traditional data carving is unlikely to produce quality results, as the testing with Foremost demonstrated. Instead, using Volatility dumpfiles can assist.

That’s all for this two-part series on analysing Tor artefacts in memory images. In traditional DFIR fashion it started as looking at Tor and ended with carving event records. You never know where you end up!


text:
# Jacobian Integration Problem

• February 23rd 2010, 11:41 AM
Jacobian Integration Problem
$\int \int e^(-r(x-2y)^2) dxdy$

x =2u+v
y=u-v

The Jacobian is -3
I've substituted for x and y in the integral to get $\int \int -3e^r(3v)^2 dudv$

where r is a positive constant

But my problem lies with the limits, I have y is greater than or equal to 0 and x is less than or equal to 0, How do I change these for the integral and proceed as a result?

Thanks
• February 23rd 2010, 12:00 PM
tonio
Quote:

$\int \int e^(-r(x-2y)^2) dxdy$

x =2u+v
y=u-v

The Jacobian is -3
I've substituted for x and y in the integral to get $\int \int -3e^r(3v)^2 dudv$

where r is a positive constant

But my problem lies with the limits, I have y is greater than or equal to 0 and x is less than or equal to 0, How do I change these for the integral and proceed as a result?

Thanks

What you wrote in the 1st integral doesn't show clearly. Anyway:

$\left\{\begin{array}{c}x=2u+v\\y=\;\;u-v\end{array}\right.$ $\Longrightarrow x+y=3u\Longrightarrow u=\frac{x+y}{3}\Longrightarrow v=\frac{x-2y}{3}$ , so $-\infty < u<\infty\,,\,\,-\infty

Tonio
• February 23rd 2010, 12:02 PM
Moo
Hello,

Hmmm the Jacobian has to be the absolute value of the determinant, so it would be 3.

now for the boundaries :

we have $x=2u+v\leq 0$ and $y=u-v\geq 0$

from this, it follows that $\boxed{v\leq u\leq -\tfrac v2}$

and for the boundaries of v : $v\leq -\tfrac v2 \Rightarrow \boxed{v\leq 0}$


text:
QUESTION

# BUS 681 Week 2 DQ 1 Seniority and Merit Pay

This file of BUS 681 Week 2 Discussion Question 1 Seniority and Merit Pay contains:

Define the concept of seniority and merit pay plans, including the strengths and limitations of such plans within an organization. Discuss the job, organizational and/or other factors that should be considered when deciding between the two. Respond to at least two of your fellow students

• @
Tutor has posted answer for $5.19. See answer's preview$5.19


text:
# Conventions¶

## Common variable names¶

Unless otherwise stated (in function or class docstrings), this package tries to adhere to using standard variable names for function and class arguments, and in example code. For shorthand, the variable w is used to represent arrays of phase-space coordinates (e.g., positions _and_ velocities). When representing only positions, the variable q is used. For just velocities or momenta, the variables p or v are used.

## Array shapes¶

The arrays and Quantity objects expected as input and returned as ouput throughout Gala have shapes that follow a particular convention, unless otherwise specified in function or class docstrings.

For arrays containing coordinate or kinematic information, axis=0 is assumed to be the coordinate dimension. For example, for representing 128 Cartesian 3D positions, the object would have shape (3,128).

For collections of orbits, arrays have 3 axes. As above, axis=0 is assumed to be the coordinate dimension, but now axis=1 is interpreted as the time axis, and axis=2 are the individual orbits.

## Energy and momentum¶

The gala documentation and functions often refer to energy and angular momentum and the respective quantities per unit mass interchangibly. Unless otherwise specified, all such quantities – e.g., energy, angular momentum, momenta, conjugate momenta – are in fact used and returned per unit mass.


text:
Hi,

Would anyone have implemented integrating with MSAccess ? I’m having trouble doing an insert operation into MSAccess97 using ODBC Adapter.

Hello … We have implemented integrations with MSAcces97. We also had some issues with the insert command using the ODBC adapter.

The problem that we ran into was that we had been using one of the ODBC reserved keywords in our insert statement.


text:
# What is 10 to the fifth power?

$\textcolor{w h i t e}{\times} {10}^{5} = 100000$
$\textcolor{w h i t e}{\times} {10}^{5} = 10 \times 10 \times 10 \times 10 \times 10$
$\textcolor{w h i t e}{\times \times} = 100000$


text:
## Numerical Methods for Engineers

$A_3=3.33 m^{2}$
$Q_{in}=Q_{out}$ $Q_1=Q_2+Q_3$ $40 \frac{m^{3}}{s}=20 \frac{m^{3}}{s}+Q_3$ $Q_3=20 \frac{m^{3}}{s}$ $Q=vA$ Since $Q_3=20 \frac{m^{3}}{s}$ and $v_3=6 \frac{m}{s}$ $20=6A_3$ $A_3\approx3.33 m^{2}$