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# September 17

## A Boolean algebra

I'm taking a discrete math class. I'm confused by some of the terminology and I've already asked my teacher for a clarification/definition and I'm still confused. So, could someone explain what "a Boolean algebra" is? For instance, one of my homework questions says:

Let B be a Boolean algebra.
Define the relation ≤ on a Boolean algebra B by x≤y if xy=x.
Prove that if x is an atom of B, and y ∈ B, then either xy=0 or xy=x.

I'm not looking for help on the problem as much as I'm looking for an explanation of what an algebra or more specifically a Boolean algebra is. My teacher basically restated what's in our articles here about fields and rings and such. I think I'd be satisfied with a fairly simplistic explanation, if one can be given, similar to how electrons around an atom are often simply explained as a particle in an orbit when the full explanation is more complicated. Thanks, Dismas|(talk) 05:14, 17 September 2014 (UTC)

Well, the long answer is that a Boolean algebra is a collection of elements and a pair of operations that satisfy a list of axioms. I assume that's the one your teacher gave. Another way to think of it is that a Boolean algebra is a collection of sets: You start with one big set, and take some of its subsets. 1 is the starting set. 0 is the empty set. xy means $x \cap y$, and x+y means $x\cup y$.--80.109.106.3 (talk) 10:40, 17 September 2014 (UTC)
Confusingly, a Boolean algebra is not an algebra. Most of the axioms for a Boolean algebra are the same as for a distributive lattice, but there is an additional operation, negation, and axioms associated with it. We have several basic articles on Boolean algebras but Boolean algebra (structure) gives an axiomatic treatment which is probably what you're looking for. Atom (order theory) defines atom in the sense used in the problem. --RDBury (talk) 14:38, 17 September 2014 (UTC)

In case it helps, our article is at Boolean algebra (structure). RDBury's (first) link goes to the wrong place, though to my mind it's a fairly natural mistake.
There are two basic notions called Boolean algebra, related but distinct, one that uses the term as a count noun (as you have used it) and one that uses it as a mass noun (roughly, the equations that are true in all Boolean algebras, and the methods of manipulating them).
There was a time that the straight search term Boolean algebra went to the article on the structure (the count-noun sense), but that resulted in too much confusion on the part of readers who were expecting the other notion. That did need correction. Unfortunately it was impossible to agree on exactly how to correct it, and it turned into a bloody war with a very unsatisfactory outcome. --Trovatore (talk) 16:51, 17 September 2014 (UTC)

Thanks, I missed that when I skimmed the articles. We also have Boolean algebras canonically defined which seems, and again I just skimmed it, to cover the subject from a constructive rather than axiomatic viewpoint. Perhaps analogous to the distinction between Group (mathematics) and Permutation group or between Synthetic geometry and Analytic geometry. --RDBury (talk) 20:49, 17 September 2014 (UTC)
The "canonically defined" article is a bit ... idiosyncratic. I really think something ought to be done about it but I don't have the belly for it anymore.
On the other hand, there certainly is an approach based on defining Boolean algebras in a particular representation rather than as models of certain axioms. This is the content of the Stone representation theorem, and yes, it's closely analogous to treating all groups as subgroups of some Sκ for sufficiently large κ, or all manifolds as submanifolds of Rn for sufficently large n. It's not always the most natural representation — but then, neither are the other two examples. --Trovatore (talk) 21:44, 17 September 2014 (UTC)
Aside: Near the top of boolean algebra, it says "exists as a core data type in all modern programming languages generally abbreviated to as type bool". That "to as type bool" is awkward and needs changing, but I can't decide what to. Any suggestions? -- SGBailey (talk) 12:44, 18 September 2014 (UTC)
Good call, I fixed it. SemanticMantis (talk) 15:19, 18 September 2014 (UTC)

# September 18

## Mathematical Problems Immediately Relevant To Medicine

Are there any open problems in mathematics the solution of which would be directly applicable to medicine - in the same way that there are problems in mathematics relevant to physics. Or, are there are any computer algorithms that, if improved in efficiency, would be of medical benefit. I'm thinking more along the lines of developing antiviral drugs and vaccines - but any relevance would be of interest. --Every time I try to do any research to answer questions like this, I end up getting loads of results about epidemiology, which isn't exactly the type of thing I had in mind. Thank you for any help:-)Phoenixia1177 (talk) 09:46, 18 September 2014 (UTC)

The issue is that most famous "open problems" in math are abstracted far away from the things they could be usefully applied to. However, just about any paper published in a mathematical biology journal sets itself up with a novel mathematical problem that will somehow shed light on a biological issues being discussed. For antivirals and vaccines, I'd think that key mathy areas would be protein folding, Molecular_dynamics, and mechanical properties of DNA. All these topics have room for research involving improvements to algorithms and smaller challenges can likely be phrased as problems in pure math. A lot of this work falls under the rubric of mathematical modeling, but once the model is specified analyzing it is just math. While not immediately drug related, here's an example of some work that uses tools from knot theory and topology to study DNA [1]. Conceivably understanding bacteria DNA better might allow better targeting of antibiotics... I'm much more familiar with the bio-math side, but the pharmacology angle also has tons of applied math. I think the best way to come up with problems in math relevant to medicine is to start scanning e.g. Journal of Mathematical Biology on a regular basis, and search in each issue for terms that interest you. That is of course much easier if you can get access to a university library. Finally, you do know you can exclude epidemiology from google scholar searches, right? E.g. this search [2] led me to this paper, which seems to be near your interest [3]. Also maybe check out some of this guy's work [4], I don't know what he's done in non-epidemiological immunology and medicine, but I'm a pretty big fan of his work in general. SemanticMantis (talk) 15:13, 18 September 2014 (UTC)
Well, in the field of artificial intelligence we have medical diagnostics. It's debatable whether it's an open problem, but those methods can always be improved. At some point I'd expect computers to diagnose problems better than doctors. And even if not as good, there's still billions of people in the world who can't afford to see a doctor, and finding a cheaper way to diagnose their medical problems would be of value.
And scans, such as MRIs, could use some sophisticated analysis, along with better resolution, to automatically detect things like cancer. StuRat (talk) 15:18, 18 September 2014 (UTC)
Thank you both for your answers:-) I'm trying to hunt down access to the sources linked. Thank you SemanticMantis for the reminder about excluding from a search...I have no idea why that did not occur to me. Stu, do you know where I could find more info on the algos used currently, or an overview of what types of problems are involved; both with automated diagnosis and scans? Despite having an interest in both mathematics and medicine, I'm having a hard time linking these two subjects - I did not have nearly as much trouble with physics, but I feel like those two are more apparently linked, perhaps? I'm not sure why this is seeming so intractable to break into; I do know that I am much stronger at pure mathematics than anything else, perhaps it is because physics has reached the level of maturity where many issues can be stated in something closer to mathematics? At any rate, thank you both very much for this, and for the many other questions you have aided me with:-)Phoenixia1177 (talk) 18:03, 21 September 2014 (UTC)
For a specific application of mathematics to drug testing and development, see Familywise_error_rate. This is an area of ongoing research, and also relevant to diagnostics such as the MRI, which must make thousands of comparisons to produce an image (see False_discovery_rate at the end of the article I just mentioned). You might also look into Genetics and Biostatistics, both of which are very mathematical. OldTimeNESter (talk) 15:49, 22 September 2014 (UTC)

## Logic Puzzles

I know how to solve a logic puzzle manually with a bit of trial and error. Is there a way to convert the statements into mathematical terms and automate a solution? EG "Bert was the first to arrive" might match in some fashion "Arrive(Bert) = Time 0" and "The person who ate the cake arrived after the flautist" from which we can derive "Person(cake) != Bert" (as Bert arrived at T0) and thus also "Food(T0) != cake". Also "Person(cake) != Person(flute)", "Instrument(cake) != Flute", "Food(flute) != cake" etc. But even if the sentences can be converted to these pseudo-maths statements, I suspect they aren't enough to automate a solution. What else needs to happen? -- SGBailey (talk) 12:57, 18 September 2014 (UTC)

I'm not sure how precisely to implement the solution to a specific problem, but this is the kind of thing that Prolog can do. If you wanted to do it by hand, I think you'd essentially be determining if there is an assignment that satisfies all of the statements, basically solving a Boolean_satisfiability_problem. SemanticMantis (talk) 15:17, 18 September 2014 (UTC)
They often require human knowledge, too, like "The person who goes last is female". A computer program would thus need to know which names are male and which are female. Perhaps a human can convert this all to math statements, then submit it to be solved by computer. StuRat (talk) 15:25, 18 September 2014 (UTC)
(edit conflict) I can't give a definitive answer, but I've heard of some success with the following models: "axioms"
foods = {cake, pie, salad} translates as
1. $(\forall x) (Food(x) = cake \or Food(x) = pie \or Food(x) = salad )$
2. $cake \ne pie$, etc.
3. $(\exists x)(Food(x) = cake)$, etc., or
$Food(cake) = cake$, etc.
and, if you don't use $(\exist!3 x)$
4. $(\forall x)(\forall y)(Food(x) = Food(y) \leftrightarrow x = y)$
For "<", you might have specific axioms:
1. $T0 < T1$, etc.
2. $T1 \nless T0$, etc.
1. $(\forall x)(x \nless x)$
The exact formalism required depends on the complexity of the problem. If you're dealing with Mr. Brown, Miss Scarlett, etc., and they have clothing items of those colors, you need function symbols. If you only have Food 1 is brought by person 2; Food 2 is brought at time 3; etc., you actually only need constant symbols, and the problem is to determine which constant symbols are the same as other constants given the axioms. If it turns out you don't need the function "Food", the axioms above become:
1. $(\forall x) (x = cake \or x = pie \or x = salad )$
2. $cake \ne pie$, etc.
In your case, the specific axioms might be:
Bert = T0
cake > flute
with consequences:
T0 ≠ cake
cake ≠ flute
Arthur Rubin (talk) 16:02, 18 September 2014 (UTC)
The TPTP problem library has a PUZ domain, which contains many different kinds of logic puzzles encoded for automated theorem provers, including some classical ones from Raymond Smullyan's books (e.g. this one from the Isle of Knights and Knaves). --Stephan Schulz (talk) 14:10, 20 September 2014 (UTC)

# September 19

## is midrange = midpoint?

2 requests:

1. Prove $(max-min)/2 + min = (max+min)/2$.
2. Is Midrange = Midpoint?174.3.125.23 (talk) 11:23, 19 September 2014 (UTC)
1)
Welcome to the Wikipedia Reference Desk. Your question appears to be a homework question. I apologize if this is a misinterpretation, but it is our aim here not to do people's homework for them, but to merely aid them in doing it themselves. Letting someone else do your homework does not help you learn nearly as much as doing it yourself. Please attempt to solve the problem or answer the question yourself first. If you need help with a specific part of your homework, feel free to tell us where you are stuck and ask for help. If you need help grasping the concept of a problem, by all means let us know.
2) The midrange is a statistical concept while the midpoint is a geometric one. So, they wouldn't normally correspond. However, if you represent data on a number line, and draw a line segment from the minimum value to the maximum to represent the range, then the midpoint of that line segment is indeed the midrange. StuRat (talk) 13:08, 19 September 2014 (UTC)
1) LHS=
$\frac{max-min}{2} + min = \frac{max}{2} +(min-\frac{min}{2}) =\frac{max+min}{2}$

AmRit GhiMire 'Ranjit' (talk) 13:30, 19 September 2014 (UTC)

1. What is "LHS"?
2. Prove $\frac{max-min}{2} = \frac{max}{2} - \frac{min}{2}$174.3.125.23 (talk) 10:44, 20 September 2014 (UTC) — Preceding unsigned comment added by 174.3.125.23 (talk) 10:43, 20 September 2014 (UTC)
1. LHS = Left Hand Side
2. See distributive law, more or less.
Arthur Rubin (talk) 15:14, 20 September 2014 (UTC)

## AntiDerivative

As we have

$\frac{dy}{dx} = \frac{dy}{du} \cdot \frac{du}{dx}.$

Is there any simple way for calculating

$\int f(u)\ \mathrm{d}x$

AmRit GhiMire 'Ranjit' (talk) 12:08, 19 September 2014 (UTC)

You can try substitution but that only trades one integral for another. In general antiderivatives are inherently more difficult than derivatives. As another example, the product rule allows you to compute the derivative of a product in terms of the derivatives of the factors, but the closest thing for antiderivatives is integration by parts which, again, only trades one integral for another. --RDBury (talk) 16:28, 19 September 2014 (UTC)
I assume you mean f(u), where u=u(x). See Integration_by_substitution. Note that the way this is usually described is taking something of the form $\int f(x)\ \mathrm{d}x$ and re-writing it as $\int g(u)\ \mathrm{d}u$. SemanticMantis (talk) 17:07, 19 September 2014 (UTC)
Actually I need Antiderivative of : $\int f(u)\ \mathrm{d}x$ in such a form that can be used without human intelligence. Actually I need it for developing algorithm or computer program for antiderivative. AmRit GhiMire 'Ranjit' (talk) 00:31, 20 September 2014 (UTC)
If the derivative of u with respect to x can be written as a function of u,
$\int f(u)\ \mathrm{d}x = \int \frac {f(u)}{u'(x)}\ \mathrm{d}u$
Is that what you are looking for? — Arthur Rubin (talk) 15:12, 20 September 2014 (UTC)

## "loops" in common-core math

A friend of mine asked: "What are 'loops' in common core math--to replace the times-tables? 8x7=56, 8x8=64, etc." I've never heard it before. What is it? Bubba73 You talkin' to me? 17:46, 19 September 2014 (UTC)

The mathematical 'loops' I know about are all covered at the disambig page loop. But I don't think any of those are likely candidates. So I don't know the answer, but let me clarify something: Common_Core_State_Standards_Initiative is a set of standards that basically try to outline what each child should be expected to know at the end of each grade. It says absolutely nothing about pedagogical technique, specific course design, textbook choice, etc. My point is, I don't think the CCSSI documents say anything at all about 'loops' in math education -- but that doesn't mean that some text book doesn't use that word as a means of teaching multiplication... Unfortunately, many weird and shoddy textbook companies have rushed to slap CCSI stickers on their newest books. This often leads to people conflating problems with the book with problems about common core. I could probably give you a better answer with a little more context, such as the name of the book and the grade level of the student. SemanticMantis (talk) 19:45, 19 September 2014 (UTC)
One type of loop in grade school math is the iterative scheme used in manually carrying out long division, see How to Do Long Division. The other kind of loop I have seen is cycling through flash cards to learn math facts--perhaps it is an alternative to staring at a multiplication table? --Mark viking (talk) 20:56, 19 September 2014 (UTC)
I suppose this method of manually multiplying 8×7 could be called "loops":
Obviously one of the numbers has to be quite small for this method to be practical. StuRat (talk) 21:20, 19 September 2014 (UTC)
Flash card: loop and say 86.146.61.61 (talk) 22:12, 19 September 2014 (UTC)
Where did you see the term being used? I would guess like the others they are referring to looping as in programming so they are counting the number of times something is added to produce the product of a multiplication. Dmcq (talk) 22:13, 19 September 2014 (UTC)

My friend thinks StuRat is right. He got it: "Some woman at the antiques show telling about the Common Core math idiocy in FLA (and elsewhere)." Bubba73 You talkin' to me? 03:04, 20 September 2014 (UTC)

# September 22

## Why does this calculates the square root of two?

Why does this calculates the square root of two?

A[0] = Y[0]/X[0]

Where both X[0] and Y[0] are positive real numbers. They can even be equal to each other.

A[k] = Y[k]/X[k]

Y[k+1] = Y[k] + 2*X[k]

X[k+1] = Y[k] + X[k]

We have A[infinity] = Sqrt(2)

But I do not understand why this works for all possible positive real numbers. It looks so deceptively simple.

Also you can extend this to

A[k] = Y[k]/X[k]

Y[k+1] = Y[k] + N*X[k]

X[k+1] = Y[k] + X[k]

We have A[infinity] = Sqrt(N)

202.177.218.59 (talk) 06:05, 22 September 2014 (UTC)

The expression corresponds to:
The formula can be written as:
$\begin{pmatrix} Y[k+1] \\ X[k+1] \end{pmatrix}= \begin{pmatrix} 1 & N \\ 1 & 1 \end{pmatrix} \begin{pmatrix} Y[k] \\ X[k] \end{pmatrix}$
The matrix $\begin{pmatrix} 1 & N \\ 1 & 1 \end{pmatrix}$ has eigenvalues $1 \pm \sqrt N$ with eigenvectors $\begin{pmatrix} \pm \sqrt N \\ 1 \end{pmatrix}.$ The answer follows from elementary matrix manipulation.
You can also see that
$A[k+1] = \frac {A[k] + N}{A[k] + 1}$
and the answer follows from properties of Mobius transformations. — Arthur Rubin (talk) 08:03, 22 September 2014 (UTC)
If you write:
$B[k] = \frac {A[k]+\sqrt N}{A[k]-\sqrt N}$
Then you get:
$B[k+1]=\frac {1+\sqrt N}{1-\sqrt N} B[k] = -\frac {N+1+2\sqrt N}{N-1} B[k]$
Arthur Rubin (talk) 19:15, 22 September 2014 (UTC)

## Real Analysis: bounded set

The question is Let A=U{(1-1/n, 1+1/n) where n is in natural numbers}. I need to find the lower and upper bounds, supA, and infA. I can find those. The problem I am having is proving that A={0,2}.

I know there are 2 parts to this proof. First, showing that A=(0,2). Second, showing that A≠anything out side (0,2). I am not sure how to begin to show that. Please help. — Preceding unsigned comment added by Pinterc (talkcontribs) 23:15, 22 September 2014 (UTC)

# September 23

## Universal algebra based on free logic

My question is about whether universal algebra, e.g. varieties, is generally based on free logic (which I understand to allow an empty universe of discourse) rather than first-order logic (which I understand to axiomatically deny such). Examples where this makes a difference would include magmas, semigroups and quasigroups, in the sense of whether the empty structure in each case would or would not be permitted. These articles suggest that these are permitted, from which I deduce that first-order logic is not the basis for their construction; in fact, it seems to me that many theorems about these structures would become convoluted with exceptions ("exceptionally convoluted"?) if the empty structure was not to be permitted. —Quondum 01:48, 23 September 2014 (UTC)

## Trig sum

Does the sum

$\sum_{n=1}^{\infty} {\frac{\sin{\frac{1}{n}}}{n}} \approx 1.472828$

have a known closed form? 70.190.182.236 (talk) 07:03, 23 September 2014 (UTC)

http://www.wolframalpha.com/input/?i=sum_{n%3D1}^infinity+sin%281%2Fn%29%2Fn confirms the value 1.472828231868503, but gives no closed form. Bo Jacoby (talk) 09:42, 23 September 2014 (UTC).