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September 18[edit]

Mathematical Problems Immediately Relevant To Medicine[edit]

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)[reply]

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)[reply]

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)[reply]

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)[reply]

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)[reply]

Logic Puzzles[edit]

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)[reply]

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)[reply]

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)[reply]

(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

$(\forall x)(Food(x)=cake\lor Food(x)=pie\lor Food(x)=salad)$
$cake\neq pie$ , etc.
$(\exists x)(Food(x)=cake)$ , etc., or
$Food(cake)=cake$ , etc.

and, if you don't use $(\exists !3x)$
$(\forall x)(\forall y)(Food(x)=Food(y)\leftrightarrow x=y)$

For "<", you might have specific axioms:

$T0<T1$ , etc.
$T1\nless T0$ , etc.

You might even add:

$(\forall x)(x\nless x)$

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:

$(\forall x)(x=cake\lor x=pie\lor x=salad)$
$cake\neq pie$ , 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)[reply]

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)[reply]