User roy maclean - MathOverflow

The unification of Mathematics via Topos Theory

2010-06-23T13:03:40Z

When the paper The unification of Mathematics via Topos Theory by Olivia Caramello, says "one can generate a huge number of new results in any mathematical field without any creative effort." is this an exaggeration, and if not is this a new idea or has it always been thought that topos theory could enable automatic generation of theorems.

Answer by Roy Maclean for analogue of a set with n binary operations

2012-10-23T01:03:40Z

The study of sets with an arbitrary number of operations is called universal algebra.

Also universal algebra isn't limited to binary operations but studies operations of any arity: nullary, unary, binary, ternary, ... , n-ary.

(Universal algebra does however typically restrict itself to axioms that are defined by equations which means fields are excluded from this way of studying algebra.)

Beware however that operations that are ostensibly independent might not be. See for example the earlier mathoverflow question: Can we unify addition and multiplication into one binary operation? To what extent can we find universal binary operations?

In a ring the distributive law connects the addition and multiplication operations, so that they cannot in a sense be independent, but there is nothing to stop a structure of n operations from being consistent if none of the axioms connect any of the operations to each other. Like Gerhard Paseman comments about, it depends what you mean by independent.

Questions along similar lines to the linked question about universal operations could be asked about any given universal-algebraic variety - given a variety in which every operation is connected to the other operations by axioms, then to what extent can the number of operations be reduced and still define the same class of structures. For questions about reducing the number of operations or axioms see https://www.cs.unm.edu/~mccune/projects/gtsax/

P.S. I'll mention here as a curiosity the topic of n,m-operations. That is operations that not only have a domain-arity but also a co-domain arity. e.g. $*:G \times G \times G \rightarrow G \times G$. Concepts such as associativity can be generalized to n,m-operations but n,m-operations haven't been studied much in universal algebra.

Answer by Roy Maclean for Euclidean inside Hyperbolic

2012-09-01T07:47:35Z

This is a response to Will Sawin's comment "What algebraic structure would you place on $\mathbb{H}^2$?"

On $\mathbb{R}^n$ there is a vector space structure where

1) The metric is given by $d(\mathbf{u},\mathbf{v})=\|\mathbf{u}-\mathbf{v}\|$

and

2) given three points $U,V,W$ the angle $U\hat WV$ satisfies $\cos \theta = \frac{(-W+U)\cdot(-W+V)}{\|-W+U\|\|-W+V\|}$

Analogously, for some models of $\mathbb{H}^n$ with points identified with a subset of $\mathbb{R}^n$ there is a vector-like structure but with a noncommutative, nonassociative binary operation $\oplus$ where

1) The metric is given by $d(\mathbf{u},\mathbf{v})=\|\mathbf{u} \ominus \mathbf{v} \| $

and

2) given three points $U,V,W$ the angle $U\hat WV$ satisfies $\cos \theta = \frac{(\ominus W\oplus U)\cdot(\ominus W\oplus V)}{\|\ominus W\oplus U\|\|\ominus W\oplus V\|}$

$\| \|$ and $\cdot $ are the vector norm and dot product inherited from $\mathbb{R}^n$.

$\ominus a$ denotes the left inverse of a.

$a\ominus b$ denotes $a\oplus (\ominus b)$.

Note the use of the trig function "cos" even though this hyperbolic geometry.

For the Beltrami-Klein model the binary operation $\oplus$ is given by the formula for relativistic velocity-addition.

Cardinality of the permutations of an infinite set

2010-06-11T05:36:18Z

If you have an infinite set X of cardinality k, then what is the cardinality of Sym(X) - the group of permutations of X ?

Elliptic curve group law, Sum of intersection points

2011-11-01T17:08:24Z

If a plane curve of degree n intersects an elliptic curve in 3n points, then do those points always sum to zero when added using the group law on the points of an elliptic curve ?

rational points on degree 4 curve

2011-11-01T18:53:08Z

If you take a conic through 5 rational points on a quartic curve, then will at least one of the remaining 3 points also be rational ?

Consolidation: Aftermathematics of fads

2010-05-24T10:41:59Z

From Frank Quinn's THE NATURE OF CONTEMPORARY CORE MATHEMATICS: "Mathematics has occasional fads, but for the most part it is a long-term solitary activity. In consequence the community lacks the customs evolved in physics to deal with the aftermathematics of fads. If mathematicians desert an area no one comes in afterwards to clean up. Lack of large-scale cleanup mechanisms makes mathematical areas vulnerable to quality control problems. There are a number of once-hot areas that did not get cleaned up and will be hard to unravel when the developers are not available. Funding agencies might watch for this and sponsor physics-style review and consolidation activity when it happens."

Can you give examples of such once-hot areas in need of consolidation ?

Time scale calculus vs Lebesgue–Stieltjes calculus

2010-10-27T14:54:49Z

About the same time, it seems, as I asked this question, a new post appeared on the wikipedia discussion page for Time scale calculus which suggests the Time scale derivative (aka Hilger derivative aka delta-derivative) is the same as the Radon-Nikodym derivative of the Lebesgue–Stieltjes integral.

Do you agree that the time scale delta-derivative is a Lebesgue–Stieltjes derivative with the appropriate weight function? (The notion of Lebesgue–Stieltjes derivative being similarly defined to the Riemann-Stieltjes derivative of my previous question.)

Answer by Roy Maclean for Time scale calculus vs Lebesgue–Stieltjes calculus

2011-02-16T10:10:02Z

A few days ago a paper was uploaded to the arXiv which answers this question:

"On the connection between the Hilger and Radon--Nikodym derivatives" by Jonathan Eckhardt and Gerald Teschl

http://arxiv.org/abs/1102.2511

Answer by Roy Maclean for What are some examples of colorful language in serious mathematics papers?

2011-01-13T07:44:02Z

A new book on sieve methods is bizzarely called Opera de Cribro with chapter subtitles in an operatic theme.

MathSciNet Reviews everyone should read

2010-12-30T16:57:26Z

Along the same lines as the question A single paper everyone should read?, which MathSciNet Reviews, Zentralblatt MATH reviews, etc should everyone read ?

Answer by Roy Maclean for Proofs that require fundamentally new ways of thinking

2010-12-09T16:11:25Z

Would the "quantum method" fit the bill here ?

"Quantum Proofs for Classical Theorems" Andrew Drucker, Ronald de Wolf

"Erdös and the Quantum Method" Richard Lipton

MathSciNet vs Google Scholar

2010-06-21T02:27:42Z

What are the pros and cons of the MathSciNet database vs Google Scholar?

I don't have access to Mathscinet so this question is out of curiosity, and also this question where MathSciNet is used to find paper counts. I reckon Google Scholar will almost always be the more comprehensive of the two with higher paper counts for any particular author and includes papers that don't use the MSC codes as well papers from other subjects that may be of mathematical interest but aren't included in mathscinet.

One definitely annoying thing about Google is that in the advanced search it doesn't have a mathematics only category, but lumps it in with computer science and engineering so you sometimes need to add something like -"computer science" -"engineering" mathematics to your search term to filter out unwanted results, which isn't ideal.

Answer by Roy Maclean for More open problems

2010-12-04T21:31:16Z

The book Research problems in discrete geometry by Peter Brass, W. O. J. Moser, János Pach, is a very large collection which describes what is known about each problem with a large reference list of papers for each problem.

Also Google Books search: "open problems in" OR "unsolved problems in" OR "research problems in" mathematics

and Google Scholar search: "open problems in" OR "unsolved problems in" OR "research problems in" mathematics

Multi-inequality

2010-12-03T00:34:31Z

Have things like $x\begin{pmatrix}< \ >\ < \end{pmatrix}y$ been used in mathematics ?

Where x and y are vectors, matrices or other component-based quantities in which the components do not all relate to each other in the same way, and the above expression means x₁ < y₁, x₂ > y₂, x₃ < y₃

e.g. the complex numbers $2+3i$ and $1+4i$ are related by $2+3i\begin{pmatrix}> \ < \end{pmatrix}1+4i$

The point of the notation $x\begin{pmatrix}< \ > \end{pmatrix}y$ is to see what happens if both sides are multiplied by $c$ where $c\begin{pmatrix}< \ < \end{pmatrix}0$ or $c\begin{pmatrix}> \ < \end{pmatrix}0$ etc in analogy with the fact that multiplying both sides of an inequality by a negative number changes the inequality symbol. Even if there aren't rules describing what happens in the case of complex numbers, perhaps there are algebras where there are such rules.

If $z$ is a complex number we could have $z\begin{pmatrix}> \ >\end{pmatrix}0$ means z is in the north-east quadrant, $z\begin{pmatrix}> \ <\end{pmatrix}0$ for $z$ in the south-east quadrant etc, or just write $z > 0$ or $z < 0$ if each component has the same inequality type.

Something like the expression $f(x)\begin{pmatrix}< \ =\ na \end{pmatrix}g(x)$ would mean the second components are equal, but the third components do not have a consistent relation.

If complex numbers are put in inequalities in place of real numbers then which known inequalities would involve inequality types that aren't just $ \begin{pmatrix}na \ na \end{pmatrix}$ ? e.g. take something like Schur's inequality and replace the real numbers in the expression with complex numbers, then what would replace the inequality symbol? would it be (> <) or (< >) etc or would there be no consistent inequality type? i.e. $ \begin{pmatrix}na \ na \end{pmatrix}$. Notwithstanding the fact that this particular example of an inequality does not hold for all real numbers.

Or to rephrase the question along the lines of http://mathoverflow.net/questions/48045/why-are-matrices-ubiquitous-but-hypermatrices-rare :

Why are inequalities ubiquitous but multi-inequalities not so ubiquitous?

Thinking about the earlier question http://mathoverflow.net/questions/24740/non-real-constants some real constants arise from inequalities, so there should be non-real constants arising from non-real inequalities: e.g. inequalities in ordered-rings or multi-inequalities in other things.

"Riemann-Stieltjes derivative" ?

2010-10-27T12:27:06Z

Can you define a "derivative" operator such that its antiderivative F(x) of f(x) can be used in the sense of F(b)-F(a) to calculate the Riemann-Stieltjes integral of f(x)?

Perhaps it would be related to the $\Delta$-derivative in time scale calculus.

This question was motivated by an edit to that wikipedia article which said that the ideas of unifying sums and integrals go back to the idea of the Riemann-Stieltjes integral. Now I'm not sure it's correct to say the RS-integral is a pre-cursor of time-scale calculus as the starting point of time-scale calculus is the derivative and the unification of difference and differential equations, but integrals on other time-scales such as the q-integral in quantum calculus can be related to the RS integral (page 7 of paper by Abreau), so maybe definite integrals on all time scales can be written in the form of RS-integrals for some suitable choice of step function depending on the time-scale.

Edit: About the same time, it seems, as I asked this question, a new post appeared on the wikipedia discussion page for "Time scale calculus" which suggests the Hilger derivative (delta-derivative) is the same as the Radon-Nikodym derivative of the Lebesgue–Stieltjes integral.

Answer by Roy Maclean for Suggestions for good notation

2010-10-22T08:13:24Z

In the notation of Time scale calculus, the ordinary calculus derivative df/dt and the forward difference operator $\Delta f $ are both written as $f^\Delta$. Indefinite sums and indefinite integrals are both written as $\int{f(t)\Delta t}$ and called indefinite integrals. The context would say $\mathbb{T}=\mathbb{Z}, \mathbb{T}=\mathbb{R}$ or other $\mathbb{T}\subset\mathbb{R}$.

Answer by Roy Maclean for What is the indefinite sum of tan(x)?

2010-10-20T21:11:43Z

Consider T(x) in T(x+1)-T(x)=tan(x) to be defined only on the integers>=0 and write it as the recurrence relation :

T(n+1)-T(n)=tan(n) and set the initial value at T(0)=0,

and then without needing a closed form expression for T(n) in terms of n,

but just plotting the values of T(n+1)=T(n)+tan(n)

we can see what the function looks like when restricted to the integers:

The top-left image shows the function plotted from n=..750

The left column of images is the plot, the middle images join the dots to get a look at the shape, and the right column shows were the image is zoomed to get the images on the next row.

The function T(n) appears to be almost periodic with period about 355.5

T(n) maximum is about 3.5, T min is about -425.

Answer by Roy Maclean for Modern algebraic geometry vs. classical algebraic geometry

2010-07-13T11:52:30Z

Depends what you mean by "modern".

"Numerical algebraic geometry" using homotopy methods is modern but concrete.

"Introduction to numerical algebraic geometry", A Sommese, J Verschelde, C Wampler http://citeseerx.ist.psu.edu/viewdoc/download?doi=10.1.1.84.8283&rep=rep1&type=pdf

"The numerical solution of systems of polynomials arising in engineering and science", Andrew John Sommese, Charles W. Wampler, World Scientific, 2005

Points at twice the distance from (-1, 0) that they are from (1, 0) in hyperbolic geometry

2010-07-02T18:01:38Z

In answer to the question Demystifying complex numbers, Charles Matthews suggests "finding the points at twice the distance from (-1, 0) that they are from (1, 0)." as a motivation for complex numbers.

Suppose you want to find these points in hyperbolic geometry instead of euclidean geometry. If this can be done with vectors or complex numbers in R^2, then I reckon it could be done with gyrovectors or gyro-complex numbers in the hyperbolic plane, but if you don't use gyro-algebra then how would you find (or describe) the points at twice the distance from (-1, 0) that they are from (1, 0)?

(Defining coordinates in hyperbolic geometry can be done with gyro-algebra, but without it just assume the origin is an arbitrary point, and that (-1,0) is a point of distance 1 from the origin and (1,0) is a point of distance 1 from the origin in the opposite direction.)

Undiscovered for a long time before it is realised it is the same concept developed under different names.

2010-06-28T12:01:11Z

Mathematics has been described as the giving of the same name to different things, but sometimes different names are given to the same thing.

Can you give examples of concepts where researchers in different areas have used the same concept under different names for a long time before it is discovered they are talking about the same thing ?

Wavelets might be an example.

EDIT: In response to Willie Wong's comment. I was thinking of the book "The World According to Wavelets. The Story of a Mathematical Technique in the Making" by Barbara Burke Hubbard.

Here's some quotes from page 26: "I have found at least 15 distinct roots of the theory, some going back to the 1930s", Meyer said. "David Marr, who worked on artificial vision and robotics at MIT, had similar ideas. The physics community was intuitively aware of wavelets dating back to a paper on renormalization by Kenneth Wilson, in 1971". Littlewood and Paley developed wavelet-like techniques. Alberto Calderon developed a continuous version of wavelets. Yet other researchers developed wavelets-which they called "self-similar Gabor functions"-to model the visual system. Jean Morlet developed wavelets as a tool for oil processing.

From page 40: "Multiresolution approximation and wavelets", Mallat. The paper made it clear that work that existed in many different guises and under many different names -- were at heart all the same.

Books you would like to see retranslated.

2010-06-22T04:04:15Z

As a follow on to this question, what books would you like to see retranslated or rewritten as the original translation wasn't very good, or can you give examples of books that have been translated more than once into the same language.

Answer by Roy Maclean for Books you would like to see translated into English.

2010-06-22T03:58:06Z

Vorlesungen über Differenzenrechnung by Niels Erik Nörlund

(Citations)

Statements reliant on conjectures

2010-06-20T07:13:16Z

There are lots of statements that have been conditionally proved on the assumption that the Riemann Hypothesis is true.

What other conjectures have a large number of proven consequences ?

Answer by Roy Maclean for Statements reliant on conjectures

2010-06-20T07:39:08Z

Clicking on Toolbox-What Link's Here in the wikipedia article Conditional proof brings up Schinzel's hypothesis H which the article says is used to prove conditional results in diophantine geometry.

Answer by Roy Maclean for How should the Math Subject Classification (MSC) be revised or improved?

2010-06-17T03:18:58Z

The top-level categories in the IMU list include one for Lie theory.

association schemes, infinite schemes, semi-schemes, quasi-schemes

2010-06-11T06:00:30Z

Some questions about possibly nonsensical ideas:

1) Can you come up with a definition of an infinite association scheme ?

2) Would infinite association schemes relate to infinite groups the way association schemes relate to finite groups ? (see http://www.ams.org/bull/2006-43-02/S0273-0979-05-01077-3/S0273-0979-05-01077-3.pdf)

3) Can you define objects that relate to semigroups and quasigroups as schemes relate to groups ?

4) What combinatorial, statistical or other properties would these infinite schemes, semi-schemes and quasi-schemes have ?

Answer by Roy Maclean for Quick proofs of hard theorems

2010-05-30T14:03:08Z

The incompressibility method based on Kolmogorov complexity is desribed in "Kolmogorov Incompressibility Method in Formal Proofs A Critical Survey", V Megalooikonomou - 1997, as often being more elegant, intuitive, simpler and shorter than counting arguments, or the probabilistic method, in areas such as lower bounds, average case complexity, random graphs or pumping lemmas in formal language theory.

Answer by Roy Maclean for Consolidation: Aftermathematics of fads

2010-05-25T20:36:21Z

In email, Frank Quinn mentions that Surgery theory from the 1970s and 80s has a mostly primary literature aimed at other experts, a lack of textbooks, and now has few new people working on it.

To me, this seems like a similar problem to properly documenting a computer program as you go along so that others (and your future self) can understand it, otherwise coming back to it can require the same or greater effort to go through it, as was required to create it in the first place, but the temptation is to skimp on that, and just plough ahead.

Cyclic order relation in Zn

2010-05-21T17:23:53Z

The ring Z_n:={0,1,..,n-1} under addition and multiplication modulo n. Suppose a,b,c,x $\in$ Z_n are nonzero and the cyclic order R(a,b,c) holds, then under what conditions does R(ax,bx,cx) hold ?

Comment by Roy Maclean

2011-09-02T15:40:51Z

Recent paper on spectral hypergraph theory, for eigenvalues of adjacency hypermatrices: "Spectra of Hypergraphs", Joshua Cooper, Aaron Dutle, arxiv.org/abs/1106.4856

Comment by Roy Maclean

2011-01-12T09:57:38Z

Taking Limits is a linear operation. en.wikipedia.org/wiki/… , so composing limits with other linear operators is also linear. e.g. say T(z) = lim z->0 (f(z)/2)

Comment by Roy Maclean

2011-01-12T09:21:23Z

The "time scale calculus" operators on complex numbers: AN INTRODUCTION TO COMPLEX FUNCTIONS ON PRODUCTS OF TWO TIME SCALES, web.mst.edu/~bohner/papers/aitcfopotts.pdf

Comment by Roy Maclean

2011-01-12T08:58:55Z

Finite Differences and antidifferences are linear.

Comment by Roy Maclean

2011-01-12T08:45:26Z

Complex numbers can be represented as matrices en.wikipedia.org/wiki/… , so functions of a complex variable are functions of a matrix variable en.wikipedia.org/wiki/Matrix_function If you have the expression f(A)+g(A) for functions f and g of a matrix A, then any linear operator T on matrices will do, although the result won't necessarily be a matrix of the same form as a complex number representation.

Comment by Roy Maclean

2010-12-31T20:12:10Z

en.wikipedia.org/wiki/Future_of_mathematics

Comment by Roy Maclean

2010-12-30T18:28:29Z

@optima: I don't see how I have a subscription, unless my ISP has subscribed to it maybe ?

Comment by Roy Maclean

2010-12-30T18:24:32Z

@optima: Hmm. That's weird. I got that message in the past, but now I can access the whole thing. For example the review you link to says: "The author suggests that the use of Cantorian spacetime ... with a point-to-point notion of time, can effect a reconciliation between quantum mechanics and gravity." review by J.S. Joel

Comment by Roy Maclean

2010-12-30T17:41:34Z

I asked the question because I noticed that MathSciNet is now freely available, but other freely available reviews would also answer the question. As Michael Greinecker pointed out in an earlier question, Zentralblatt reviews are also freely available if you know exactly which item you want rather than searching.

Comment by Roy Maclean

2010-12-11T10:50:01Z

Look for the terms: dendriform algebra, mould calculus, pil, pal, til, tal, loma, lomi, roma, romi, zig, zag, ari, gari and alien calculus, in Ecalle's work on multizeta arithmetic.

Comment by Roy Maclean

2010-12-05T10:39:51Z

@Robin Chapman: That book is about solving inclusions, not about inclusions that always hold.

Comment by Roy Maclean

2010-12-05T10:09:29Z

I'm not looking for static inclusions like $\mathbb{Q}\subset\mathbb{R}$, but variable inclusions like $f(X)\subseteq g(X)$. There are whole books written about inequalities, why not books about inclusions ?

Comment by Roy Maclean

2010-12-05T08:07:58Z

Something like the Cauchy–Schwarz inequality appears in many places. So are there any particularly important inclusions that are used frequently ?

Comment by Roy Maclean

2010-12-04T21:04:44Z

Hmm, yes the hypervextex construction doesn't work very well. Need to change the question to be about hypergraphs made of ordinary vertices connected by simplices.

Comment by Roy Maclean

2010-12-04T14:31:21Z

The edges in these hypergraphs only join two hypervertices. The hypervertices are straight lines in R^n, although now you mention it I suppose that for the secondary question about manifolds the lines have to be curves homeomorphic to the real line, so the main question is really two questions: one with straight lines and the same question again allowing curves. "not line segments" means proper infinite-length lines, not finite-length segments.