User miroslav korbelar - MathOverflow most recent 30 from http://mathoverflow.net 2013-05-24T00:29:58Z http://mathoverflow.net/feeds/user/18376 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/105026/semirings-with-subtractive-primes/115756#115756 Answer by Miroslav Korbelar for Semirings with subtractive primes Miroslav Korbelar 2012-12-07T23:08:15Z 2012-12-08T16:08:11Z <p>I think the answer is no. Let $S={0}\cup[1,\infty)$ be the subsemiring of the (usual) reals. A non-zero ideal of $S$ is of the form ${0}\cup[a,\infty)$ where $a\geq 1$. Clearly, the only prime ideal of $S$ (according to your definition) is ${0}$ and it is subtractive. But no proper non-zero ideal of $S$ is subtractive.</p> <p>Correction: My argument is not right: actually every non-zero ideal of $S$ ie either of the form ${0}\cup[a,\infty)$ or ${0}\cup(a,\infty)$ where $a\geq 1$. Hence $P={0}\cup(1,\infty)$ is a non-zero prime ideal which is not subtractive. And in general a unitary semiring has to always have a maximal ideal (as the unitary ring does.) </p> http://mathoverflow.net/questions/91827/application-of-polynomials-with-non-negative-coefficients Application of polynomials with non-negative coefficients Miroslav Korbelar 2012-03-21T14:39:40Z 2012-12-03T00:19:02Z <p><strong>Question 1:</strong> Are there any deeper applications (in any field of mathematics) of polynomials (with possibly more than one variable) over the real numbers whose coefficients are non-negative? So far I have found only some vague mentions of control theory and maybe some approximations of functions, but don't know the details (e.g. www.math.ttu.edu/~barnard/poly.pdf). </p> <p><strong>Question 2:</strong> There is a similar notion, namely "positive polynomials", but they are defined as "polynomial functions that are non-negative". Does anybody know some connection between these two kinds of polynomials? Thanks.</p> <p>All comments and suggested answers here (thanks for them:)) have certainly something to do with the mentioned type of polynomials. To specify more what was my intention - a desired answer (to Question 1) should fullfil the following: "Is the application in you proposed area of such a kind that it forces an (independent) research of polynomials with non-negative coefficients on themselves?"</p> http://mathoverflow.net/questions/91827/application-of-polynomials-with-non-negative-coefficients/115040#115040 Answer by Miroslav Korbelar for Application of polynomials with non-negative coefficients Miroslav Korbelar 2012-12-01T00:33:03Z 2012-12-01T00:38:41Z <p>Thanks to all for your comments and suggestions. The original motivation for this question was a connection between polynomials with non-negative coefficients and commutative semirings. After some search we found some papers asking for determining the least degree of a polynomial with non-negative coefficients that is divisible by a given (general) polynomial. Suprisingly similar questions were investigated repeatedly and independetly but without any deeper motivation. We deceided hence to make an overview, improvements of some results and suggested a few conjectures. The result (made before the latest updates of this webpage) can be found here <a href="http://arxiv.org/abs/1210.6868" rel="nofollow">http://arxiv.org/abs/1210.6868</a>. (Suggestions and comments are welcome.)</p> http://mathoverflow.net/questions/89307/higher-categories-in-logic/89341#89341 Answer by Miroslav Korbelar for Higher categories in logic Miroslav Korbelar 2012-02-23T22:32:54Z 2012-02-23T22:32:54Z <p>There should be a lot of stuff concerning this in papers and books of Michael Makkai.</p> http://mathoverflow.net/questions/80609/algorithm-for-the-intersection-of-a-vector-subspace-with-a-cone-of-non-negative-v Algorithm for the intersection of a vector subspace with a cone of non-negative vectors Miroslav Korbelar 2011-11-10T17:04:25Z 2011-11-12T22:20:48Z <p>Hi, I would like to know whether there is some more effective way of how to compute an intersection of a vector subspace of $\mathbb{R}^{n}$ with a cone of vectors with non-negative entries than the following one:</p> <p>Let $W$ be a subspace of $\mathbb{R}^{n}$ and $(e_1,\dots,e_n)$ be the standard basis of $\mathbb{R}^{n}$. Find all $F\subseteq{1,\dots,n}$ such that $W_{F}:=W\cap \left\langle e_{i}|i\in F\right\rangle$ is 1-dimensional and intersects non-trivially the cone of vectors with non-negative entries (let $u_{F}$ be such non-trivial vector). Then our desired set is generated (as a cone) by all such $u_{F}$ 's for appropriate $F$ 's.</p> <p>Thanks.</p> http://mathoverflow.net/questions/77536/sbn-and-ibn-rings SBN and IBN rings Miroslav Korbelar 2011-10-08T14:30:31Z 2011-10-08T20:10:55Z <p>Hello, I can not figure out why a ring that is not IBN (invariant basis number) must be SBN (single basis number). More precisely: Let $R$ be a ring (with unit, generally non-commutative) such that the free $R$-module $R^n$ is isomorphic to another free $R$-module $R^m$, where $n, m$ are different natural numbers. How does this imply that $R$ is isomorphic to $R^2$ as an $R$-module? </p> http://mathoverflow.net/questions/77536/sbn-and-ibn-rings/77555#77555 Answer by Miroslav Korbelar for SBN and IBN rings Miroslav Korbelar 2011-10-08T20:10:55Z 2011-10-08T20:10:55Z <p>It seems that the implication does not hold. Thanks to Lukas Vokrinek for noticing this: According the example "Tom Leinster (mathoverflow.net/users/586), when is A isomorphic to A^3?" there is an abelian group $A$ such that $A$ is isomorphic to $A^3$ but not to $A^2$. Now, let $R=End(A)$. Then R is isomorphic to $R^3$ (as $R$-modules - that is easy), but not to $R^2$ (otherwise one could construct an isomorphism between $A$ and $A^2$ using the matrices with endomorphism entries, which would arise from the isomorphism $R$ and $R^2$). </p> http://mathoverflow.net/questions/91827/application-of-polynomials-with-non-negative-coefficients/91874#91874 Comment by Miroslav Korbelar Miroslav Korbelar 2012-03-22T14:49:59Z 2012-03-22T14:49:59Z The previous proof works if $p(1)\neq 1$ (i.e. $p(x)=x^n$). But one can just take $p(2)$ and $p(p(2))$ for instance. http://mathoverflow.net/questions/91827/application-of-polynomials-with-non-negative-coefficients/91863#91863 Comment by Miroslav Korbelar Miroslav Korbelar 2012-03-22T09:18:29Z 2012-03-22T09:18:29Z In fact I had some more systematic usage in mind, not only the fact that some polynomials are just of this kind. Or does one really use this property in the examples you mentioned? http://mathoverflow.net/questions/80609/algorithm-for-the-intersection-of-a-vector-subspace-with-a-cone-of-non-negative-v/80664#80664 Comment by Miroslav Korbelar Miroslav Korbelar 2011-11-11T11:39:25Z 2011-11-11T11:39:25Z I don't know still how the algorithm in the paper above works (it has 60 pages), but the intersection I am looking for (i.e. the cone) is generated by the &quot;corner&quot; edges which of course don't need to be orthogonal. But maybe I don't understand your comments right.. http://mathoverflow.net/questions/80609/algorithm-for-the-intersection-of-a-vector-subspace-with-a-cone-of-non-negative-v Comment by Miroslav Korbelar Miroslav Korbelar 2011-11-10T20:08:27Z 2011-11-10T20:08:27Z The algorithm above requires to test (generally) $2^n$ cases of $W_{F}$ (or at least till one doesn't meet the 1-dimensional ones). This seems to be quite much to me. That is why I have asked whether there is known some better implementation or a completely different approach. http://mathoverflow.net/questions/77536/sbn-and-ibn-rings Comment by Miroslav Korbelar Miroslav Korbelar 2011-10-08T20:34:35Z 2011-10-08T20:34:35Z Oh, thank you, just now I have noticed your comment. I discussed it with someone else, but the idea for counterexample (see below) comes obviously from you.