## Recursos de colección

#### Project Euclid (Hosted at Cornell University Library) (192.320 recursos)

Experimental Mathematics

2. #### Random Walks on Barycentric Subdivisions and the Strichartz Hexacarpet

Begue, Matthew; Kelleher, Daniel J.; Nelson, Aaron; Panzo, Hugo; Pellico, Ryan; Teplyaev, Alexander
We investigate simple random walks on graphs generated by repeated barycentric subdivisions of a triangle. We use these random walks to study the diffusion on the self-similar fractal known as the Strichartz hexacarpet, which is generated as the limit space of these graphs. We make this connection rigorous by establishing a graph isomorphism between the hexacarpet approximations and graphs produced by repeated barycentric subdivisions of the triangle. This includes a discussion of various numerical calculations performed on these graphs and their implications to the diffusion on the limiting space. In particular, we prove that equilateral barycentric subdivisions—a metric space generated by replacing the metric on each 2-simplex of the subdivided triangle with...

3. #### Universal Gröbner Bases of Colored Partition Identities

Bogart, Tristram; Hemmecke, Ray; Petrovíc, Sonja
Associated to any toric ideal are two special generating sets: the universal Gröbner basis and the Graver basis, which encode polyhedral and combinatorial properties of the ideal, or equivalently, its defining matrix. If the two sets coincide, then the complexity of the Graver bases of the higher Lawrence liftings of the toric matrices is bounded. ¶ While a general classification of all matrices for which both sets agree is far from known, we identify all such matrices within two families of nonunimodular matrices, namely, those defining rational normal scrolls and those encoding homogeneous primitive colored partition identities. This also allows us to show that higher Lawrence liftings of matrices with fixed...

4. #### Decomposition of Semigroup Algebras

Böhm, Janko; Eisenbud, David; Nitsche, Max J.
Let $A \supseteq B$ be cancellative abelian semigroups, and let $R$ be an integral domain. We show that the semigroup ring $R[B]$ can be decomposed, as an $R[A]$-module, into a direct sum of $R[A]$-submodules of the quotient ring of $R[A]$. In the case of a finite extension of positive affine semigroup rings, we obtain an algorithm computing the decomposition. When $R[A]$ is a polynomial ring over a field, we explain how to compute many ring-theoretic properties of $R[B]$ in terms of this decomposition. In particular, we obtain a fast algorithm to compute the Castelnuovo–Mumford regularity of homogeneous semigroup rings. As an application we confirm the Eisenbud–Goto conjecture in a...

5. #### An Empirical Approach to the Normality of π

Bailey, David H.; Borwein, Jonathan M.; Calude, Cristian S.; Dinneen, Michael J.; Dumitrescu, Monica; Yee, Alex
Using the results of several extremely large recent computations, we tested positively the normality of a prefix of roughly four trillion hexadecimal digits of $\pi$. This result was used by a Poisson process model of normality of $\pi$: in this model, it is extraordinarily unlikely that $\pi$ is not asymptotically normal base 16, given the normality of its initial segment.

6. #### Experimental Data for Goldfeld’s Conjecture over Function Fields

Baig, Salman; Hall, Chris

20. #### A Markov Chain on the Symmetric Group That Is Schubert Positive?

Lam, Thomas; Williams, Lauren
We study a multivariate Markov chain on the symmetric group with remarkable enumerative properties. We conjecture that the stationary distribution of this Markov chain can be expressed in terms of positive sums of Schubert polynomials. This Markov chain is a multivariate generalization of a Markov chain introduced by the first author in the study of random affine Weyl group elements.

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