Mostrando recursos 1 - 20 de 731

  1. Corrigendum to: "The Alternative Operad Is Not Koszul" by Askar Dzhumadil’daev and Pasha Zusmanovich

    Dzhumadil’daev, Askar; Zusmanovich, Pasha

  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
    This paper presents empirical evidence supporting Goldfeld’s conjecture on the average analytic rank of a family of quadratic twists of a fixed elliptic curve in the function field setting. In particular, we consider representatives of the four classes of nonisogenous elliptic curves over $\mathbb{F}_q(t) with $(q, 6) = 1$ possessing two places of multiplicative reduction and one place of additive reduction. The case of $q = 5$ provides the largest data set as well as the most convincing evidence that the average analytic rank converges to 1/2, which we also show is a lower bound following an argument of Kowalski. The data were generated via explicit computation of the $L$-function...

  7. Discriminants, Symmetrized Graph Monomials, and Sums of Squares

    Alexandersson, Per; Shapiro, Boris
    In 1878, motivated by the requirements of the invariant theory of binary forms, J. J. Sylvester constructed, for every graph with possible multiple edges but without loops, its symmetrized graph monomial, which is a polynomial in the vertex labels of the original graph. We pose the question for which graphs this polynomial is nonnegative or a sum of squares. This problem is motivated by a recent conjecture of F. Sottile and E. Mukhin on the discriminant of the derivative of a univariate polynomial and by an interesting example of P. and A. Lax of a graph with four edges whose symmetrized graph monomial is nonnegative but not a sum of...

  8. Twisted Alexander Polynomials of Hyperbolic Knots

    Dunfield, Nathan M.; Friedl, Stefan; Jackson, Nicholas
    We study a twisted Alexander polynomial naturally associated to a hyperbolic knot in an integer homology 3-sphere via a lift of the holonomy representation to $\mathrm{SL}(2, \mathbb{C})$. It is an unambiguous symmetric Laurent polynomial whose coefficients lie in the trace field of the knot. It contains information about genus, fibering, and chirality, and moreover, is powerful enough to sometimes detect mutation. ¶ We calculated this invariant numerically for all $313\, 209$ hyperbolic knots in $S^3$ with at most 15 crossings, and found that in all cases it gave a sharp bound on the genus of the knot and determined both fibering and chirality. ¶ We also study how such twisted Alexander...

  9. Conjectures and Experiments Concerning the Moments of $L (1/2, \chi_d)$

    Alderson, Matthew W.; Rubinstein, Michael O.
    We report on some extensive computations and experiments concerning the moments of quadratic Dirichlet $L$-functions at the critical point. We computed the values of $L (1/2, \chi_d)$ for $−5 × 10^{10} \lt d \lt 1.3 × 10^{10}$ in order to numerically test conjectures concerning the moments $\sum_{|d|\lt X} L (1/2, \chi_d)^k$. Specifically, we tested the full asymptotics for the moments conjectured by Conrey, Farmer, Keating, Rubinstein, and Snaith, as well as the conjectures of Diaconu, Goldfeld, Hoffstein, and Zhang concerning additional lower-order terms in the moments. We also describe the algorithms used for this large-scale computation.

  10. A Note on Beauville $p$-Groups

    Barker, Nathan; Boston, Nigel; Fairbairn, Ben
    We examine which $p$-groups of order $\le p^6$ are Beauville. We completely classify them for groups of order $\le p^4$. We also show that the proportion of 2-generated groups of order $p^5$ that are Beauville tends to 1 as $p$ tends to infinity; this is not true, however, for groups of order $p^6$. For each prime $p$ we determine the smallest nonabelian Beauville $p$-group.

  11. The $T$-Graph of a Multigraded Hilbert Scheme

    Hering, Milena; Maclagan, Diane
    The $T$-graph of a multigraded Hilbert scheme records the zeroand one-dimensional orbits of the $T = (K^*)^n$ action on the Hilbert scheme induced from the $T$-action on $\mathbb{A}^n$. It has vertices the $T$-fixed points, and edges the one-dimensional $T$-orbits. We give a combinatorial necessary condition for the existence of an edge between two vertices in this graph. For the Hilbert scheme of points in the plane, we give an explicit combinatorial description of the equations defining the scheme parameterizing all one-dimensional torus orbits whose closures contain two given monomial ideals. For this Hilbert scheme we show that the $T$-graph depends on the ground field, resolving a question of Altmann and Sturmfels.

  12. Zero Cells of the Siegel–Gottschling Fundamental Domain of Degree 2

    Hayata, Takahiro; Oda, Takayuki; Yatougo, Tomoki
    Let $\mathcal{F}_n$ be a fundamental domain of the Siegel upper half-space of degree $n$ with respect to the Siegel modular group $\operatorname{Sp}(n, \mathbb{Z})$. According to Siegel himself, $\mathcal{F}_n$ is determined by only finitely many polynomial inequalities. In case of degree $n = 2$, Gottschling determined the minimal set of inequalities. The boundary of $\mathcal{F}_2$ is of great concern in the literature not only from a homological point of view but also from the geometry of numbers. In this paper we compute the vertices of $\mathcal{F}_2$ under the condition that the defining ideal is zero-dimensional (“0-cells”). We also discuss an equivalence relation among 0-cells.

  13. The Secant Conjecture in the Real Schubert Calculus

    García-Puente, Luis D.; Hein, Nickolas; Hillar, Christopher; del Campo, Abraham Martín; Ruffo, James; Sottile, Frank; Teitler, Zach
    We formulate the secant conjecture, which is a generalization of the Shapiro conjecture for Grassmannians. It asserts that an intersection of Schubert varieties in a Grassmannian is transverse with all points real if the flags defining the Schubert varieties are secant along disjoint intervals of a rational normal curve. We present theoretical evidence for this conjecture as well as computational evidence obtained in over one terahertz-year of computing, and we discuss some of the phenomena we observed in our data.

  14. The Noncommutative A-Polynomial of $(−2, 3, n)$ Pretzel Knots

    Garoufalidis, Stavros; Koutschan, Christoph
    We study $q$-holonomic sequences that arise as the colored Jones polynomial of knots in 3-space. The minimal-order recurrence for such a sequence is called the (noncommutative) A-polynomial of a knot. Using the method of guessing, we obtain this polynomial explicitly for the $K_p = (−2, 3, 3 + 2p)$ pretzel knots for $p= −5, \dots , 5$. This is a particularly interesting family, since the pairs $(K_p,−K_{−p})$ are geometrically similar (in particular, scissors congruent) with similar character varieties. Our computation of the noncommutative $A$-polynomial complements the computation of the $A$-polynomial of the pretzel knots done by the first author and Mattman, supports the AJ conjecture for knots with reducible $A$-polynomial, and...

  15. The Riemann Zeta Function on Arithmetic Progressions

    Steuding, J örn; Wegert, Elias
    We prove asymptotic formulas for the first discrete moment of the Riemann zeta function on certain vertical arithmetic progressions inside the critical strip. The results give some heuristic arguments for a stochastic periodicity that we observed in the phase portrait of the zeta function.

  16. Ramanujan-like Series for $1/\pi^2$ and String Theory

    Almkvist, Gert; Guillera, Jesús
    Using the machinery from the theory of Calabi–Yau differential equations, we find formulas for $1/\pi^2$ of hypergeometric and nonhypergeometric types.

  17. Critical Values of Higher Derivatives of Twisted Elliptic $L$-Functions

    Fearnley, Jack; Kisilevsky, Hershy
    Let $L (E /\mathbb{Q} , s)$ be the $L$-function of an elliptic curve $E$ defined over the rational field $\mathbb{Q}$. Assuming the Birch–Swinnerton-Dyer conjectures, we examine special values of the $r$th derivatives, $L^{(r)}(E , 1, \chi)$, of twists by Dirichlet characters of $L (E /\mathbb{Q} , s)$ when $L (E , 1, \chi) = • • • = L^{(r−1)} (E , 1, \chi) = 0$.

  18. Sums of Three Squareful Numbers

    Browning, T. D.; Van Valckenborgh, K.
    We investigate the frequency of positive squareful numbers $x, y, z \le B$ for which $x + y = z$ and present a conjecture concerning its asymptotic behavior.

  19. Extended Torelli Map to the Igusa Blowup in Genus 6, 7, and 8

    Alexeev, Valery; Livingston, Ryan; Tenini, Joseph; Arap, Maxim; Hu, Xiaoyan; Huckaba, Lauren; McFaddin, Patrick; Musgrave, Stacy; Shin, Jaeho; Ulrich, Catherine
    It was conjectured in Yukihiko Namikawa, “On the Canonical Holomorphic Map from the Moduli Space of Stable Curves to the Igusa Monoidal Transform,” that the Torelli map $M_g \to A_g$ associating to a curve its Jacobian extends to a regular map from the Deligne–Mumford moduli space of stable curves $\bar{M}_g$ to the (normalization of the) Igusa blowup $\bar{A}^{\rm cent}_g$. A counterexample in genus $g = 9$ was found in Valery Alexeev and Adrian Brunyate, “Extending Torelli Map to Toroidal Compactifications of Siegel Space.” Here, we prove that the extended map is regular for all $g \le 8$, thus completely solving the problem in every genus.

  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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