• Volume 126, Issue 4

      October 2016,   pages  461-653

    • Generalized $r$-Lah numbers


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      In this paper, we consider a two-parameter polynomial generalization, denoted by ${\mathcal G}_{a,b}(n, k; r)$, of the $r$-Lah numbers which reduces to these recently introduced numbers when $a = b = 1$. We present several identities for ${\mathcal G}_{a,b}(n, k; r)$ that generalize earlier identities given for the $r$-Lah and $r$-Stirling numbers. We also provide combinatorial proofs of some earlier identities involving the $r$-Lah numbers by defining appropriate sign-changing involutions. Generalizing these arguments yields orthogonality-type relations that are satisfied by ${\mathcal G}_{a,b}(n, k; r)$.

    • Certain variants of multipermutohedron ideals


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      Multipermutohedron ideals have rich combinatorial properties. An explicit combinatorial formula for the multigraded Betti numbers of a multipermutohedron ideal and their Alexander duals are known. Also, the dimension of the Artinian quotient of an Alexander dual of a multipermutohedron ideal is the number of generalized parking functions. In this paper, monomial ideals which are certain variants of multipermutohedron ideals are studied. Multigraded Betti numbers of these variant monomial ideals and their Alexander duals are obtained. Further, many interesting combinatorial properties of multipermutohedron ideals are extended to these variant monomial ideals.

    • On conjectures of Minkowski and Woods for $n = 9$


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      Let $\mathbb{R}^n$ be the $n$-dimensional Euclidean space with $O$ as the origin. Let $\wedge$ be a lattice of determinant 1 such that there is a sphere $\mid X \mid \lt R$ which contains no point of $\wedge$ other than $O$ and has $n$ linearly independent points of $\wedge$ on its boundary. A well known conjecture in the geometry of numbers asserts that any closed sphere in $\mathbb{R}^n$ of radius $\sqrt{n/4}$ contains a point of $\wedge$. This is known to be true for $n\leq 8$. Here we prove a more general conjecture of Woods for $n = 9$ from which this conjecture follows in $\mathbb{R}^9$. Together with a result of McMullen (J. Amer. Math. Soc. 18 (2005) 711–734), the long standing conjecture of Minkowski follows for $n = 9$.

    • Combinatorics of tenth-order mock theta functions


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      In this paper, we provide the combinatorial interpretations of two tenth order mock theta functions which appeared in some identities given in Ramanujan’s lost notebook ((1988) Narosa Publishing House, New Delhi).

    • Unitary representations of the fundamental group of orbifolds


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      Let $X$ be a smooth complex projective variety of dimension $n$ and $\mathcal{L}$ an ample line bundle on it. There is a well known bijective correspondence between the isomorphism classes of polystable vector bundles $E$ on $X$ with $c_{1}(E) = 0 = c_{2}(E) \cdot c_{1} \mathcal (L)^{n−2}$ and the equivalence classes of unitary representations of $\pi_{1}(X)$. We show that this bijective correspondence extends to smooth orbifolds.

    • Eigenfunction statistics for Anderson model with Hölder continuous single site potential


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      We consider random Schrödinger operators on ${\mathcal l}^{2} ({\mathbb Z}^d)$ with $\alpha$-Hölder continuous $(0 \lt \alpha \leq 1)$ single site distribution. In localized regime, we study the distribution of eigenfunctions in space and energy simultaneously. In a certain scaling limit, we prove limit points are Poisson.

    • Elliptic ovoids and their rosettes in a classical generalized quadrangle of even order


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      Let $\mathcal{Q}_0$ be the classical generalized quadrangle of order $q = 2^{n}(n \geq 2)$ arising from a non-degenerate quadratic form in a 5-dimensional vector space defined over a finite field of order $q$. We consider the rank two geometry $\mathcal{X}$ having as points all the elliptic ovoids of $\mathcal{Q}^0$ and as lines the maximal pencils of elliptic ovoids of $\mathcal{Q}_0$ pairwise tangent at the same point. We first prove that $\mathcal{X}$ is isomorphic to a 2-fold quotient of the affine generalized quadrangle $\mathcal{Q} \backslash \mathcal{Q}_0$, where $\mathcal{Q}$ is the classical $(q, q^2)$- generalized quadrangle admitting $\mathcal{Q}_0$ as a hyperplane. Further, we classify the cliques in the collinearity graph $\Gamma$ of $\mathcal{X}$. We prove that any maximal clique in $\Gamma$ is either a line of $\mathcal{X}$ or it consists of 6 or 4 points of $\mathcal{X}$ not contained in any line of $\mathcal{X}$, accordingly as $n$ is odd or even.We count the number of cliques of each type and show that those cliques which are not contained in lines of $\mathcal{X}$ arise as subgeometries of $\mathcal{Q}$ defined over $\mathbb{F}_2$

    • Axioms of spheres in lightlike geometry of submanifolds


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      We prove that if an indefinite Kaehler manifold $\bar{M}$ with lightlike submanifolds satisfies the axioms of holomorphic 2$r$-spheres, axioms of holomorphic 2$r$-planes, axioms of transversal $r$-spheres and axioms of transversal $r$-planes, then it is an indefinite complex space form.

    • Strictly convex functions on complete Finsler manifolds


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      The purpose of the present paper is to investigate the influence of strictly convex functions on the metric structures of complete Finsler manifolds. More precisely we discuss the properties of the group of isometries and the exponential maps on a complete Finsler manifold admitting strictly convex functions.

    • An algorithmic approach to construct crystallizations of 3-manifolds from presentations of fundamental groups


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      We have defined the weight of the pair $(\langle S \mid R \rangle,\,R)$ for a given presentation $\langle S \mid R \rangle$ of a group, where the number of generators is equal to the number of relations. We present an algorithm to construct crystallizations of 3-manifolds whose fundamental group has a presentation with two generators and two relations. If the weight of $(\langle S \mid R \rangle,\,R)$ is $n$, then our algorithm constructs all the $n$-vertex crystallizations which yield $(\langle S\mid R\rangle,\,R)$. As an application, we have constructed some new crystallizations of 3-manifolds. We have generalized our algorithm for presentations with three generators and a certain class of relations. For $m \geq 3$ and $m \geq n \geq k \geq 2$, our generalized algorithm gives a $2(2m+2n+2k−6+ \delta^2_n+ \delta^2_k)$-vertex crystallization of the closed connected orientable 3-manifold $M\langle m,n,k\rangle$ having fundamental group $\langle x1,x2,x3 \mid x^m_1 = x^n_2 = x^k_3 = x1x2x3 \rangle$. These crystallizations are minimal and unique with respect to the given presentations. If ‘$n = 2$’ or ‘$k \geq 3$ and $m \geq 4$’ then our crystallization of $M\langle m,n,k \rangle$ is vertex-minimal for all the known cases.

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