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

We have defined the weight of the pair (〈S∣R〉,R) for a given presentation 〈S∣R〉 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 (〈S∣R〉,R) is n, then our algorithm constructs all the n-vertex crystallizations which yield (〈S∣R〉,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≥3 and m≥n≥k≥2, our generalized algorithm gives a \(2(2m+2n+2k-6+{\delta _{n}^{2}} + {\delta _{k}^{2}})\)-vertex crystallization of the closed connected orientable 3-manifold M〈m,n,k〉 having fundamental group \(\langle x_{1},x_{2},x_{3} \mid {x_{1}^{m}}={x_{2}^{n}}={x_{3}^{k}}=x_{1}x_{2}x_{3} \rangle \). These crystallizations are minimal and unique with respect to the given presentations. If ‘ n=2’ or ‘ k≥3 and m≥4’ then our crystallization of M〈m,n,k〉 is vertex-minimal for all the known cases.     

Searching for a Best Least Absolute Deviations Solution of an Overdetermined System of Linear Equations Motivated by Searching for a Best Least Absolute Deviations Hyperplane on the Basis of Given Data

We consider the problem of searching for a best LAD-solution of an overdetermined system of linear equations Xa=z, X∈?^m×n, m≥n, \(\mathbf{a}\in \mathbb{R}^{n}, \mathbf {z}\in\mathbb{R}^{m}\). This problem is equivalent to the problem of determining a best LAD-hyperplane x?a ^T x, x∈? ⁿ on the basis of given data \((\mathbf{x}_{i},z_{i}), \mathbf{x}_{i}= (x_{1}^{(i)},\ldots,x_{n}^{(i)})^{T}\in \mathbb{R}^{n}, z_{i}\in\mathbb{R}, i=1,\ldots,m\), whereby the minimizing functional is of the form

$F(\mathbf{a})=\|\mathbf{z}-\mathbf{Xa}\|_1=\sum_{i=1}^m|z_i-\mathbf {a}^T\mathbf{x}_i|.$

An iterative procedure is constructed as a sequence of weighted median problems, which gives the solution in finitely many steps. A criterion of optimality follows from the fact that the minimizing functional F is convex, and therefore the point a ^?∈? ⁿ is the point of a global minimum of the functional F if and only if 0∈?F(a ^?).

Motivation for the construction of the algorithm was found in a geometrically visible algorithm for determining a best LAD-plane (x,y)?αx+βy, passing through the origin of the coordinate system, on the basis of the data (x _i ,y _i ,z _i ),i=1,…,m.     

Kähler geometry of hyperbolic type on the manifold of nondegenerate <Emphasis Type="Italic">m</Emphasis>-pairs

A nondegenerate m-pair (A, Ξ) in an n-dimensional projective space ?P _n consists of an m-plane A and an (n ? m ? 1)-plane Ξ in ?P _n , which do not intersect. The set \(\mathfrak{N}_m^n \) of all nondegenerate m-pairs ?P _n is a 2(n ? m)(n ? m ? 1)-dimensional, real-complex manifold. The manifold \(\mathfrak{N}_m^n \) is the homogeneous space \(\mathfrak{N}_m^n = {{GL(n + 1,\mathbb{R})} \mathord{\left/ {\vphantom {{GL(n + 1,\mathbb{R})} {GL(m + 1,\mathbb{R})}}} \right. \kern-\nulldelimiterspace} {GL(m + 1,\mathbb{R})}} \times GL(n - m,\mathbb{R})\) equipped with an internal Kähler structure of hyperbolic type. Therefore, the manifold \(\mathfrak{N}_m^n \) is a hyperbolic analogue of the complex Grassmanian ?G _m,n = U(n+1)/U(m+1) × U(n?m). In particular, the manifold of 0-pairs \(\mathfrak{N}_m^n {{GL(n + 1,\mathbb{R})} \mathord{\left/ {\vphantom {{GL(n + 1,\mathbb{R})} {GL(1,\mathbb{R})}}} \right. \kern-\nulldelimiterspace} {GL(1,\mathbb{R})}} \times GL(n,\mathbb{R})\) is a hyperbolic analogue of the complex projective space ?P _n = U(n+1)/U(1) × U(n). Similarly to ?P _n , the manifold \(\mathfrak{N}_m^n \) is a Kähler manifold of constant nonzero holomorphic sectional curvature (relative to a hyperbolic metrics). In this sense, \(\mathfrak{N}_0^n \) is a hyperbolic spatial form. It was proved in [6] that the manifold of 0-pairs \(\mathfrak{N}_0^n \) is globally symplectomorphic to the total space T*?P _n of the cotangent bundle over the projective space ?P _n . A generalization of this result (see [7]) is as follows: the manifold of nondegenerate m-pairs \(\mathfrak{N}_m^n \) is globally symplectomorphic to the total space T*?G _m,n of the cotangent bundle over the Grassman manifold ?G _m,n of m-dimensional subspaces of the space ?P _n .In this paper, we study the canonical Kähler structure on \(\mathfrak{N}_m^n \). We describe two types of submanifolds in \(\mathfrak{N}_m^n \), which are natural hyperbolic spatial forms holomorphically isometric to manifolds of 0-pairs in ?P _m +1 and in ?P _n?m , respectively. We prove that for any point of the manifold \(\mathfrak{N}_m^n \), there exist a 2(n ? m)-parameter family of 2(m + 1)-dimensional hyperbolic spatial forms of first type and a 2(m + 1)-parameter family of 2(n ? m)-dimensional hyperbolic spatial forms of second type passing through this point. We also prove that natural hyperbolic spatial forms of first type on \(\mathfrak{N}_m^n \) are in bijective correspondence with points of the manifold \(\mathfrak{N}_{m + 1}^n \) and natural hyperbolic spatial forms of second type on \(\mathfrak{N}_m^n \) are in bijective correspondence with points of the manifolds \(\mathfrak{N}_{m + 1}^n \).     

Hyperbolic prime number theorem

We count the number S(x) of quadruples \( {\left( {x_{1} ,x_{2} ,x_{3} ,x_{4} } \right)} \in \mathbb{Z}^{4} \) for which

$ p = x^{2}_{1} + x^{2}_{2} + x^{2}_{3} + x^{2}_{4} \leqslant x $

is a prime number and satisfying the determinant condition: x ₁ x ₄???x ₂ x ₃?=?1. By means of the sieve, one shows easily the upper bound S(x)???x/log x. Under a hypothesis about prime numbers, which is stronger than the Bombieri–Vinogradov theorem but is weaker than the Elliott–Halberstam conjecture, we prove that this order is correct, that is S(x)???x/log x.

     

Meromorphic Jost functions and asymptotic expansions for Jacobi parameters

We show that the parameters a _n , b _n of a Jacobi matrix have a complete asymptotic expansion

$a_n^2 - 1 = \sum\limits_{k = 1}^{K(R)} {p_k (n)\mu _k^{ - 2n} + O(R^{ - 2n} ),} b_n = \sum\limits_{k = 1}^{K(R)} {p_k (n)\mu _k^{ - 2n + 1} + O(R^{ - 2n} )} $

, where 1 < |µ_j| < R for j ? K(R) and all R, if and only if the Jost function, u, written in terms of z (where E = z + z ^?1) is an entire meromorphic function. We relate the poles of u to the µ_j’s.

     

Polynomiality of some hook-length statistics

We prove a conjecture of Okada giving an exact formula for a certain statistic for hook-lengths of partitions:

$\frac{1}{n!} \sum_{\lambda \vdash n} f_{\lambda}^2 \sum_{u \in \lambda} \prod_{i=1}^{r}\bigl(h_u^2 - i^2\bigr) = \frac{1}{2(r+1)^2} \binom{2r}{r}\binom{2r+2}{ r+1} \prod_{j=0}^{r} (n-j),$

where f _λ is the number of standard Young tableaux of shape λ and h _u is the hook length of the square u of the Young diagram of λ. We also obtain other similar formulas.

     

Divided differences for functions of two variables for irregularly spaced arguments

Divided differences forf (x, y) for completely irregular spacing of points (x _i ,y _i ) are developed here by a natural generalization of Newton's scheme. Existing bivariate schemes either iterate the one-dimensional scheme, thus constraining (x _i ,y _i ) to be at corners of rectangles, or give polynomials Σa _jk x ^j y ^k having more coefficients than interpolation conditions. Here the generalizedn ^th divided difference is defined by (1)\(\left[ {01... n} \right] = \sum\limits_{i = 0}^n {A_i f\left( {x_i , y_i } \right)} \) where (2)\(\sum\limits_{i = 0}^n {A_i x_i^j , y_i^k = 0} \), and 1 for the last or (n+1)^th equation, for every (j, k) wherej+k=0, 1, 2,... in the usual ascending order. The gen. div. diff. [01...n] is symmetric in (x _i ,y _i ), unchanged under translation, 0 forf (x, y) an, ascending binary polynomial as far asn terms, degree-lowering with respect to (X, Y) whenf(x, y) is any polynomialP(X+x, Y+y), and satisfies the 3-term recurrence relation (3) [01...n]=λ{[1...n]?[0...n?1]}, where (4) λ= |1...n|·|01...n?1|/|01...n|·|1...n?1|, the |...i...| denoting determinants inx _i ^j y _i ^k . The generalization of Newton's div. diff. formula is (5)

$$\begin{gathered} f\left( {x, y} \right) = f\left( {x_0 , y_0 } \right) - \frac{{\left| {\alpha 0} \right|}}{{\left| 0 \right|}}\left[ {01} \right] + \frac{{\left| {\alpha 01} \right|}}{{\left| {01} \right|}}\left[ {012} \right] - \frac{{\left| {\alpha 012} \right|}}{{\left| {012} \right|}}\left[ {0123} \right] + \cdots + \hfill \\ + \left( { - 1} \right)^n \frac{{\left| {\alpha 01 \ldots n - 1} \right|}}{{\left| {01 \ldots n - 1} \right|}}\left[ {01 \ldots n} \right] + \left( { - 1} \right)^{n + 1} \frac{{\left| {\alpha 01 \ldots n} \right|}}{{\left| {01 \ldots n} \right|}}\left[ {01 \ldots n} \right], \hfill \\ \end{gathered} $$     

Congruences for bipartitions with odd parts distinct

Hirschhorn and Sellers studied arithmetic properties of the number of partitions with odd parts distinct. In another direction, Hammond and Lewis investigated arithmetic properties of the number of bipartitions. In this paper, we consider the number of bipartitions with odd parts distinct. Let this number be denoted by pod _?2(n). We obtain two Ramanujan-type identities for pod _?2(n), which imply that pod _?2(2n+1) is even and pod _?2(3n+2) is divisible by 3. Furthermore, we show that for any α≥1 and n≥0, \(\mathit{pod}_{-2}(3^{2\alpha+1}n+\frac{23\times 3^{2\alpha}-7}{8}) \) is a multiple of 3 and \(\mathit{pod}_{-2}(5^{\alpha+1}n+\frac{11\times5^{\alpha}+1}{4})\) is divisible by 5. We also find combinatorial interpretations for the two congruences modulo 2 and 3.     

Unit-Regularity of Regular Nilpotent Elements

Let a be a regular element of a ring R. If either K:=r _R (a) has the exchange property or every power of a is regular, then we prove that for every positive integer n there exist decompositions

$$R_{R} = K \oplus X_{n} \oplus Y_{n} = E_{n} \oplus X_{n} \oplus aY_{n}, $$

where \(Y_{n} \subseteq a^{n}R\) and E _n ?R/a R. As applications we get easier proofs of the results that a strongly π-regular ring has stable range one and also that a strongly π-regular element whose every power is regular is unit-regular.

     

Permutations of finite fields for check digit systems

Let q be a prime power. For a divisor n of q ? 1 we prove an asymptotic formula for the number of polynomials of the form

$f(X)=\frac{a-b}{n}\left(\sum_{j=1}^{n-1}X^{j(q-1)/n}\right)X+\frac{a+b(n-1)}{n}X\in\mathbb{F}_q[X]$

such that the five (not necessarily different) polynomials f(X), f(X)±X and f(f(X))±X are all permutation polynomials over \({\mathbb{F}_q}\) . Such polynomials can be used to define check digit systems that detect the most frequent errors: single errors, adjacent transpositions, jump transpositions, twin errors and jump twin errors.