Irreducible Integer Research Articles

A note on surgery obstructions and hyperbolic integer homology spheres

Auckly gave two examples of irreducible integer homology spheres (one toroidal and one hyperbolic) which are not surgery on a knot in the three-sphere. Using Heegaard Floer homology, the authors and Karakurt provided infinitely many small Seifert fibered examples. In this note, we extend those results to give infinitely many hyperbolic examples, as well as infinitely many examples with arbitrary JSJ decomposition.

Non-L–space integral homology 3–spheres with no nice orderings

This paper gives infinitely many examples of non L-space irreducible integer homology 3-spheres whose fundamental groups do not have nontrivial $\widetilde{PSL_2(\mathbb{R})}$ representations.

Surgery obstructions and Heegaard Floer homology

Using Taubes' periodic ends theorem, Auckly gave examples of toroidal and hyperbolic irreducible integer homology spheres which are not surgery on a knot in the three-sphere. We give an obstruction to a homology sphere being surgery on a knot coming from Heegaard Floer homology. This is used to construct infinitely many small Seifert fibered examples.

On the Number of Irreducible Points in Polyhedra

An integer point in a polyhedron is called irreducible iff it is not the midpoint of two other integer points in the polyhedron. We prove that the number of irreducible integer points in $n$-dimensional polytope with radius $k$ given by a system of $m$ linear inequalities is at most $O(m^{\lfloor\frac{n}{2}\rfloor}\log^{n-1} k)$ if $n$ is fixed. Using this result we prove the hypothesis asserting that the teaching dimension in the class of threshold functions of $k$-valued logic in $n$ variables is $\Theta(\log^{n-2} k)$ for any fixed $n\ge 2$.

The Farrell–Jones conjecture for some nearly crystallographic groups

In this paper, we prove the K-theoretical and L-theoretical Farrell-Jones Conjecture with coefficients in an additive category for nearly crystallographic groups of the form $\mathbb{Q}^n \rtimes \mathbb{Z}$, where $\mathbb{Z}$ acts on $\mathbb{Q}^n$ as an irreducible integer matrix with determinant $d$, $|d |>1$.

Finding Degree-16 Monic Irreducible Integer Polynomials of Minimal Trace by Optimization Methods

In this paper, we give a list of monic integer polynomials of smallest possible trace, irreducible, of degree 16, and having all roots real and positive. Moreover, we think that this list may be complete.

Fractal property of generalized M-set with rational number exponent

Dynamic systems described by fc(z)=z2+c is called Mandelbrot set (M-set), which is important for fractal and chaos theories due to its simple expression and complex structure. fc(z)=zk+c is called generalized M set (k–M set). This paper proposes a new theory to compute the higher and lower bounds of generalized M set while exponent k is rational, and proves relevant properties, such as that generalized M set could cover whole complex number plane when k<1, and that boundary of generalized M set ranges from complex number plane to circle with radius 1 when k ranges from 1 to infinite large. This paper explores fractal characteristics of generalized M set, such as that the boundary of k–M set is determined by k, when k=p/q, where p and q are irreducible integers, (GCD(p,q)=1, k>1), and that k–M set can be divided into |p–q| isomorphic parts.

On Lagrangian formulations for arbitrary bosonic HS fields on Minkowski backgrounds

We review the details of unconstrained Lagrangian formulations for Bose particles propagated on an arbitrary dimensional flat space-time and described by the unitary irreducible integer higher-spin representations of the Poincare group subject to Young tableaux Y(s 1, ..., s k ) with k rows. The procedure is based on the construction of scalar auxiliary oscillator realizations for the symplectic sp(2k) algebra which encodes the second-class operator constraints subsystem in the HS symmetry algebra. Application of an universal BRST approach reproduces gauge-invariant Lagrangians with reducible gauge symmetries describing the free dynamics of both massless and massive bosonic fields of any spin with appropriate number of auxiliary fields.

General Lagrangian formulation for higher spin fields with arbitrary index symmetry. I. Bosonic fields

We construct a Lagrangian description of irreducible integer higher spin representations of the Poincaré-group with an arbitrary Young tableaux having k rows, on a basis of the universal BRST approach. Starting with a description of bosonic mixed-symmetry higher spin fields in a flat space of any dimension in terms of an auxiliary Fock space associated with special Poincaré module, we realize a conversion of the initial operator constraint system (constructed with respect to the relations extracting irreducible Poincaré-group representations) into a first-class constraint system. For this purpose, we find, for the first time, auxiliary representations of the constraint subalgebra, to be isomorphic due to Howe duality to sp(2k) algebra, and containing the subsystem of second-class constraints in terms of new oscillator variables. We propose a universal procedure of constructing unconstrained gauge-invariant Lagrangians with reducible gauge symmetries describing the dynamics of both massless and massive bosonic fields of any spin. It is shown that the space of BRST cohomologies with a vanishing ghost number is determined only by the constraints corresponding to an irreducible Poincaré-group representation. As examples of the general procedure, we formulate the method of Lagrangian construction for bosonic fields subject to arbitrary Young tableaux having 3 rows and derive the gauge-invariant Lagrangian for new model of massless rank-4 tensor field with spin (2,1,1) and second stage reducible gauge symmetries.

Ω-estimates related to irreducible algebraic integers

We study large values of the remainder term EK (x) in the asymptotic formula for the number of irreducible integers in an algebraic number field K. We show that EK (x) = Ω± (√(x)(log x)) for certain positive constant BK, improving in that way the previously best known estimate EK (x) = Ω± (x(1/2)-ε) for every ε > 0, due to A. Perelli and the present author. Assuming that no entire L-function from the Selberg class vanishes on the vertical line σ = 1, we show that EK (x) = Ω± (√(x)(log log x)D (K)-1(log x)-1), supporting a conjecture raised recently by the author. In particular, it follows that the last omega estimate is a consequence of the Selberg Orthonormality Conjecture (© 2010 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim)

On the distribution of irreducible algebraic integers

We study large values of the remainder term in the asymptotic formula for the number of irreducible integers in an algebraic number field K. In the case when the class number h of K is larger than 1, under certain technical condition on multiplicities of non-trivial zeros of Hecke L-functions, we detect oscillations larger than what one could expect on the basis of the classical Littlewood’s omega estimate for the remainder term in the prime number formula. In some cases the main result is unconditional. It is proved that this is always the case when h = 2.

The values of Mahler measures

The set ℳ* of numbers which occur as Mahler measures of integer polynomials and the subset ℳ of Mahler measures of algebraic numbers (that is, of irreducible integer polynomials) are investigated. It is proved that every number α of degree d in ℳ* is the Mahler measure of a separable integer polynomial of degree at most with all its roots lying in the Galois closure F of ℚ(α), and every unit in ℳ is the Mahler measure of a unit in F of degree at most over ℚ This is used to show that some numbers considered earlier by Boyd are not Mahler measures. The set of numbers which occur as Mahler measures of both reciprocal and nonreciprocal algebraic numbers is also investigated. In particular, all cubic units in this set are described and it is shown that the smallest Pisot number is not the measure of a reciprocal number.

Extended formulations for Gomory Corner polyhedra

We present several types of extended formulations for integer programs, based on irreducible integer solutions to Gomory's group relaxations. We present algorithmic schemes based on an iterative reformulation technique using these extended formulations. We give computational results for benchmark problems, which illustrate the primal and dual effect of the reformulation.

Upper bounds for the coefficients of irreducible integer polynomials in several variables

Upper bounds for the coefficients of irreducible integer polynomials in several variables

Tilted irreducible representations of the permutation group

A fast algorithm to compute irreducible integer representations of the symmetric group is described. The representation is called tilted because the identity is not represented by a unit matrix, but a matrix β satisfying a reduced characteristic equation of the form ( β − I) k = 0. A distinctive feature of the approach is that the non-zero matrix elements are restricted to ±1. A so called natural representation is obtained by multiplying each representation matrix by β −1. Alternatively the representation property of the matrices is mantained by inserting the matrix β −1 between two representation matrices.

Universal jumps of conductance at the metal-insulator transition in one dimension

The transition for a one-dimensional collection of spinless interacting charged particles from metal to a Mott insulator phase is shown to exhibit two kinds of universal jump of conductance: when the filling factor is p/q, where p and q are irreducible integers and q gives the order of backward scattering responsible for localization, there is a jump ${\mathit{e}}^{2}$/${\mathit{q}}^{2}$h along the metal-insulator phase boundary, but at special points with particle-hole symmetry the universal jump of conductance is exactly twice as large. Even far inside the metal phase, where the Mott insulator does not occur, its existence is manifested by a singularity in the conductance at the corresponding rational filling. Most of the results hold also for the case of Mott localization in a quasiperiodic potential. The only essential difference here is that insulator phases are characterized by filling factors which are, in general, irrational numbers: for a quasiperiodic potential with a ratio of fundamental incommensurate length \ensuremath{\sigma} the insulator phase is characterized by an irrational filling factor (${\mathit{m}}_{0}$+${\mathit{m}}_{1}$\ensuremath{\sigma})/q, where ${\mathit{m}}_{0,1}$ are integers.

Number of representations and maximum dimensions for S(N)

A fast algorithm to compute irreducible integer representations of the symmetric group is described. The representation is called tilted because the identity is not represented by a unit matrix, but a matrix β satisfying a reduced characteristic equation of the form (β − I)k = 0. A distinctive feature of the approach is that the non-zero matrix elements are restricted to ±1. A so called natural representation is obtained by multiplying each representation matrix by β−1. Alternatively the representation property of the matrices is mantained by inserting the matrix β−1 between two representation matrices.