On a Lower Bound for ∥(4/3)k∥\|(4/3)^k\|

We prove, that $$\biggl\|\biggl(\frac43\biggr)^k\biggr\| >\biggl(\frac49\biggr)^k\qquad\text{for}\quad k\ge6,$$ where $\|\cdot\|$ is a distance to the nearest prime

Lamination exact relations and their stability under homogenization

Relations between components of the effective tensors of composites that hold regardless of composite's microstructure are called exact relations. Relations between components of the effective tensors of all laminates are called lamination exact relations. The question of existence of sets of effective tensors of composites that are stable under lamination, but not homogenization was settled by Milton with an example in 3D elasticity. In this paper we discuss an analogous question for exact relations, where in a wide variety of physical contexts it is known (a posteriori) that all lamination exact relations are stable under homogenization. In this paper we consider 2D polycrystalline multi-field response materials and give an example of an exact relation that is stable under lamination, but not homogenization. We also shed some light on the surprising absence of such examples in most other physical contexts (including 3D polycrystalline multi-field response materials). The methods of our analysis are algebraic and lead to an explicit description (up to orthogonal conjugation equivalence) of all representations of formally real Jordan algebras as symmetric $n\times n$ matrices. For each representation we examine the validity of the 4-chain relation|a 4th degree polynomial identity, playing an important role in the theory of special Jordan algebras

Upper bound on list-decoding radius of binary codes

Consider the problem of packing Hamming balls of a given relative radius subject to the constraint that they cover any point of the ambient Hamming space with multiplicity at most $L$. For odd $L\ge 3$ an asymptotic upper bound on the rate of any such packing is proven. Resulting bound improves the best known bound (due to Blinovsky'1986) for rates below a certain threshold. Method is a superposition of the linear-programming idea of Ashikhmin, Barg and Litsyn (that was used previously to improve the estimates of Blinovsky for $L=2$) and a Ramsey-theoretic technique of Blinovsky. As an application it is shown that for all odd $L$ the slope of the rate-radius tradeoff is zero at zero rate.Comment: IEEE Trans. Inform. Theory, accepte

Difference Sturm--Liouville problems in the imaginary direction

We consider difference operators in $L^2$ on $\R$ of the form $$L f(s)=p(s)f(s+i)+q(s) f(s)+r(s) f(s-i) ,$$ where $i$ is the imaginary unit. The domain of definiteness are functions holomorphic in a strip with some conditions of decreasing at infinity. Problems of such type with discrete spectra are well known (Meixner--Pollaszek, continuous Hahn, continuous dual Hahn, and Wilson hypergeometric orthogonal polynomials). We write explicit spectral decompositions for several operators $L$ with continuous spectra. We also discuss analogs of 'boundary conditions' for such operators.Comment: 27p