Kronecker's Theorem Research Articles

Is Uniqueness Lost for Under-Sampled Continuous-Time Auto-Regressive Processes?

We consider the problem of sampling continuous-time auto-regressive processes on a uniform grid. We investigate whether a given sampled process originates from a single continuous-time model, and address this uniqueness problem by introducing an alternative description of poles in the complex plane. We then utilize Kronecker's approximation theorem and prove that the set of non-unique continuous-time AR(2) models has Lebesgue measure zero in this plane. This is a key aspect in current estimation algorithms that use sampled data, as it allows one to remove the sampling rate constraint that is imposed currently.

An equivalent of Kronecker's theorem for powers of an algebraic number and structure of linear recurrences of fixed length

After defining a notion of $\epsilon$-density, we provide for any real algebraic number $\alpha$ an estimate of the smallest $\epsilon$ such that for each $m>1$ the set of vectors of the form $(t,t\alpha,...,t\alpha^{m-1})$ for $t\in\R$ is $\epsilon$-dense modulo 1, in terms of the multiplicative Mahler measure $M(A(x))$ of the minimal integral polynomial $A(x)$ of $\alpha$, and independently of $m$. In particular, we show that if $\alpha$ has degree $d$ it is possible to take $\epsilon = 2^{[d/2]}/M(A(x))$. On the other hand using asymptotic estimates for Toeplitz determinants we show that for sufficiently large $m$ we cannot have $\epsilon$-density if $\epsilon$ is a fixed number strictly smaller than $1/M(A(x))$. As a byproduct of the proof we obtain a result of independent interest about the structure of the $\Z$-module of integral linear recurrences of fixed length determined by a non-monic polynomial.

Strongly central sets and sets of polynomial returns mod 1

Central sets in N were introduced by Furstenberg and are known to have substantial combinatorial structure. For example, any central set con- tains arbitrarily long arithmetic progressions, all finite sums of distinct terms of an infinite sequence, and solutions to all partition regular systems of ho- mogeneous linear equations. We introduce here the notions of strongly central and very strongly central, which is as the names suggest are strictly stronger than the notion of central. They are also strictly stronger than syndetic, which in the case of N means that gaps are bounded. Given x 2 R, let w(x) = x b x + 1 c. Kronecker's Theorem says that if 1, 1, 2,..., v are linearly independent over Q and U is a nonempty open subset of ( 1 , 1 ) v , then {x 2 N : (w( 1x),...,w( vx)) 2 U} is nonempty and Weyl showed that this set has positive density. We show here that if 0 is in the closure of U, then this set is strongly central. More generally, let P1,P2,...,Pv be real polynomials with zero constant term. We show that {x 2 N : (w(P1(x)),...,w(Pv(x))) 2 U} is nonempty for every open U with 0 2 c'U if and only if it is very strongly central for every such U and we show that these conclusions hold if and only if any nontrivial rational linear combination of P1,P2,...,Pv has at least one irrational coecient.

Stripes on Penrose tilings

We study the existence of one-dimensional quasicrystal structures on the vertex set ΛP of a Penrose tiling, in an arbitrary direction . If , ζ = e2πi/5, then is a discrete family of lines that has a one-dimensional quasicrystal structure. Conversely, if w ≠ 0 and , is a dense subset of . We also have a weak analog of Kronecker's approximation theorem.

On the distribution of zeros of a sequence of entire functions approaching the Riemann zeta function

In this paper we study the distribution of zeros of each entire function of the sequence { G n ( z ) ≡ 1 + 2 z + ⋯ + n z : n ⩾ 2 } , which approaches the Riemann zeta function for Re z < − 1 , and is closely related to the solutions of the functional equations f ( z ) + f ( 2 z ) + ⋯ + f ( n z ) = 0 . We determine the density of the zeros of G n ( z ) on the critical strip where they are situated by using almost-periodic functions techniques. Furthermore, by using a theorem of Kronecker, we also establish a formula for the number of zeros of G n ( z ) inside certain rectangles in the critical strip.

DIRICHLET TRUNCATIONS OF THE RIEMANN ZETA FUNCTION IN THE CRITICAL STRIP POSSESS ZEROS NEAR EVERY VERTICAL LINE

We study the zeros of the finite truncations of the alternating Dirichlet series expansion of the Riemann zeta function in the critical strip. We do this with an (admittedly highly) ambitious goal in mind. Namely, that this series converges to the zeta function (up to a trivial term) in the critical strip and our hope is that if we can obtain good estimates for the zeros of these approximations it may be possible to generalize some of the results to zeta itself. This paper is a first step towards this goal. Our results show that these finite approximations have zeros near every vertical line (so no vertical strip in this region is zero-free). Furthermore, we give upper bounds for the imaginary parts of the zeros (the real parts are pinned). The bounds are numerically very large. Our tools are: the inverse mapping theorem (for a perturbative argument), the prime number theorem (for counting primes), elementary geometry (for constructing zeros of a related series), and a quantitative version of Kronecker's theorem to go from one series to another.

A simple constructive proof of Kronecker's Density Theorem

Douglas Bridges lehrt seit 1999 als Professor fur Reine Mathematik an der Universitat von Canterbury in Christchurch, Neuseeland. Neben seinem Wirken in der konstruktiven Mathematik hat er unter anderem uber mathematische Fragen der Nationalokonomie publiziert. Er ist Mitherausgeber von Zeitschriften wie dem ” Mathematical Logic Quarterly“. Zusammen mit Errett Bishop schrieb er den Grundlehrenband ” Constructive Analysis“.

Overlapping unit cells in 3D quasicrystal structure

A 3-dimensional quasiperiodic lattice, with overlapping unit cells and periodic in one direction, is constructed using grid and projection methods pioneered by de Bruijn. Each unit cell consists of 26 points, of which 22 are the vertices of a convex polytope P, and 4 are interior points also shared with other neighboring unit cells. Using Kronecker's theorem the frequencies of all possible types of overlapping are found.

Two variations of a theorem of Kronecker

We present two variations of Kronecker's classical result that every nonzero algebraic integer that lies with its conjugates in the closed unit disc is a root of unity. The first is an analogue for algebraic nonintegers, while the second is a several variable version of the result, valid over any field.

The set of common fixed points of an n-parameter continuous semigroup of mappings

In this paper, using Kronecker's theorem, we discuss the set of common fixed points of an n-parameter continuous semigroup { T ( p ) : p ∈ R + n } of mappings. We also discuss some convergence theorems to a common fixed point of an n-parameter nonexpansive semigroup { T ( p ) : p ∈ R + n } without using the Bochner integral.

Positive extensions, Fejér–Riesz factorization and autoregressive filters in two variables

In this paper we treat the two-variable positive extension problem for trigonometric polynomials where the extension is required to be the reciprocal of the absolute value squared of a stable polynomial. This problem may also be interpreted as an autoregressive filter design problem for bivariate stochastic processes. We show that the existence of a solution is equivalent to solving a finite positive definite matrix completion problem where the completion is required to satisfy an additional low rank condition. As a corollary of the main result a necessary and sufficient condition for the existence of a spectral FejerRiesz factorization of a strictly positive two-variable trigonometric polynomial is given in terms of the Fourier coefficients of its reciprocal. Tools in the proofs include a specific two-variable Kronecker theorem based on certain elements from algebraic geometry, as well as a two-variable Christoffel-Darboux like formula. The key ingredient is a matrix valued polynomial that appears in a parametrized version of the Schur-Cohn test for stability. The results also have consequences in the theory of two-variable orthogonal polynomials where a spectral matching result is obtained, as well as in the study of inverse formulas for doubly-indexed Toeplitz matrices. Finally, numerical results are presented for both the autoregressive filter problem and the factorization problem.

The independence of characters on non-abelian groups

We show that there are characters of compact, connected, non-abelian groups that approximate random choices of signs. The work was motivated by Kronecker's theorem on the independence of exponential functions and has applications to thin sets.

Extremal Values of L -Functions Associated with Newforms in the Half-Plane of Absolute Convergence

We prove estimates for extremal values of L-functions associated with newforms f in the half-plane of absolute convergence of their Dirichlet series expansion. The proof is based on an effective version of Kronecker's approximation theorem and estimates for the Fourier coefficients of the newform f.

Hidden constructions in abstract algebra (1): integral dependance

We give an elementary method, hidden in a theorem of abstract algebra, for constructing integral dependance relations. We apply this method in order to give a constructive proof of a theorem of Kronecker.

The Best Quantitative Kronecker's Theorem

The Best Quantitative Kronecker's Theorem

A Dynamical Interpretation of the Global Canonical Height on an Elliptic Curve

There is a well-understood connection between polynomials and certain simple algebraic dynamical systems. In this connection, the Mahler measure corresponds to the topological entropy, Kronecker's Theorem relates ergodicity to positivity of entropy, approximants to the Mahler measure are related to growth rates of periodic points, and Lehmer's problem is related to the existence of algebraic models for Bernoulli shifts. Thereare similar relationships for higher-dimensional algebraic dynamical systems. We review this connection, and indicate a possible analogous connection between the global canonical height attached to points on elliptic curves and a possible ‘ell iptic’ dynamical system.

Numbers and shapes revisited: more problems for young mathematicians

1. The Fibonacci sequence, generalized Fibonacci sequences, and related topics 2. Patterns of dots and partitions of integers 3. Patterns, related to rational numbers: periodic decimal fractions, repunits, and visible lattice points 4. Reflected light rays and real numbers: a theorem of Kronecker 5. The Chinese remainder theorem and invisible lattice points 6. Two famous inequalities and some related problems 7. Mysteries of the their dimension: on cubes, pyramids, and spheres 8. A Glimpse of higher-dimensional spaces: hypercubes, lattice paths, and related number patterns 9. Do it with groups 10. From puzzles to research topics: selected problems of combinatorics

Kronecker type theorems, normality and continuity of the multivariate Padé operator

For univariate functions the Kronecker theorem, stating the equivalence between the existence of an infinite block in the table of Pade approximants and the approximated function \(f\) being rational, is well-known. In [Lubi88] Lubinsky proved that if \(f\) is not rational, then its Pade table is normal almost everywhere: for an at most countable set of points the Taylor series expansion of \(f\) is such that it generates a non-normal Pade table. This implies that the Pade operator is an almost always continuous operator because it is continuous when computing a normal Pade approximant [Wuyt81]. In this paper we generalize the above results to the case of multivariate Pade approximation. We distinguish between two different approaches for the definition of multivariate Pade approximants: the general order one introduced in [Levi76, CuVe84] and the so-called homogeneous one discussed in [Cuyt84].

View-Obstruction Problems and Kronecker's Theorem

View-Obstruction Problems and Kronecker's Theorem

A Kronecker Theorem for Higher Order Hankel Forms

A Kronecker Theorem for Higher Order Hankel Forms