Kronecker's Theorem Research Articles

Method for estimating the number of zeros of the spatially one-dimensional Pauli–Jordan–Dirac function on spatial intervals using the Kronecker theorem

Method for estimating the number of zeros of the spatially one-dimensional Pauli–Jordan–Dirac function on spatial intervals using the Kronecker theorem

Modular orbits on the representation spaces of compact abelian Lie groups

Let S be a closed surface of genus g greater than zero. In the present paper, we study the topological-dynamical action of the mapping class group on the \mathbb T^n -character variety giving necessary and sufficient conditions for \mathrm{Mod}(S) -orbits to be dense. As an application, such a characterisation provides a dynamical proof of the Kronecker's theorem concerning inhomogeneous Diophantine approximation.

Computing zeta functions of large polynomial systems over finite fields

We improve the algorithms of Lauder-Wan [11] and Harvey [8] to compute the zeta function of a system of m polynomial equations in n variables, over the q element finite field Fq, for large m. The dependence on m in the original algorithms was exponential in m. Our main result is a reduction of the dependence on m from exponential to polynomial. As an application, we speed up a doubly exponential algorithm from a recent software verification paper [3] (on universal equivalence of programs over finite fields) to singly exponential time. One key new ingredient is an effective, finite field version of the classical Kronecker theorem which (set-theoretically) reduces the number of defining equations for a polynomial system over Fq when q is suitably large.

On construction of almost periodic sequences and applications to some discrete population models

In this work, we develop a novel approximation strategy for building almost periodic sequences in the theory of almost periodic functions. Here, we create a different perspective for the argument of Dirichlet in the theory of numbers and design an integer approximation strategy in this regard. The idea behind the strategy comes from Kronecker's theorem and it is proven that for given an almost periodic function, it is possible to design its corresponding almost periodic sequence. Moreover, we provide two population models in both continuous and discrete cases where almost periodic sequence solutions are designed under suitable circumstances.

Heights of polynomials over lemniscates

We consider a family of heights defined by the $L_p$ norms of polynomials with respect to the equilibrium measure of a lemniscate for $0 \le p \le \infty$, where $p=0$ corresponds to the geometric mean (the generalized Mahler measure) and $p=\infty$ corresponds to the standard supremum norm. This special choice of the measure allows to find an explicit form for the geometric mean of a polynomial, and estimate it via certain resultant. For lemniscates satisfying appropriate hypotheses, we establish explicit polynomials of lowest height, and also show their uniqueness. We discuss relations between the standard results on the Mahler measure and their analogues for lemniscates that include generalizations of Kronecker's theorem on algebraic integers in the unit disk, as well as of Lehmer's conjecture.

On construction of sampling patterns for preserving observability/controllability of linear sampled-data systems

ABSTRACT In this paper, we design a new nonequidistant sampling pattern such that the property of controllability and observability of the linear continuous time systems is preserved during the procedure of the sample and zero-order hold operation. The idea behind the strategy comes from Kronecker's theorem and it is proven that new construction is more effective by requiring less number of sampling times which guarantees controllability and observability of sampled-data system in two dimensional space. Moreover, we provide one application in the concept of controllability of the pendulum in two dimensional space and obtain corresponding digital controls which make the corresponding sampled-data system controllable. Finally, the research is extended to different type of sampling patterns which are compared with the constructed sampling pattern separately.

On Nudel’man’s Problem and Indefinite Interpolation in the Generalized Schur and Nevanlinna Classes

This work is a revised and corrected version of the authors' joint paper with T. Constantinescu (Trans Am Math Soc 355:813-836, 2003). Some of the theorems from the original paper that are withdrawn are recast as new open problems for indefinite interpolation. Partial results are obtained by other methods, including Kronecker's theorem for Hankel operators.

On homogeneous and inhomogeneous Diophantine approximation over the fields of formal power series

We prove over fields of power series the analogues of several Diophantine approximation results obtained over the field of real numbers. In particular we establish the power series analogue of Kronecker's theorem for matrices, together with a quantitative form of it, which can also be seen as a transference inequality between uniform approximation and inhomogeneous approximation. Special attention is devoted to the one dimensional case. Namely, we give a necessary and sufficient condition on an irrational power series $\alpha$ which ensures that, for some positive $\eps$, the set $$ \liminf_{Q \in \mathbb{F}_q[z], \,\, \deg Q \to \infty} \| Q \| \cdot |\langle Q \alpha - \theta \rangle| \geq \eps $$ has full Hausdorff dimension.

On an effective variation of Kronecker’s approximation theorem avoiding algebraic sets

Let Λ ⊂ R n \Lambda \subset \mathbb R^n be an algebraic lattice coming from a projective module over the ring of integers of a number field K K . Let Z ⊂ R n \mathcal Z \subset \mathbb R^n be the zero locus of a finite collection of polynomials such that Λ ⊈ Z \Lambda \nsubseteq \mathcal Z or a finite union of proper full-rank sublattices of Λ \Lambda . Let K 1 K_1 be the number field generated over K K by coordinates of vectors in Λ \Lambda , and let L 1 , … , L t L_1,\dots ,L_t be linear forms in n n variables with algebraic coefficients satisfying an appropriate linear independence condition over K 1 K_1 . For each ε > 0 \varepsilon > 0 and a ∈ R n \boldsymbol a \in \mathbb R^n , we prove the existence of a vector x ∈ Λ ∖ Z \boldsymbol x \in \Lambda \setminus \mathcal Z of explicitly bounded sup-norm such that ‖ L i ( x ) − a i ‖ > ε \begin{equation*} \| L_i(\boldsymbol x) - a_i \| > \varepsilon \end{equation*} for each 1 ≤ i ≤ t 1 \leq i \leq t , where ‖ ‖ \|\ \| stands for the distance to the nearest integer. The bound on sup-norm of x \boldsymbol x depends on ε \varepsilon , as well as on Λ \Lambda , K K , Z \mathcal Z , and heights of linear forms. This presents a generalization of Kronecker’s approximation theorem, establishing an effective result on density of the image of Λ ∖ Z \Lambda \setminus \mathcal Z under the linear forms L 1 , … , L t L_1,\dots ,L_t in the t t -torus R t / Z t \mathbb R^t/\mathbb Z^t .

Polynomial–Exponential Decomposition From Moments

We analyze the decomposition problem of multivariate polynomial–exponential functions from their truncated series and present new algorithms to compute their decomposition. Using the duality between polynomials and formal power series, we first show how the elements in the dual of an Artinian algebra correspond to polynomial–exponential functions. They are also the solutions of systems of partial differential equations with constant coefficients. We relate their representation to the inverse system of the isolated points of the characteristic variety. Using the properties of Hankel operators, we establish a correspondence between polynomial–exponential series and Artinian Gorenstein algebras. We generalize Kronecker theorem to the multivariate case, by showing that the symbol of a Hankel operator of finite rank is a polynomial–exponential series and by connecting the rank of the Hankel operator with the decomposition of the symbol. A generalization of Prony’s approach to multivariate decomposition problems is presented, exploiting eigenvector methods for solving polynomial equations. We show how to compute the frequencies and weights of a minimal polynomial–exponential decomposition, using the first coefficients of the series. A key ingredient of the approach is the flat extension criteria, which leads to a multivariate generalization of a rank condition for a Caratheodory–Fejer decomposition of multivariate Hankel matrices. A new algorithm is given to compute a basis of the Artinian Gorenstein algebra, based on a Gram–Schmidt orthogonalization process and to decompose polynomial–exponential series. A general framework for the applications of this approach is described and illustrated in different problems. We provide Kronecker-type theorems for convolution operators, showing that a convolution operator (or a cross-correlation operator) is of finite rank, if and only if, its symbol is a polynomial–exponential function, and we relate its rank to the decomposition of its symbol. We also present Kronecker-type theorems for the reconstruction of measures as weighted sums of Dirac measures from moments and for the decomposition of polynomial–exponential functions from values. Finally, we describe an application of this method for the sparse interpolation of polylog functions from values.

On Helson matrices: moment problems, non-negativity, boundedness, and finite rank

We study Helson matrices (also known as multiplicative Hankel matrices), i.e. infinite matrices of the form $M(\alpha) = \{\alpha(nm)\}_{n,m=1}^\infty$, where $\alpha$ is a sequence of complex numbers. Helson matrices are considered as linear operators on $\ell^2(\mathbb{N})$. By interpreting Helson matrices as Hankel matrices in countably many variables we use the theory of multivariate moment problems to show that $M(\alpha)$ is non-negative if and only if $\alpha$ is the moment sequence of a measure $\mu$ on $\mathbb{R}^\infty$, assuming that $\alpha$ does not grow too fast. We then characterize the non-negative bounded Helson matrices $M(\alpha)$ as those where the corresponding moment measures $\mu$ are Carleson measures for the Hardy space of countably many variables. Finally, we give a complete description of the Helson matrices of finite rank, in parallel with the classical Kronecker theorem on Hankel matrices.

The Hankel matrix rank theorem revisited

We give a new short proof of a version of a Hankel matrix rank theorem. That version expresses the rank of H by the smallest possible rank of an infinite Hankel matrix containing H. The new approach is based on application of the Kronecker theorem.

On the Structure of Positive Semi-Definite Finite Rank General Domain Hankel and Toeplitz Operators in Several Variables

Multivariate versions of the Kronecker theorem in the continuous multivariate setting has recently been published, that characterize the generating functions that give rise to finite rank multidimensional Hankel and Toeplitz type operators defined on general domains. In this paper we study how the additional assumption of positive semi-definite affects the characterization of the corresponding generating functions. We show that these theorems become particularly transparent in the continuous setting, by providing elegant if-and-only-if statements connecting the rank with sums of exponential functions. We also discuss how these operators can be discretized, giving rise to an interesting class of structured matrices that inherit desirable properties from their continuous analogs. In particular we describe how the continuous Kronecker theorem also applies to these structured matrices, given sufficient sampling. We also provide a new proof for the Carathéodory-Fejér theorem for block Toeplitz matrices, based on tools from tensor algebra.

Fractional descriptor continuous–time linear systems described by the Caputo–Fabrizio derivative

AbstractThe Weierstrass–Kronecker theorem on the decomposition of the regular pencil is extended to fractional descriptor continuous-time linear systems described by the Caputo–Fabrizio derivative. A method for computing solutions of continuous-time systems is presented. Necessary and sufficient conditions for the positivity and stability of these systems are established. The discussion is illustrated with a numerical example.

Positivity and stability of fractional descriptor time–varying discrete–time linear systems

Abstract The Weierstrass–Kronecker theorem on the decomposition of the regular pencil is extended to fractional descriptor time-varying discrete-time linear systems. A method for computing solutions of fractional systems is proposed. Necessary and sufficient conditions for the positivity of these systems are established.

A New Frequency Estimation Method for Equally and Unequally Spaced Data

Spectral estimation is an important classical problem that has received considerable attention in the signal processing literature. In this contribution, we propose a novel method for estimating the parameters of sums of complex exponentials embedded in additive noise from regularly or irregularly spaced samples. The method relies on Kronecker's theorem for Hankel operators, which enables us to formulate the nonlinear least squares problem associated with the spectral estimation problem in terms of a rank constraint on an appropriate Hankel matrix. This matrix is generated by sequences approximating the underlying sum of complex exponentials. Unequally spaced sampling is accounted for through a proper choice of interpolation matrices. The resulting optimization problem is then cast in a form that is suitable for using the alternating direction method of multipliers (ADMM). The method can easily include either a nuclear norm or a finite rank constraint for limiting the number of complex exponentials. The usage of a finite rank constraint makes, in contrast to the nuclear norm constraint, the method heuristic in the sense that the problem is non-convex and convergence to a global minimum can not be guaranteed. However, we provide a large set of numerical experiments that indicate that usage of the finite rank constraint nevertheless makes the method converge to minima close to the global minimum for reasonably high signal to noise ratios, hence essentially yielding maximum-likelihood parameter estimates. Moreover, the method does not seem to be particularly sensitive to initialization and performs substantially better than standard subspace-based methods.

Late-Time Estimation for Resonance-Based Radar Target Identification

This communication presents a novel method for determining the commencement of the late-time resonant response of a radar target, without a priori knowledge of the target geometry or orientation. Insights from Kronecker's theorem for Hankel matrices are used to determine the offset for the processing window for extraction of the true complex natural resonances by exploiting characteristic changes in the Hankel matrix, including the matrix rank and the distribution of eigenvalues. Results are presented for both simulated and measured targets.

Computing constraint sets for differential fields

Kronecker's Theorem and Rabin's Theorem are fundamental results about computable fields F and the decidability of the set of irreducible polynomials over F. We adapt these theorems to the setting of differential fields K, with constrained pairs of differential polynomials over K assuming the role of the irreducible polynomials. We prove that two of the three basic aspects of Kronecker's Theorem remain true here, and that the reducibility in one direction (but not the other) from Rabin's Theorem also continues to hold.

Isometric Reflection Vectors and Characterizations of Hilbert Spaces

A known characterization of Hilbert spaces via isometric reflection vectors is based on the following implication: if the set of isometric reflection vectors in the unit sphereSXof a Banach spaceXhas nonempty interior inSX, thenXis a Hilbert space. Applying a recent result based on well-known theorem of Kronecker from number theory, we improve this by substantial reduction of the set of isometric reflection vectors needed in the hypothesis.

Almost perfect state transfer in quantum spin chains

The natural notion of almost perfect state transfer (APST) is examined. It is applied to the modelling of efficient quantum wires with the help of $XX$ spin chains. It is shown that APST occurs in mirror-symmetric systems, when the 1-excitation energies of the chains are linearly independent over rational numbers. This result is obtained as a corollary of the Kronecker theorem in Diophantine approximation. APST happens under much less restrictive conditions than perfect state transfer (PST) and moreover accommodates the unavoidable imperfections. Some examples are discussed.