Unit Circle Research Articles

Co-Axial Metrics on the Sphere and Algebraic Numbers

In this paper, we consider the following curvature equation $$\Delta u+{\rm e}^u=4\pi\biggl((\theta_0-1)\delta_0+(\theta_1-1)\delta_1 +\sum_{j=1}^{n+m}\bigl(\theta_j'-1\bigr)\delta_{t_j}\biggr)\qquad \text{in}\ \mathbb R^2,$$ $$u(x)=-2(1+\theta_\infty)\ln|x|+O(1)\qquad \text{as} \ |x|\to\infty,$$ where $\theta_0$, $\theta_1$, $\theta_\infty$, and $\theta_{j}'$ are positive non-integers for $1\le j\le n$, while $\theta_{j}'\in\mathbb{N}_{\geq 2}$ are integers for $n+1\le j\le n+m$. Geometrically, a solution $u$ gives rise to a conical metric ${\rm d}s^2=\frac12 {\rm e}^u|{\rm d}x|^2$ of curvature $1$ on the sphere, with conical singularities at $0$, $1$, $\infty$, and $t_j$, $1\le j\le n+m$, with angles $2\pi\theta_0$, $2\pi\theta_1$, $2\pi\theta_\infty$, and $2\pi\theta_{j}'$ at $0$, $1$, $\infty$, and $t_j$, respectively. The metric ${\rm d}s^2$ or the solution $u$ is called co-axial, which was introduced by Mondello and Panov, if there is a developing map $h(x)$ of $u$ such that the projective monodromy group is contained in the unit circle. The sufficient and necessary conditions in terms of angles for the existence of such metrics were obtained by Mondello-Panov (2016) and Eremenko (2020). In this paper, we fix the angles and study the locations of the singularities $t_1,\dots,t_{n+m}$. Let $A\subset\mathbb{C}^{n+m}$ be the set of those $(t_1,\dots,t_{n+m})$'s such that a co-axial metric exists, among other things we prove that (i) If $m=1$, i.e., there is only one integer $\theta_{n+1}'$ among $\theta_j'$, then $A$ is a finite set. Moreover, for the case $n=0$, we obtain a sharp bound of the cardinality of the set $A$. We apply a result due to Eremenko, Gabrielov, and Tarasov (2016) and the monodromy of hypergeometric equations to obtain such a bound. (ii) If $m\ge 2$, then $A$ is an algebraic set of dimension $\leq m-1$.

Green's functions for anisotropic elastic plates containing polygonal holes

Two sets of Green's functions for anisotropic elastic media containing a polygonal hole with rounded corners are presented. Analyzing non-elliptical holes poses a formidable mathematical challenge for anisotropic materials because existing mapping functions generally fail to meet certain criteria. The nonconformal mapping scheme and the perturbation method with conformal mapping are applied to derive two sets of solutions, respectively. They are solved using the method of analytical continuation. For the nonconformal approach, the hole is mapped onto a unit circle with nonconformal mapping functions, and the solution form is identical to elliptical holes only with a different mapping function. The perturbation solution is expressed in a series form based on the conformal mapping functions and solutions for elliptical holes. Proper adjustments of branch cuts of complex logarithm functions are introduced for both solutions. These adjustments are essential in numerical computation, as they ensure the continuity of an elastic body across branch cuts. Also, the two solutions presented in this paper are compared in-depth. Based upon their pros and cons, a combined use of these two solutions is suggested for application in boundary element method.

Phase retrieval on circles and lines

Abstract Let f and g be analytic functions on the open unit disk ${\mathbb D}$ such that $|f|=|g|$ on a set A. We give an alternative proof of the result of Perez that there exists c in the unit circle ${\mathbb T}$ such that $f=cg$ when A is the union of two lines in ${\mathbb D}$ intersecting at an angle that is an irrational multiple of $\pi $ , and from this, deduce a sequential generalization of the result. Similarly, the same conclusion is valid when f and g are in the Nevanlinna class and A is the union of the unit circle and an interior circle, tangential or not. We also provide sequential versions of this result and analyze the case $A=r{\mathbb T}$ . Finally, we examine the most general situation when there is equality on two distinct circles in the disk, proving a result or counterexample for each possible configuration.

Harer–Zagier transform of the HOMFLY–PT polynomial for families of twisted hyperbolic knots

In an attempt to generalise knot matrix models for non-torus knots, which currently remains an open problem, we derived expressions for the Harer–Zagier transform—a discrete Laplace transform—of the HOMFLY–PT polynomial for some infinite families of twisted hyperbolic knots. Among them, we found a family of pretzel knots for which, like for torus knots, the transform has a fully factorised form, while for the remaining families considered it consists of sums of factorised terms. Their zero loci show a remarkable structure, which mostly lies on the unit circle and deviates from it only in pairs.

On rotated CMV operators and orthogonal polynomials on the unit circle

Split-step quantum walk operators can be expressed as a generalized version of CMV operators with complex transmission coefficients, which we call rotated CMV operators. Following the idea of Cantero, Moral and Velazquez's original construction of the original CMV operators from the theory of orthogonal polynomials on the unit circle (OPUC), we show that rotated CMV operators can be constructed similarly via a rotated version of OPUCs with respect to the same measure, and admit an analogous LM -factorization as the original CMV operators. We also develop the rotated second kind polynomials corresponding to the rotated OPUCs. We then use the LM -factorization of rotated alternate CMV operators to compute the Gesztesy–Zinchenko transfer matrices for rotated CMV operators.

Applications of multiple orthogonal polynomials with hypergeometric moment generating functions

We investigate several families of multiple orthogonal polynomials associated with weights for which the moment generating functions are hypergeometric series with slightly varying parameters. The weights are supported on the unit interval, the positive real line, or the unit circle and the multiple orthogonal polynomials are generalizations of the Jacobi, Laguerre or Bessel orthogonal polynomials. We give explicit formulas for the type I and type II multiple orthogonal polynomials and study some of their properties. In particular, we describe the asymptotic distribution of the (scaled) zeros of the type II multiple orthogonal polynomials via the free convolution. Essential to our overall approach is the use of the Mellin transform. Finally, we discuss two applications. First, we show that the multiple orthogonal polynomials appear naturally in the study of the squared singular values of (mixed) products of truncated unitary random matrices and Ginibre matrices. Secondly, we use the multiple orthogonal polynomials to simultaneously approximate certain hypergeometric series and to provide an explicit proof of their Q-linear independence.

Convolution quadrature time-domain boundary element method for antiplane anisotropic viscoelastic wave propagation

This paper presents a Convolution Quadrature Time-Domain Boundary Element Method (CQBEM) for antiplane anisotropic viscoelastodynamics. The proposed CQBEM formulation employs the standard linear viscoelastic model and the fundamental solution developed by Wang and Achenbach for expressing viscoelastic and anisotropic properties, respectively. This fundamental solution, which involves integration over the unit circle, is evaluated numerically. After offering a comprehensive overview of the proposed CQBEM formulation, we demonstrate the effectiveness of the proposed method by analyzing the scattering of elastic waves by cavities in certain types of anisotropic viscoelastic materials. Additionally, the use of group velocity curves further validates the numerical results, confirming the anisotropic and viscoelastic effects and thus substantiates the proposed CQBEM for antiplane anisotropic viscoelastic wave propagation.

Chebyshev polynomials corresponding to a vanishing weight

We consider weighted Chebyshev polynomials on the unit circle corresponding to a weight of the form (z−1)s where s>0. For integer values of s this corresponds to prescribing a zero of the polynomial on the boundary. As such, we extend findings of Lachance et al. (1979), to non-integer s. Using this generalisation, we are able to relate Chebyshev polynomials on lemniscates and other, more established, categories of Chebyshev polynomials. An essential part of our proof involves the broadening of the Erdős–Lax inequality to encompass powers of polynomials. We believe that this particular result holds significance in its own right.

Shift invariant subspaces in growth spaces and sets of finite entropy

AbstractWe investigate shift invariant subspaces within the realm of analytic functions in the unit disc, whose radial growth is determined by a majorant . Our main result offers a complete characterization of the shift invariant subspaces generated by Nevanlinna class functions within the aforementioned class of growth spaces. Our description is expressed in terms of a certain ‐entropy condition, which naturally arises in connection with boundary zero sets for analytic functions in the unit disc, having modulus of continuity not exceeding on the unit circle. Furthermore, we utilize our findings to establish a structural theorem on approximation in model spaces.

The symplectic structure of a toric conic transform

Suppose that a compact r-dimensional torus Tr acts in a holomorphic and Hamiltonian manner on polarized complex d-dimensional projective manifold M, with nowhere vanishing moment map Φ. Assuming that Φ is transverse to the ray through a given weight ν, associated to these data there is a complex (d−r+1)-dimensional polarized projective orbifold Mˆν (referred to as the ν-th conic transform of M). Namely, Mˆν is a suitable quotient of the inverse image of the ray in the unit circle bundle of the polarization of M. With the aim to clarify the geometric significance of this construction, we consider the special case where M is toric, and show that Mˆν is itself a Kähler toric orbifold, whose (marked) moment polytope is obtained from the one of M by a certain ‘transform’ operation (depending on Φ and ν).

The β maps: Strong clustering and distribution results on the complex unit circle

In the current work, we study the eigenvalue distribution results of a class of non-normal matrix-sequences which may be viewed as a low rank perturbation, depending on a parameter β>1, of the basic Toeplitz matrix-sequence {Tn(eiθ)}n∈N, i2=−1. The latter of which has obviously all eigenvalues equal to zero for any matrix order n, while for the matrix-sequence under consideration we will show a strong clustering on the complex unit circle. A detailed discussion on the outliers is also provided. The problem appears mathematically innocent, but it is indeed quite challenging since all the classical machinery for deducing the eigenvalue clustering does not cover the considered case. In the derivations, we resort to a trick used for the spectral analysis of the Google matrix plus several tools from complex analysis. We only mention that the problem is not an academic curiosity and in fact stems from problems in dynamical systems and number theory. Additionally, we also provide numerical experiments in high precision, a distribution analysis in the Weyl sense concerning both eigenvalues and singular values is given, and more results are sketched for the limit case of β=1.

Gradient Flow of the Sinai–Ruelle–Bowen Entropy

Motivated by an extension to Gallavotti–Cohen Chaotic Hypothesis, we study local and global existence of a gradient flow of the Sinai–Ruelle–Bowen entropy functional in the space of transitive Anosov maps. For the space of expanding maps on the unit circle, we equip it with a Hilbert manifold structure using a Sobolev norm in the tangent space of the manifold. Under the additional measure-preserving assumption and a slightly modified metric, we show that the gradient flow exists globally and every trajectory of the flow converges to a unique limiting map where the SRB entropy attains the maximal value. In a simple case, we obtain an explicit formula for the flow’s ordinary differential equation representation. This gradient flow has close connection to a nonlinear partial differential equation, a gradient-dependent diffusion equation.

A note on logarithmic equidistribution

For every algebraic number [Formula: see text] on the unit circle which is not a root of unity we prove the existence of a strict sequence of algebraic numbers whose height tends to zero, such that the averages of the evaluation of [Formula: see text] at the conjugates are essentially bounded from above by [Formula: see text]. This completes a characterization on functions [Formula: see text] initiated by Autissier and Baker–Masser, who cover the cases [Formula: see text] and [Formula: see text], respectively. Using the same ideas we also prove analogues in the [Formula: see text]-adic setting.

Resonance expansion for quantum walks and its applications to the long-time behavior

In this paper, resonances are introduced to a class of quantum walks on \mathbb{Z} . Resonances are defined as poles of the meromorphically extended resolvent of the unitary time evolution operator. In particular, they appear inside the unit circle. Some analogous properties to those of quantum resonances for Schrödinger operators are shown. Especially, the resonance expansion, an analogue of the eigenfunction expansion, indicates the long-time behavior of quantum walks. The decaying rate, the asymptotic probability distribution, and the weak limit of the probability density are described by resonances and associated (generalized) resonant states. The generic simplicity of resonances is also investigated.

Using the Golden Section to Approximate π

Summary We derive one specific example of a Machin-like formula for approximating π using the unit circle and the reciproval of the golden section. The efficiency of our derived formula is compared to other, similar formulas using Lehmer’s measure.

Function theory off the complexified unit circle: Fréchet space structure and automorphisms

Motivated by recent work on strict deformation quantization of the unit disk and the Riemann sphere, we study the Fréchet space structure of the set of holomorphic functions on the complement \(\Omega:=\{(z,w)\in \hat{\mathbb{C}}^2\colon z\cdot w\not=1\}\) of the complexified unit circle \(\{(z,w) \in \hat{\mathbb{C}}^2 \colon z\cdot w=1\}\). We also characterize the subgroup of all biholomorphic automorphisms of \(\Omega\) which leave the canonical Laplacian on \(\Omega\) invariant.

Asymptotics of the optimal values of potentials generated by greedy energy sequences on the unit circle

For the Riesz and logarithmic potentials, we consider greedy energy sequences (an)n=0∞ on the unit circle S1, constructed in such a way that for every n≥1, the discrete potential generated by the first n points a0,…,an−1 of the sequence attains its minimum value (say Un) at an. We obtain asymptotic formulae that describe the behavior of Un as n→∞, in terms of certain bounded arithmetic functions with a doubling periodicity property. As previously shown in [8], after properly translating and scaling Un, one obtains a new sequence (Fn) that is bounded and divergent. We find the exact value of liminfFn (the value of limsupFn was already given in [8]), and show that the interval [liminfFn,limsupFn] comprises all the limit points of the sequence (Fn).

Integrability of rank-two web models

We continue our work on lattice models of webs, which generalise the well-known loop models to allow for various kinds of bifurcations [1,2]. Here we define new web models corresponding to each of the rank-two spiders considered by Kuperberg [3]. These models are based on the A2, G2 and B2 Lie algebras, and their local vertex configurations are intertwiners of the corresponding q-deformed quantum algebras. In all three cases we define a corresponding model on the hexagonal lattice, and in the case of B2 also on the square lattice. For specific root-of-unity choices of q, we show the equivalence to a number of three- and four-state spin models on the dual lattice.The main result of this paper is to exhibit integrable manifolds in the parameter spaces of each web model. For q on the unit circle, these models are critical and we characterise the corresponding conformal field theories via numerical diagonalisation of the transfer matrix.In the A2 case we find two integrable regimes. The first one contains a dense and a dilute phase, for which we have analytic control via a Coulomb gas construction, while the second one is more elusive and likely conceals non-compact physics. Three particular points correspond to a three-state spin model with plaquette interactions, of which the one in the second regime appears to present a new universality class. In the G2 case we identify four regimes numerically. The B2 case is too unwieldy to be studied numerically in the general case, but it found analytically to contain a simpler sub-model based on generators of the dilute Birman-Murakami-Wenzl algebra.

On Complex Pisot Numbers That Are Roots of Borwein Trinomials

Let n>m be positive integers. Polynomials of the form zn±zm±1 are called Borwein trinomials. Using an old result of Bohl, we derive explicit formulas for the number of roots of a Borwein trinomial inside the unit circle |z|<1. Based on this, we determine all Borwein trinomials that have a complex Pisot number as a root. There are exactly 29 such trinomials.

Hausdorff dimension of recurrence sets

We consider linear mappings on the 2-dimensional torus, defined by T(x)=Ax (mod 1) , where A is an invertible 2×2 integer matrix, with no eigenvalues on the unit circle. In the case detA=±1 , we give a formula for the Hausdorff dimension of the set {x∈T2:d(Tn(x),x)<e−αn for infinitely many n}.