Obius Transformations Research Articles

Möbius transformation and coupled-wave theory: Complete identification of the transfer matrix

A M\"obius transformation which conformally maps the unit circle onto itself is applied to the scalar coupled-wave equations, describing electromagnetic wave propagation in Bragg gratings, and reduces them to a first-order nonlinear differential equation of a single real variable. This equation is analytically integrated for linear detuning and numerically for more complicated refractive index modulation scenarios, e.g., chirped and apodized Bragg gratings, offering a platform for identifying both the amplitude and phase of all elements of the transfer matrix of arbitrarily complex cases. A link between coupled-wave theory and coupled oscillators is established, and exploring the transformation's geometrical properties leads to alternative definitions of the photonic band gap.

Two stability results for the Kawahara equation with a time-delayed boundary control

In this paper, we consider the Kawahara equation in a bounded interval and with a delay term in one of the boundary conditions. Using two different approaches, we prove that this system is exponentially stable under a condition on the length of the spatial domain. Specifically, the first result is obtained by introducing a suitable energy and using the Lyapunov approach, to ensure that the unique solution of the Kawahara system exponentially decays. The second result is achieved by means of a compactness-uniqueness argument, which reduces our study to prove an observability inequality. Furthermore, the main novelty of this work is to characterize the critical set phenomenon for this equation by showing that the stability results hold whenever the spatial length is related to the M\"obius transformations.

On differential equations invariant under two-variable M\"obius transformations

We compute invariants for the two-variable M\"obius transformation. In particular we are interested in partial differential equations in two dependent and two independent variables that are kept invariant under this transformation.

Some properties of a Cauchy family on the sphere derived from the Möbius transformations

We present some properties of a Cauchy family of distributions on the sphere, which is a spherical extension of the wrapped Cauchy family on the circle. The spherical Cauchy family is closed under the M\"obius transformation on the sphere and there is a similar induced transformation on the parameter space. Stereographic projection transforms the the spherical Cauchy family into a multivariate $t$-family with a certain degree of freedom on Euclidean space. Many tractable properties of the spherical Cauchy are derived using the M\"obius transformation and stereographic projection. A method of moments estimator and an asymptotically efficient estimator are expressed in closed form. The maximum likelihood estimation is also straightforward.

Emergent Spatial Structure and Entanglement Localization in Floquet Conformal Field Theory

We study the energy and entanglement dynamics of $(1+1)$D conformal field theories (CFTs) under a Floquet drive with the sine-square deformed (SSD) Hamiltonian. Previous work has shown this model supports both a non-heating and a heating phase. Here we analytically establish several robust and `super-universal' features of the heating phase which rely on conformal invariance but not on the details of the CFT involved. First, we show the energy density is concentrated in two peaks in real space, a chiral and anti-chiral peak, which leads to an exponential growth in the total energy. The peak locations are set by fixed points of the M\"obius transformation. Second, all of the quantum entanglement is shared between these two peaks. In each driving period, a number of Bell pairs are generated, with one member pumped to the chiral peak, and the other member pumped to the anti-chiral peak. These Bell pairs are localized and accumulate at these two peaks, and can serve as a source of quantum entanglement. Third, in both the heating and non-heating phases we find that the total energy is related to the half system entanglement entropy by a simple relation $E(t)\propto c \exp \left( \frac{6}{c}S(t) \right)$ with $c$ being the central charge. In addition, we show that the non-heating phase, in which the energy and entanglement oscillate in time, is unstable to small fluctuations of the driving frequency in contrast to the heating phase. Finally, we point out an analogy to the periodically driven harmonic oscillator which allows us to understand global features of the phases, and introduce a quasiparticle picture to explain the spatial structure, which can be generalized to setups beyond the SSD construction.

Critical points of the classical Eisenstein series of weight two

In this paper, we completely determine the critical points of the normalized Eisenstein series $E_2(\tau)$ of weight $2$. Although $E_2(\tau)$ is not a modular form, our result shows that $E_2(\tau)$ has at most one critical point in every fundamental domain of $\Gamma_{0}(2)$. We also give a criteria for a fundamental domain containing a critical point of $E_2(\tau)$. Furthermore, under the M\"obius transformation of $\Gamma_{0}(2)$ action, all critical points can be mapped into the basic fundamental domain $F_0$ and their images are contained densely on three smooth curves. A geometric interpretation of these smooth curves is also given. It turns out that these smooth curves coincide with the degeneracy curves of trivial critical points of a multiple Green function related to flat tori.

Random ideal hyperbolic quadrilaterals, the cross ratio distribution and punctured tori

Earlier work introduced a geometrically natural probability measure on the group of all M\"obius transformations of the hyperbolic plane so as to be able to study "random" groups of M\"obius transformations, and in particular random two-generator groups. Here we extend these results to consider random punctured tori. These Riemann surfaces have finite hyperbolic area $2\pi$ and fundamental group the free group of rank 2. They can be obtained by pairing (identifying) the opposite sides of an ideal hyperbolic quadrilateral. There is a natural distribution on ideal quadrilateral given by the cross ratio of their vertices. We identify this distribution and then calculate the distributions of various geometric quantities associated with random punctured tori such as the base of the geodesic length spectrum and the conformal modulus, along with more subtle things such as the distribution of the distance in Teichm\"uller space to the central "square" punctured torus.

Reconstructing a Lattice Equation: a Non-Autonomous Approach to the Hietarinta Equation

In this paper we construct a non-autonomous version of the Hietarinta equation [Hietarinta J., J. Phys. A: Math. Gen. 37 (2004), L67-L73] and study its integrability properties. We show that this equation possess linear growth of the degrees of iterates, generalized symmetries depending on arbitrary functions, and that it is Darboux integrable. We use the first integrals to provide a general solution of this equation. In particular we show that this equation is a sub-case of the non-autonomous $Q_{\rm V}$ equation, and we provide a non-autonomous M\"obius transformation to another equation found in [Hietarinta J., J. Nonlinear Math. Phys. 12 (2005), suppl. 2, 223-230] and appearing also in Boll's classification [Boll R., Ph.D. Thesis, Technische Universit\"at Berlin, 2012].

Limit sets for modules over groups onCAT(0) spaces: from the Euclidean to the hyperbolic

The observation that the 0-dimensional Geometric Invariant $\Sigma ^{0}(G;A)$ of Bieri-Neumann-Strebel-Renz can be interpreted as a horospherical limit set opens a direct trail from Poincar\'e's limit set $\Lambda (\Gamma)$ of a discrete group $\Gamma $ of M\"obius transformations (which contains the horospherical limit set of $\Gamma $) to the roots of tropical geometry (closely related to $\Sigma ^{0}(G;A)$ when G is abelian). We explore this trail by introducing the horospherical limit set, $\Sigma (M;A)$, of a G-module A when G acts by isometries on a proper CAT(0) metric space M. This is a subset of the boundary at infinity of M. On the way we meet instances where $\Sigma (M;A)$ is the set of all conical limit points, the complement of a spherical building, the complement of the radial projection of a tropical variety, or (via the Bieri-Neumann-Strebel invariant) where it is closely related to the Thurston norm.

Möbius Transformation for Left-Derivative Quaternion Holomorphic Functions

Holomorphic quaternion functions only admit affine functions; thus, the M\"obius transformation for these functions, which we call quaternionic holomorphic transformation (QHT), only comprises similarity transformations. We determine a general group $\mathsf{X}$ which has the group $\mathsf{G}$ of QHT as a particular case. Furthermore, we observe that the M\"obius group and the Heisenberg group may be obtained by making $\mathsf{X}$ more symmetric. We provide matrix representations for the group $\mathsf{X}$ and for its algebra $\mathfrak{x}$. The Lie algebra is neither simple nor semi-simple, and so it is not classified among the classical Lie algebras. They prove that the group $\mathsf{G}$ comprises $\mathsf{SU}(2,\mathbb{C})$ rotations, dilations and translations. The only fixed point of the QHT is located at infinity, and the QHT does not admit a cross-ratio. Physical applications are addressed at the conclusion.

Geometry of the Cassinian Metric and Its Inner Metric

The Cassinian metric and its inner metric have been studied for subdomains of the $n$-dimensional Euclidean space $\mathbb{R}^n$ ($n\ge 2$) by the first named author. In this paper we obtain various inequalities between the Cassinian metric and other related metrics in some specific subdomains of $\mathbb{R}^n$. Also, a sharp distortion property of the Cassinian metric under M\"obius transformations of the unit ball is obtained.

An algorithm for the Baker-Campbell-Hausdorff formula

A simple algorithm, which exploits the associativity of the BCH formula, and that can be generalized by iteration, extends the remarkable simplification of the Baker-Campbell-Hausdorff (BCH) formula, recently derived by Van-Brunt and Visser. We show that if $[X,Y]=uX+vY+cI$, $[Y,Z]=wY+zZ+dI$, and, consistently with the Jacobi identity, $[X,Z]=mX+nY+pZ+eI$, then $$ \exp(X)\exp(Y)\exp(Z)=\exp({aX+bY+cZ+dI}) $$ where $a$, $b$, $c$ and $d$ are solutions of four equations. In particular, the Van-Brunt and Visser formula $$\exp(X)\exp(Z)=\exp({aX+bZ+c[X,Z]+dI}) $$ extends to cases when $[X,Z]$ contains also elements different from $X$ and $Z$. Such a closed form of the BCH formula may have interesting applications both in mathematics and physics. As an application, we provide the closed form of the BCH formula in the case of the exponentiation of the Virasoro algebra, with ${\rm SL}_2({\rm C})$ following as a subcase. We also determine three-dimensional subalgebras of the Virasoro algebra satisfying the Van-Brunt and Visser condition. It turns out that the exponential form of ${\rm SL}_2({\rm C})$ has a nice representation in terms of its eigenvalues and of the fixed points of the corresponding M\"obius transformation. This may have applications in Uniformization theory and Conformal Field Theories.

Real-time correction of panoramic images using hyperbolic Möbius transformations

Wide-angle images gained a huge popularity in the last years due to the development of computational photography and imaging technological advances. They present the information of a scene in a way which is more natural for the human eye but, on the other hand, they introduce artifacts such as bent lines. These artifacts become more and more unnatural as the field of view increases. In this work, we present a technique aimed to improve the perceptual quality of panorama visualization. The main ingredients of our approach are, on one hand, considering the viewing sphere as a Riemann sphere, what makes natural the application of M\"obius (complex) transformations to the input image, and, on the other hand, a projection scheme which changes in function of the field of view used. We also introduce an implementation of our method, compare it against images produced with other methods and show that the transformations can be done in real-time, which makes our technique very appealing for new settings, as well as for existing interactive panorama applications.

Möbius photogrammetry

Motivated by results on the mobility of mechanical devices called pentapods, this paper deals with a mathematically freestanding problem, which we call M\"obius Photogrammetry. Unlike traditional photogrammetry, which tries to recover a set of points in three-dimensional space from a finite set of central projection, we consider the problem of reconstructing a vector of points in $\mathbb{R}^3$ starting from its orthogonal parallel projections. Moreover, we assume that we have partial information about these projections, namely that we know them only up to M\"obius transformations. The goal in this case is to understand to what extent we can reconstruct the starting set of points, and to prove that the result can be achieved if we allow some uncertainties in the answer. Eventually, the techniques developed in the paper allow us to show that for a pentapod with mobility at least two either some anchor points are collinear, or platform and base are similar, or they are planar and affine equivalent.

Reversed Radial SLE and the Brownian Loop Measure

The Brownian loop measure is a conformally invariant measure on loops in the plane that arises when studying the Schramm-Loewner evolution (SLE). When an SLE curve in a domain evolves from an interior point, it is natural to consider the loops that hit the curve and leave the domain, but their measure is infinite. We show that there is a related normalized quantity that is finite and invariant under M\"obius transformations of the plane. We estimate this quantity when the curve is small and the domain simply connected. We then use this estimate to prove a formula for the Radon-Nikodym derivative of reversed radial SLE with respect to whole-plane SLE.

Optical similaritons in a tapered graded-index nonlinear-fiber amplifier with an external source

We analytically explore a wide class of optical similariton solutions to the nonlinear Schr\"odinger equation appropriately modified to model beam propagation in a tapered, graded-index nonlinear-fiber amplifier with an external source. Under certain physical conditions, we reduce the coupled nonlinear Schr\"odinger equations to a single-wave equation that aptly describes similariton propagation through asymmetric twin-core fiber amplifiers. The asymmetric twin-core fiber is composed of two adjoining, closely spaced, single-mode fibers in which the active one is a tapered, graded-index nonlinear-fiber and the passive one is a step-index fiber. We obtain these self-similar waves for different choices of tapered index profile, using a M\"obius transformation. Our procedure is applicable for both self-focusing and self-defocusing nonlinearities.

Nonautonomous matter waves in a waveguide

We present a physical model that describes the transport of Bose-Einstein-condensed atoms from a reservoir to a waveguide. By using the similarity and M\"obius transformations, we study nonautonomous matter waves in Bose-Einstein condensates in the presence of an inhomogeneous source. Then, we find its various types of exact nonautonomous matter-wave solutions, including the W-shaped bright solitary waves, W-shaped and U-shaped dark solitary waves, periodic wave solutions, and rational solitary waves. The results show that these different types of matter-wave structures can be generated and effectively controlled by modulating the amplitude of the source. Our results may raise the possibility of some experiments and potential applications related to Bose-Einstein condensates in the presence of an inhomogeneous source.

Convex expansion of condenser plates

In this paper we introduce an ew notion of the con vex expansion (convex hull) of condenser plates invariant with respect to M ¨ obius transformations of the space. If one plate is the point at infinity, then the convex expansion of another bounded plate coincides with its usual convex hull. The convex expansion of plates does not decrease the condenser capacity. The main theorem shows that in the case of a flat condenser with connected plates it cannot increase more than threefold. In particular, if the monitoring of the hidden deformation of a condenser with flat plates (with the constancy of its M ¨ obius invariant hull) shows the decrease of its capacity more than twice, then this indicates to the violation of the connectivity, i.e., the destruction of the plate. This result is applicable to nondestructive testing.

Pólya-Schur master theorems for circular domains and their boundaries

We characterize all linear operators on finite or infinite-dimensional polynomial spaces that preserve the property of having the zero set inside a prescribed region $\Omega\subseteq \mathbb{C}$ for arbitrary closed circular domains $\Omega$ (i.e., images of the closed unit disk under a M\"obius transformation) and their boundaries. This provides a natural framework for dealing with several long-standing fundamental problems, which we solve in a unified way. In particular, for $\Omega=\mathbb{R}$ our results settle open questions that go back to Laguerre and P\'olya-Schur.

Rigidity of Schottky sets

We call the complement of a union of at least three disjoint (round) open balls in the unit sphere ${\Bbb S}^n$ a Schottky set. We prove that every quasisymmetric homeomorphism of a Schottky set of spherical measure zero to another Schottky set is the restriction of a M\"obius transformation on ${\Bbb S}^n$. In the other direction we show that every Schottky set in ${\Bbb S}^2$ of positive measure admits nontrivial quasisymmetric maps to other Schottky sets. These results are applied to establish rigidity statements for convex subsets of hyperbolic space that have totally geodesic boundaries.