Riemann Sphere Research Articles

Asymptotics for the Expected Number of Nodal Components for Random Lemniscates

Abstract We determine the true asymptotic behaviour for the expected number of connected components for a model of random lemniscates proposed recently by Lerario and Lundberg. These are defined as the subsets of the Riemann sphere, where the absolute value of certain random, $\textrm{SO}(3)$-invariant rational function of degree $n$ equals to $1$. We show that the expected number of the connected components of these lemniscates, divided by $n$, converges to a positive constant defined in terms of the quotient of two independent plane Gaussian analytic functions. A major obstacle in applying the novel non-local techniques due to Nazarov and Sodin on this problem is the underlying non-Gaussianity, intrinsic to the studied model.

A Lightcone Embedding of the Twin Building of a Hyperbolic Kac-Moody Group

Let A be a symmetrizable hyperbolic generalized Cartan matrix with Kac-Moody algebra g = g(A) and (adjoint) Kac-Moody group G = G(A)=$\langle\exp(ad(t e_i)), \exp(ad(t f_i)) \,|\, t\in C\rangle$ where $e_i$ and $f_i$ are the simple root vectors. Let $(B^+, B^-, N)$ be the twin BN-pair naturally associated to G and let $(\mathcal B^+,\mathcal B^-)$ be the corresponding twin building with Weyl group W and natural G-action, which respects the usual W-valued distance and codistance functions. This work connects the twin building of G and the Kac-Moody algebra g in a new geometrical way. The Cartan-Chevalley involution, $\omega$, of g has fixed point real subalgebra, k, the 'compact' (unitary) real form of g, and k contains the compact Cartan t = k $\cap$ h. We show that a real bilinear form $(\cdot,\cdot)$ is Lorentzian with signatures $(1, \infty)$ on k, and $(1, n -1)$ on t. We define $\{x\in {\rm k} \,|\, (x, x) \leq 0\}$ to be the lightcone of k, and similarly for t. Let K be the compact (unitary) real form of G, that is, the fixed point subgroup of the lifting of $\omega$ to G. We construct a K-equivariant embedding of the twin building of G into the lightcone of the compact real form k of g. Our embedding gives a geometric model of part of the twin building, where each half consists of infinitely many copies of a W-tessellated hyperbolic space glued together along hyperplanes of the faces. Locally, at each such face, we find an $SU(2)$-orbit of chambers stabilized by $U(1)$ which is thus parametrized by a Riemann sphere $SU(2)/U(1)\cong S^2$. For n = 2 the twin building is a twin tree. In this case, we construct our embedding explicitly and we describe the action of the real root groups on the fundamental twin apartment. We also construct a spherical twin building at infinity, and construct an embedding of it into the set of rays on the boundary of the lightcone.

Winding numbers and full extendibility in holomorphic motions

We construct an example of a holomorphic motion of a five-point subset of the Riemann sphere over an annulus such that it satisfies the zero winding number condition but is not fully extendable.

On Rational Functions with More than Three Branch Points

Let d be a positive integer and Λ be a collection of partitions of d of the form (a1, …, ap), (b1, …, bq), (m1+ 1, 1, …, 1), …, (ml+ 1, 1, …, 1), where (m1, …, ml) is a partition of p + q − 2 > 0. We prove that there exists a rational function on the Riemann sphere with branch data Λ if and only if max(m1, …, ml) < d/GCD(a1, …, ap, b1, …, bq). As an application, we give a new class of branch data which can be realized by Belyi functions on the Riemann sphere.

Extended Complex Plane and Riemann Sphere

Extended Complex Plane and Riemann Sphere

Conformal Invariance of CLEΚ on the Riemann Sphere for Κ ∈(4,8)

Abstract The conformal loop ensemble (${\textrm{CLE}}$) is the canonical conformally invariant probability measure on non-crossing loops in a simply connected domain in $\mathbbm{C}$ and is indexed by a parameter $\kappa \in (8/3,8)$. We consider ${\textrm{CLE}}_\kappa $ on the whole-plane in the regime in which the loops are self-intersecting ($\kappa \in (4,8)$) and show that it is invariant under the inversion map $z \mapsto 1/z$. This shows that whole-plane ${\textrm{CLE}}_\kappa $ for $\kappa \in (4,8)$ defines a conformally invariant measure on loops on the Riemann sphere. The analogous statement in the regime in which the loops are simple ($\kappa \in (8/3,4]$) was proven by Kemppainen and Werner and together with the present work covers the entire range $\kappa \in (8/3,8)$ for which ${\textrm{CLE}}_\kappa $ is defined. As an intermediate step in the proof, we show that ${\textrm{CLE}}_\kappa $ for $\kappa \in (4,8)$ on an annulus, with any specified number of inner-boundary-surrounding loops, is well defined and conformally invariant.

On the Monodromy of Meromorphic Cyclic Opers on the Riemann Sphere

Abstract We study the monodromy of meromorphic cyclic $\textrm{SL}(n,{\mathbb{C}})$-opers on the Riemann sphere with a single pole. We prove that the monodromy map, sending such an oper to its Stokes data, is an immersion in the case where the order of the pole is a multiple of $n$. To do this, we develop a method based on the work of Jimbo, Miwa, and Ueno from the theory of isomonodromic deformations. Specifically, we introduce a system of equations that is equivalent to the isomonodromy equations of Jimbo–Miwa–Ueno, but which is adapted to the decomposition of the Lie algebra $\mathfrak{sl}(n,\mathbb{C})$ as a direct sum of irreducible representations of $\mathfrak{sl}(2,\mathbb{C})$. Using properties of some structure constants for $\mathfrak{sl}(n,\mathbb{C})$ to analyze this system of equations, we show that deformations of certain families of cyclic $\textrm{SL}(n,\mathbb{C})$-opers on the Riemann sphere with a single pole are never infinitesimally isomonodromic.

The locus of the representation of logarithmic connections by Fuchsian equations

The generic element of the moduli space of logarithmic connections with parabolic points on holomorphic vector bundle over the Riemann sphere can be represented by a Fuchsian equation with some singularities and some apparent singularities. We analyze the case of rank 3 vector bundle which leads to third order Fuchsian equation. We find coordinates on an open subset of the moduli space and we construct a non-trivial part of the moduli space by blowing up along a variety in a special case.

Wall-crossing and recursion formulae for tropical Jucys covers

Hurwitz numbers enumerate branched genus g g covers of the Riemann sphere with fixed ramification data or equivalently certain factorisations of permutations. Double Hurwitz numbers are an important class of Hurwitz numbers, obtained by considering ramification data with a specific structure. They exhibit many fascinating properties, such as a beautiful piecewise polynomial structure, which has been well-studied in the last 15 years. In particular, using methods from tropical geometry, it was possible to derive wall-crossing formulae for double Hurwitz numbers in arbitrary genus. Further, double Hurwitz numbers satisfy an explicit recursive formula. In recent years several related enumerations have appeared in the literature. In this work, we focus on two of those invariants, so-called monotone and strictly monotone double Hurwitz numbers. Monotone double Hurwitz numbers originate from random matrix theory, as they appear as the coefficients in the asymptotic expansion of the famous Harish-Chandra–Itzykson–Zuber integral. Strictly monotone double Hurwitz numbers are known to be equivalent to an enumeration of Grothendieck dessins d’enfants. These new invariants share many structural properties with double Hurwitz numbers, such as piecewise polynomiality. In this work, we enlarge upon this study and derive new explicit wall-crossing and recursive formulae for monotone and strictly monotone double Hurwitz numbers. The key ingredient is a new interpretation of monotone and strictly monotone double Hurwitz numbers in terms of tropical covers, which was recently derived by the authors. An interesting observation is the fact that monotone and strictly monotone double Hurwitz numbers satisfy wall-crossing formulae, which are almost identical to the classical double Hurwitz numbers.

Genus one Belyi maps by quadratic correspondences

We present a method of obtaining a Belyi map on an elliptic curve from that on the Riemann sphere. This is done by writing the former as a radical of the latter, which we call a quadratic correspondence, with the radical determining the elliptic curve. With a host of examples of various degrees, we demonstrate that the correspondence is an efficient way of obtaining genus one Belyi maps. As applications, we find the Belyi maps for the dessins d’enfant which have arisen as brane-tilings in the physics community, including ones, such as the so-called suspended pinched point, which have been a standing challenge for a number of years.

Bi-pruned Hurwitz numbers

Hurwitz numbers enumerate ramified coverings of the Riemann sphere with fixed ramification data. Certain kinds of ramification data are of particular interest, such as double Hurwitz numbers, which count covers with fixed arbitrary ramification over 0 and ∞ and simple ramification over b points, where b is given by the Riemann-Hurwitz formula. In this work, we introduce the notion of bi-pruned double Hurwitz numbers. This is a new enumerative problem, which yields smaller numbers but completely determines double Hurwitz numbers. They count a relevant subset of covers and share many properties with double Hurwitz numbers, such as piecewise polynomial behaviour and an expression in the symmetric group. Thus, we may view them as a core of the double Hurwitz numbers problem. This work is built on and generalises previous results of Zvonkine [18], Irving [13], Irving-Rattan [12], Do–Norbury [3] and the author [9].

Supersymmetric S-matrices from the worldsheet in 10 & 11d

We obtain compact formulae for tree super-amplitudes for 10 and 11-dimensional supergravity and 10-dimensional supersymmetric Yang-Mills and Born-Infeld. These are based on the polarised scattering equations. These incorporate polarization data into a spinor field on the Riemann sphere and arise from a twistorial representation of ambitwistor strings in 10 and 11 dimensions. They naturally extend amplitude formulae to manifest maximal supersymmetry. The framework is the natural generalization of twistorial ambitwistor string formulae found previously in four and six dimensions and is informally motivated from a vertex operator prescription for a family of supersymmetric worldsheet ambitwistor string models.

Understanding truncated non-commutative geometries through computer simulations

When aiming to apply mathematical results of non-commutative geometry to physical problems, the following question arises: How they translate to a context in which only a part of the spectrum is known? In this article, we aim to detect when a finite-dimensional triple is the truncation of the Dirac spectral triple of a spin manifold. To this end, we numerically investigate the restriction that the higher Heisenberg equation [A. H. Chamseddine et al., J. High Energy Phys. 2014, 98] places on a truncated Dirac operator. We find a bounded perturbation of the Dirac operator on the Riemann sphere that induces the same Chern class.

Pareto optimization of resonances and minimum-time control

The aim of the paper is to reduce one spectral optimization problem, which involves the minimization of the decay rate |Imκ| of a resonance κ, to a collection of optimal control problems on the Riemann sphere Cˆ. This reduction allows us to apply methods of extremal synthesis to the structural optimization of layered optical cavities. We start from a dual problem of minimization of the resonator length and give several reformulations of this problem that involve Pareto optimization of the modulus |κ| of a resonance, minimum-time control problems on Cˆ, and associated Hamilton-Jacobi-Bellman equations. Various types of controllability properties are studied in connection with the existence of optimizers and with the relationship between the Pareto optimal frontiers of minimal decay and minimal modulus. We give explicit examples of optimal resonances and describe qualitatively properties of the Pareto frontiers near them. A special representation of bang-bang controlled trajectories is combined with the analysis of extremals to obtain various bounds on optimal widths of layers. We propose a new method of computation of optimal symmetric resonators based on minimum-time control and compute with high accuracy several Pareto optimal frontiers and high-Q resonators.

Connected components of affine primitive permutation groups

For a finite group G, the Hurwitz space Hr,gin(G) is the space of genus g covers of the Riemann sphere with r branch points and the monodromy group G.In this paper, we give a complete list of primitive genus one systems of affine type. That is, we assume that G is a primitive group of affine type. Under this assumption we determine the braid orbits on the suitable Nielsen classes, which is equivalent to finding connected components in Hr,1in(G). Furthermore, we give a new algorithm for computing large braid orbits on Nielsen classes. This algorithm utilizes a correspondence between the components of Hr,1in(G) and Hr,1in(M), where M is the point stabilizer in G.

Constructing Strebel Differentials via Belyi Maps on the Riemann Sphere

In this manuscript, by using Belyi maps and dessin d'enfants, we construct some concrete examples of Strebel differentials with four double poles on the Riemann sphere. As an application, we could give some explicit cone spherical metrics on the Riemann sphere.

β-flatness condition in CR spheres multiplicity results

We give multiplicity results for the problem of prescribing the scalar curvature on Cauchy–Riemann spheres under [Formula: see text]-flatness condition. To find a lower bound for the number of solutions, we use Bahri’s methods based on the theory of critical points at infinity and a Poincaré–Hopf-type formula.

Institutional Designing of Food Security by Instruments of Matrix Modelling and Value Flows Synchronization

There are some theoretically-methodological aspects of food security institute and functioning of the goods market of agro-industrial complex in article. It is shown that the concept "food security" is considered by modern domestic agroeconomists from different attitudes. At the same time characteristics of the institutional analysis didn't receive worthy reflection. In this regard, authors made a contribution to interpretation of the concept "food security" within the system of institutional knowledge.
 There are shown institutional bases of ensuring food security on the basis of the approved techniques of matrix vector modeling and cost flows synchronization. Authors made evaluation of food security institute development and the market of agroindustrial complex of such countries as Belarus, Brazil, Germany, India, China, Russia, Poland and the United States of America on the basis of official statistical data for 2005-2015.
 There are provided strengths (production and economic characteristics) of regions in the sphere of agroindustrial complex. There was made a group of regions on progressive and regressive, with allocation of the regions which are at a transitional stage of market development of agroindustrial complex – on the system of the standardized indicators. On the basis of estimative and analytical procedures are offered the directions of institutional designing of food security and functioning of the market of agroindustrial complex.
 There is formulated a number of recommendations of increasing the efficiency of Russia food security institute, including: 1) to use the system of vertical or horizontal integration of the entities of agro-industrial complex more active; 2) to modernize the entities of agroindustrial complex taking into account specialized, but not universal requirements of institutional designing; 3) to make active policy of rational import substitution in agroindustrial complex.
 The conclusion is drawn that institutional designing of "multispeed" cost flows is substantially determined by researches on development of the standard rates providing overcoming negative market events. Achievement of the set proportions will allow institutional units to receive preferences in the field of the taxation and providing with credit resources for the development. These directions are seems priority to authors and determine the agenda for the subsequent researches

Long hitting times for expanding systems

We prove a new result in the area of hitting time statistics. Currently, there is a lot of papers showing that the first entry times into cylinders or balls are often faster than the Birkhoff’s Ergodic theorem would suggest. We provide an opposite counterpart to these results by proving that the hitting times into shrinking balls are also often much larger than these theorems would suggest, by showing that for many expanding dynamical systemsfor an appropriately large, at least of full measure, set of points y and x.We first do this for all transitive open distance expanding maps and Gibbs/equilibrium states of Hölder continuous potentials; in particular for all irreducible subshifts of finite type with a finite alphabet. Then we prove such result for all finitely irreducible subshifts of finite type with a countable alphabet and Gibbs/equilibrium states for Hölder continuous summable potentials. Next, we show that the limsup result holds for all graph directed Markov systems (far going natural generalizations of iterated function systems) and projections of aforementioned Gibbs states on their limit sets. By utilizing the first return map techniques, we then prove the limsup result for all tame topological Collect–Eckmann multimodal maps of an interval, all tame topological Collect–Eckmann rational functions of the Riemann sphere, and all dynamically semi-regular transcendental meromorphic functions from to .

Connected Components of the Hurwitz Space for the Symmetric Group of Degree 7

The Hurwitz space is the space of genus covers of the Riemann sphere with branch points and the monodromy group . Let be the symmetric group . In this paper, we enumerate the connected components of . Our approach uses computational tools, relying on the computer algebra system GAP and the MAPCLASS package, to find the connected components of . This work gives us the complete classification of primitive genus zero symmetric group of degree seven.