Lower Half-plane Research Articles

Construction of rational interpolations using Mamquist-Takenaka systems

Rational functions have deep system-theoretic significance. They represent the natural way of modeling linear dynamical systems in the frequency (Laplace) domain. Using rational functions, the goal of this paper to compute models that match (interpolate) given data sets of measurements. In this paper, the authors show that using special rational orthonormal systems, the Malmquist-Takenaka systems, it is possible to write the rational interpolant $r_{(n, m)}$, for $n=N-1, m=N$ using only $N$ sampling nodes (instead of $2N$ nodes) if the interpolating nodes are in the complex unit circle or on the upper half-plane. Moreover, the authors prove convergence results related to the rational interpolant. They give an efficient algorithm for the determination of the rational interpolant.

On generalizations of the upper half plane in a multidimensional complex space

On generalizations of the upper half plane in a multidimensional complex space

Classical solution of the initial-value problem for a one-dimensional quasilinear wave equation

For a one-dimensional mildly quasilinear wave equation given in the upper half-plane, we consider the Cauchy problem. The solution is constructed by the method of characteristics in an implicit analytical form as a solution of some integro-differential equation. The solvability of this equation, as well the smoothness of its solution, is studied. For the problem in question, the uniqueness of the solution is proved and the conditions under which its classical solution exists are established. When given data is not enough smooth a mild solution is constructed.

Burgers’ equation in the complex plane

Burgers’ equation is a well-studied model in applied mathematics with connections to the Navier–Stokes equations in one spatial direction and traffic flow, for example. Following on from previous work, we analyse solutions to Burgers’ equation in the complex plane, concentrating on the dynamics of the complex singularities and their relationship to the solution on the real line. For an initial condition with a simple pole in each of the upper- and lower-half planes, we apply formal asymptotics in the small- and large-time limits in order to characterise the initial and later motion of the singularities. The small-time limit highlights how infinitely many singularities are born at t=0 and how they orientate themselves to lie increasingly close to anti-Stokes lines in the far field of the inner problem. This inner problem also reveals whether or not the closest singularity to the real axis moves toward the axis or away. For intermediate times, we use the exact solution, apply method of steepest descents, and implement the AAA approximation to track the complex singularities. Connections are made between the motion of the closest singularity to the real axis and the steepness of the solution on the real line. While Burgers’ equation is integrable (and has an exact solution), we deliberately apply a mix of techniques in our analysis in an attempt to develop methodology that can be applied to other nonlinear partial differential equations that do not.

Proportion of modular forms with transcendental zeros for general levels

Let $\Gamma$ be a congruence subgroup such that $\Gamma_{1}(N)\subset\Gamma\subset\Gamma_{0}(N)$ for some positive integer $N$. For a positive integer $k$, let $M_{k,\mathbf{Z}}(\Gamma)$ be the set of modular forms of weight $k$ on $\Gamma$ with integral Fourier coefficients. Let $R_{k}(\Gamma)$ be the set of common zeros in the upper half plane $\mathbf{H}$ of all the modular forms of weight $k$ on $\Gamma$. In this note, we prove that the density of modular forms in $M_{k,\mathbf{Z}}(\Gamma)$ with an algebraic zero $z \notin R_{k}(\Gamma)$ is zero.

Moduli of quadrilaterals and quasiconformal reflection

We study the interior and exterior moduli of polygonal quadrilaterals. The main result is a formula for a conformal mapping of the upper half plane onto the exterior of a convex polygonal quadrilateral. We prove this by a careful analysis of the Schwarz-Christoffel transformation and obtain the so-called accessory parameters and then the result in terms of the Lauricella hypergeometric function. This result enables us to understand the dissimilarities of the exterior and interior of a convex polygonal quadrilateral. We also give a Mathematica algorithm for the computation. Then we study the special case of an isosceles trapezoidal polygon L and obtain two-sided estimates for the coefficient of quasiconformal reflection over L in terms of special functions and geometric parameters of L.

Multi-point passage probabilities and Green’s functions for SLE$${}_{8/3}$$

We consider a loop representation of the $$O(n)$$ model at the critical point. In the case $$n=0$$ , the model reduces to statistical ensembles of self-avoiding loops, which can be described by Schramm–Loewner evolution (SLE) with $$\kappa=8/3$$ . In this limit, the $$O(n=0)$$ model corresponds to a logarithmic conformal field theory (LCFT) with the central charge $$c=0$$ . We study the LCFT correlation functions in the upper half-plane containing several twist operators in the bulk and a pair of the $$\Phi_{1,2}$$ boundary operators. By using a Coulomb gas representation for the correlation functions, we obtain explicit results for the probabilities of the SLE $${}_{8/3}$$ trace to pass in various ways about $$N\geq 1$$ marked points. When the points approach each other pairwise, the probabilities reduce to multipoint SLE Green’s functions. We propose an explicit representation for the Green’s functions in terms of the correlation functions of the bulk $$\Phi_{3,1}$$ and boundary $$\Phi_{1,2}$$ operators.

The Heston stochastic volatility model has a boundary trace at zero volatility

We establish boundary regularity results in Hölder spaces for the degenerate parabolic problem obtained from the Heston stochastic volatility model in Mathematical Finance set up in the spatial domain (upper half-plane) mathbb {H} = mathbb {R}times (0,infty )subset mathbb {R}^2. Starting with nonsmooth initial data u_0in H, we take advantage of smoothing properties of the parabolic semigroup textrm{e}^{-t{mathcal {A}}}:,Hrightarrow H, tin mathbb {R}_+, generated by the Heston model, to derive the smoothness of the solution u(t) = textrm{e}^{-t{mathcal {A}}} u_0 for all t>0. The existence and uniqueness of a weak solution is obtained in a Hilbert space H = L^2(mathbb {H};mathfrak {w}) with very weak growth restrictions at infinity and on the boundary partial mathbb {H}= mathbb {R}times { 0}subset mathbb {R}^2 of the half-plane mathbb {H}. We investigate the influence of the boundary behavior of the initial data u_0in H on the boundary behavior of u(t) for t>0.

Explosive growth for a constrained Hastings–Levitov aggregation model

We consider a constrained version of the HL(0) Hastings-Levitov model of aggregation in the complex plane, in which particles can only attach to the part of the cluster that has already been grown. Although one might expect that this gives rise to a non-trivial limiting shape, we prove that the cluster grows explosively: in the upper half plane, the aggregate accumulates infinite diameter as soon as it reaches positive capacity. More precisely, we show that after $nt$ particles of (half-plane) capacity $1/(2n)$ have attached, the diameter of the shape is highly concentrated around $\sqrt{t\log n}$, uniformly in $t\in [0,T]$. This illustrates a new instability phenomenon for the growth of single trees/fjords in unconstrained HL(0).

О задаче Дирихле для обобщенного уравнения Лапласа

This paper considers the Dirichlet problem for a second-order partial differential equation with the Riemann-Liouville derivative with respect to one of two independent order variables, less than two, in the upper half-plane. The equation under study turns into a two-dimensional Laplace equation if the order of the fractional derivative coincides with an integer. The main result of this work is the proof of theorems on the existence and uniqueness of a solution to the problem posed. An explicit form of representation of the solution is obtained. Сorresponding asymptotic estimates аre given.

Walls and asymptotics for Bridgeland stability conditions on 3-folds

We consider Bridgeland stability conditions for three-folds conjectured by Bayer-Macr\`i-Toda in the case of Picard rank one. We study the differential geometry of numerical walls, characterizing when they are bounded, discussing possible intersections, and showing that they are essentially regular. Next, we prove that walls within a certain region of the upper half plane that parametrizes geometric stability conditions must always intersect the curve given by the vanishing of the slope function and, for a fixed value of s, have a maximum turning point there. We then use all of these facts to prove that Gieseker semistability is equivalent to asymptotic semistability along a class of paths in the upper half plane, and to show how to find large families of walls. We illustrate how to compute all of the walls and describe the Bridgeland moduli spaces for the Chern character (2,0,-1,0) on complex projective 3-space in a suitable region of the upper half plane.

Taylor Series of Biharmonic Poisson Integral for Upper Half-Plane

The fourth-order partial differential equation for the biharmonic Poisson integral is presented in the case of the upper half-plane (y > 0). To solve this equation, two boundary conditions must be taken into account. The boundary-value problem is solved by transforming the presented boundary-value problem for the biharmonic Poisson integral into two boundary-value problems for some two-dimensional functions A(q, y) and B (q, y). After that, the biharmonic Poisson integral for the upper half-plane is obtained. It was found that the derived Taylor series of biharmonic Poisson integral for the upper half-plane contains the remainder in the integral form.

Triangular modular curves of small genus

Triangular modular curves are a generalization of modular curves that arise from quotients of the upper half-plane by congruence subgroups of hyperbolic triangle groups. These curves also arise naturally as a source of Belyi maps with monodromy text {PGL}_2(mathbb {F}_q) or text {PSL}_2(mathbb {F}_q). We present a computational approach to enumerate Borel-type triangular modular curves of low genus, and we carry out this enumeration for prime level and small genus.

Two-Dimensional Beam Steering Based on Compact Programmable Coding Metasurface

A programmable coding metasurface provides unprecedented flexibility to manipulate electromagnetic waves dynamically. By controlling the peculiarity of subwavelength artificial atoms, devices with metasurfaces perform various functionalities. In this paper, a compact programmable coding metasurface with PIN diodes is proposed to realize the beam steering in the Ka band. The phase distribution on the metasurface can be actively controlled by switching the states of each meta-atom. By tuning the phase gradient along the metasurface plane, the reflective beam can scan all directions in the upper half-plane. In addition, the compact metasurface is easier to integrate, which could expand the fields of applications. The full-wave simulation results show that the radiation direction of the main lobe is consistent with the theoretical calculation results, and the maximum steering angle of simulation is 60°. As experimental verification, a prototype was processed and the functionality of beam steering in the xoz plane and in the yoz plane was tested. Experimental results show that the designed metasurface can achieve beam steering in both planes, and the maximum scan angle is 45° in the xoz plane. The proposed metasurface opens a new way of beam steering in half space, which may have potential applications in sensing and wireless communications in millimeter waves.

Norms of Basic Operators in Vector Valued Model Spaces and de Branges Spaces

Let \(\Omega _+\) be either the open unit disc or the open upper half plane or the open right half plane. In this paper, we compute the norm of the basic operator \(A_\alpha =\Pi _\Theta T_{b_\alpha }|_{\mathcal {H}(\Theta )}\) in the vector valued model space \(\mathcal {H}(\Theta )=H^m_2 \ominus \Theta H^m_2\) associated with an \(m\times m\) matrix valued inner function \(\Theta \) in \(\Omega _+\) and show that the norm is attained. Here \(\Pi _\Theta \) denotes the orthogonal projection from the Lebesgue space \(L^m_2\) onto \(\mathcal {H}(\Theta )\) and \(T_{b_\alpha }\) is the operator of multiplication by the elementary Blaschke factor \(b_{\alpha }\) of degree one with a zero at a point \(\alpha \in \Omega _+\). We show that if \(A_\alpha \) is strictly contractive, then its norm may be expressed in terms of the singular values of \(\Theta (\alpha )\). We then extend this evaluation to the more general setting of vector valued de Branges spaces.

A CLASSIFICATION OF SURDS OF NON-ARITHMETICAL GROUPS

Definitions and classifications of reduced surds for non-arithmetical groups on the hyperbolic plane is proposed, as well as those of some congruence subgroups. The considered groups are the non-arithmetical desymmetrized group, its Γ0 congruence subgroup, and a generalized Hecke group according the choice of the commutator subgroup of the desymmetrized group. The definitions of the reduced surds are studied according to the ordering of the generators of the conjugacy subclasses of the considered groups. The construction of tori is defined as obtained after the tessellation of the Upper Hyperbolic Half Plane and discussed. Cutting trajectories and cutting-trajectories sequences are uniquely defined after the new definition of reduced surds.

Products of Toeplitz operators with angular symbols

Abstract We study products of Toeplitz operators with angular symbols on the Bergman space over the upper half-plane. We establish necessary and sufficient conditions for the product of two such Toeplitz operators to give rise to a Toeplitz operator, especially in case one of the symbols is absolutely continuous or with bounded variation. Our conditions make appeal to Volterra and Fredholm integral equations, and to Duhamel–Mikusiński convolution products as well as to functional equations involving Stieltjes integrals. We illustrate our results by concrete examples showing that there are many angular symbols satisfying our natural conditions and ensuring Toeplitzness of such Toeplitz products.

Quantized Nonlinear Transport with Ultracold Atoms

In this letter, we propose how to measure the quantized nonlinear transport using two-dimensional ultracold atomic Fermi gases in a harmonic trap. This scheme requires successively applying two optical pulses in the left and lower half-planes and then measuring the number of extra atoms in the first quadrant. In ideal situations, this nonlinear density response to two successive pulses is quantized, and the quantization value probes the Euler characteristic of the local Fermi sea at the trap center. We investigate the practical effects in experiments, including finite pulse duration, finite edge width of pulses, and finite temperature, which can lead to deviation from quantization. We propose a method to reduce the deviation by averaging measurements performed at the first and third quadrants, inspired by symmetry considerations. With this method, the quantized nonlinear response can be observed reasonably well with experimental conditions readily achieved with ultracold atoms.

{U}_{mathfrak{q}}{mathfrak{sl}}_2 -invariant non-compact boundary conditions for the XXZ spin chain

We introduce new {U}_{mathfrak{q}}{mathfrak{sl}}_2 -invariant boundary conditions for the open XXZ spin chain. For generic values of mathfrak{q} we couple the bulk Hamiltonian to an infinite-dimensional Verma module on one or both boundaries of the spin chain, and for mathfrak{q}={e}^{frac{ipi}{p}} a 2p-th root of unity — to its p-dimensional analogue. Both cases are parametrised by a continuous “spin” α ∈ ℂ.To motivate our construction, we first specialise to mathfrak{q} = i, where we obtain a modified XX Hamiltonian with unrolled quantum group symmetry, whose spectrum and scaling limit is computed explicitly using free fermions. In the continuum, this model is identified with the (η, ξ) ghost CFT on the upper-half plane with a continuum of conformally invariant boundary conditions on the real axis. The different sectors of the Hamiltonian are identified with irreducible Virasoro representations.Going back to generic q we investigate the algebraic properties of the underlying lattice algebras. We show that if mathfrak{q} α ∉± mathfrak{q} ℤ, the new boundary coupling provides a faithful representation of the blob algebra which is Schur-Weyl dual to {U}_{mathfrak{q}}{mathfrak{sl}}_2 . Then, modifying the boundary conditions on both the left and the right, we obtain a representation of the universal two-boundary Temperley-Lieb algebra. The generators and parameters of these representations are computed explicitly in terms of mathfrak{q} and α. Finally, we conjecture the general form of the Schur-Weyl duality in this case.This paper is the first in a series where we will study, at all values of the parameters, the spectrum and its continuum limit, the representation content of the relevant lattice algebras and the fusion properties of these new spin chains.

Hamiltonian systems on almost cosymplectic manifolds

We determine the Hamiltonian vector field on an odd dimensional manifold endowed with almost cosymplectic structure. This is a generalization of the corresponding Hamiltonian vector field on manifolds with almost transitive contact structures, which extends the contact Hamiltonian systems. Applications are presented to the equations of motion on a particular five-dimensional manifold, the extended Siegel-Jacobi upper-half plane X˜1J. The X˜1J manifold is endowed with a generalized transitive almost cosymplectic structure, an almost cosymplectic structure, more general than transitive almost contact structure and cosymplectic structure. The equations of motion on X˜1J extend the Riccati equations of motion on the four-dimensional Siegel-Jacobi manifold X1J attached to a linear Hamiltonian in the generators of the real Jacobi group G1J(R).