Half Of Complex Plane Research Articles

A symbolic decision procedure for relations arising among Taylor coefficients of classical Jacobi theta functions

The class of objects we consider are algebraic relations between the four kinds of classical Jacobi theta functions θj(z|τ), j=1,…,4, and their derivatives. We present an algorithm to prove such relations automatically where the function argument z is zero, but where the parameter τ in the upper half complex plane is arbitrary.

Laplace inversion for the solution of an abstract heat equation without the forward transform of the source term

AbstractWe consider the discretization in time of inhomogeneous parabolic equations, using the technique of Laplace inversion along a contour located in the complex left half-plane which, after transformation to a finite interval, is then evaluated to high accuracy by a contour quadrature rule. This reduces the problem to a finite set of elliptic equations with complex coefficients, which may be solved in parallel. A serious problem is how to treat the source term

Multipliers of Hilbert spaces of analytic functions on the complex half-plane

It follows, from a generalised version of Paley-Wiener theorem, that the Laplace transform is an isometry between certain spaces of weighted $L^2$ functions defined on $(0, \infty)$ and (Hilbert) spaces of analytic functions on the right complex half-plane (for example Hardy, Bergman or Dirichlet spaces). We can use this fact to investigate properties of multipliers and multiplication operators on the latter type of spaces. In this paper we present a full characterisation of multipliers in terms of a generalised concept of a Carleson measure. Under certain conditions, these spaces of analytic functions are not only Hilbert spaces but also Banach algebras, and are therefore contained within their spaces of multipliers. We provide some necessary as well as sufficient conditions for this to happen and look at its consequences.

Multipliers of Hilbert spaces of analytic functions on the complex half-plane

It follows, from a generalised version of Paley-Wiener theorem, that the Laplace transform is an isometry between certain spaces of weighted $L^2$ functions defined on $(0, \infty)$ and (Hilbert) spaces of analytic functions on the right complex half-plane (for example Hardy, Bergman or Dirichlet spaces). We can use this fact to investigate properties of multipliers and multiplication operators on the latter type of spaces. In this paper we present a full characterisation of multipliers in terms of a generalised concept of a Carleson measure. Under certain conditions, these spaces of analytic functions are not only Hilbert spaces but also Banach algebras, and are therefore contained within their spaces of multipliers. We provide some necessary as well as sufficient conditions for this to happen and look at its consequences.

Outer transfer functions of differential-algebraic systems

We consider differential-algebraic systems (DAEs) whose transfer function is outer : i . e ., it has full row rank and all transmission zeros lie in the closed left half complex plane. We characterize outer, with the aid of the Kronecker structure of the system pencil and the Smith–McMillan structure of the transfer function, as the following property of a behavioural stabilizable and detectable realization: each consistent initial value can be asymptotically controlled to zero while the output can be made arbitrarily small in the ℒ 2 -norm. The zero dynamics of systems with outer transfer functions are analyzed. We further show that our characterizations of outer provide a simple and very structured analysis of the linear-quadratic optimal control problem.

Model-Free Closed-Loop Stability Analysis: A Linear Functional Approach

Performing a stability analysis during the design of any electronic circuit is critical to guarantee its correct operation. A closed-loop stability analysis can be performed by analysing the impedance presented by the circuit at a well-chosen node without internal access to the simulator. If any of the poles of this impedance lie in the complex right half-plane, the circuit is unstable. The classic way to detect unstable poles is to fit a rational model on the impedance. In this paper, a projection-based method is proposed which splits the impedance into a stable and an unstable part by projecting on an orthogonal basis of stable and unstable functions. When the unstable part lies significantly above the interpolation error of the method, the circuit is considered unstable. Working with a projection provides one, at small cost, with a first appraisal of the unstable part of the system. Both small-signal and large-signal stability analysis can be performed with this projection-based method. In the small-signal case, a low-order rational approximation can be fitted on the unstable part to find the location of the unstable poles.

On decay properties of solutions to the Stokes equations with surface tension and gravity in the half space

In this paper, we proved decay properties of solutions to the Stokes equations with surface tension and gravity in the half space $\mathbf{R}_{+}^{N}=\{(x',x_N)\mid x'\in\mathbf{R}^{N-1},\ x_N>0\}$ $(N\geq 2)$. In order to prove the decay properties, we first show that the zero points $\lambda_\pm$ of Lopatinskii determinant for some resolvent problem associated with the Stokes equations have the asymptotics: $\lambda_\pm=\pm i c_g^{1/2}|\xi'|^{1/2} -2|\xi'|^2+O(|\xi'|^{5/2})$ as $|\xi'|\to0$, where $c_g>0$ is the gravitational acceleration and $\xi'\in\mathbf{R}^{N-1}$ is the tangential variable in the Fourier space. We next shift the integral path in the representation formula of the Stokes semi-group to the complex left half-plane by Cauchy's integral theorem, and then it is decomposed into closed curves enclosing $\lambda_\pm$ and the remainder part. We finally see, by the residue theorem, that the low frequency part of the solution to the Stokes equations behaves like the convolution of the $(N-1)$-dimensional heat kernel and $\mathcal{F}_{\xi'}^{-1}[e^{\pm i c_g^{1/2}|\xi'|^{1/2}t}](x')$ formally, where $\mathcal{F}_{\xi'}^{-1}$ is the inverse Fourier transform with respect to $\xi'$. However, main task in our approach is to show that the remainder part in the above decomposition decay faster than the residue part.

Acceleration of contour integration techniques by rational Krylov subspace methods

We suggest a rational Krylov subspace approximation for products of matrix functions and a vector appearing in exponential integrators. We consider matrices with a field-of-values in a sector lying in the left complex half-plane. The choice of the poles for our method is suggested by a fixed rational approximation based on contour integration along a hyperbola around the sector. Compared to the fixed approximation, our rational Krylov subspace method exhibits an accelerated and more stable convergence of order O(e−Cn).

Determination of the modular elliptic function in problems of free-flow filtration

The calculated dependences in elementary functions for determining the modular elliptic function λ(τ) =λ1 + iλ2 obtained on the basis of consecutive (six) conformal mappings of a curvilinear triangle to a complex half-plane are presented. Comparison of the values of λ(τ) from the proposed dependences with the results of the Hamel–Gunter exact analytical solution for the boundary contour of the curvilinear triangle, i.e., the real axis of the complex half-plane, gives a very close coincidence (with the largest error of ≤1%). The use of the complex values of the function λ(τ) for the entire internal region of the curvilinear triangle makes it possible to solve one of the most difficult problems of the theory of filtration (filtration through a rectangular dam) in the direct formulation and, for the first time, to construct the pattern of an equal filtration-rate field (the family of isotaches) over the entire internal region of the dam. In this case, the boundary values of filtration rates for special cases (along the sides and along the base of the dam) completely coincide with the results of the Masket exact analytical calculations.

Carleson measures for Hilbert spaces of analytic functions on the complex half-plane

The notion of a Carleson measure was introduced by Lennart Carleson in his proof of the Corona Theorem for H∞(D). In this paper we will define it for certain type of reproducing kernel Hilbert spaces of analytic functions of the complex half-plane, C+, which will include Hardy, Bergman and Dirichlet spaces. We will obtain several necessary or sufficient conditions for a positive Borel measure to be Carleson by preforming tests on reproducing kernels, weighted Bergman kernels, and studying the tree model obtained from a decomposition of the complex half-plane. The Dirichlet space will be investigated in detail as a special case. Finally, we will present a control theory application of Carleson measures in determining admissibility of controls in well-posed linear evolution equations.

Structure and location of branch point singularities for Stokes waves on deep water

The Stokes wave is a finite-amplitude periodic gravity wave propagating with constant velocity in an inviscid fluid. The complex analytical structure of the Stokes wave is analysed using a conformal mapping of a free fluid surface of the Stokes wave onto the real axis with the fluid domain mapped onto the lower complex half-plane. There is one square root branch point per spatial period of the Stokes wave located in the upper complex half-plane at a distance $v_{c}$ from the real axis. The increase of Stokes wave height results in $v_{c}$ approaching zero with the limiting Stokes wave formation at $v_{c}=0$. The limiting Stokes wave has a $2/3$ power-law singularity forming a $2{\rm\pi}/3$ radians angle on the crest which is qualitatively different from the square root singularity valid for arbitrary small but non-zero $v_{c}$, making the limit of zero $v_{c}$ highly non-trivial. That limit is addressed by crossing a branch cut of a square root into the second and subsequently higher sheets of the Riemann surface to find coupled square root singularities at distances $\pm v_{c}$ from the real axis at each sheet. The number of sheets is infinite and the analytical continuation of the Stokes wave into all of these sheets is found together with the series expansion in half-integer powers at singular points within each sheet. It is conjectured that a non-limiting Stokes wave at the leading order consists of an infinite number of nested square root singularities which also implies the existence in the third and higher sheets of additional square root singularities away from the real and imaginary axes. These nested square roots form a $2/3$ power-law singularity of the limiting Stokes wave as $v_{c}$ vanishes.

Reprint of “The Kalman–Yakubovich–Popov inequality for differential-algebraic systems: Existence of nonpositive solutions”

The Kalman–Yakubovich–Popov lemma is a central result in systems and control theory which relates the positive semidefiniteness of a Popov function on the imaginary axis to the solvability of a linear matrix inequality. In this paper we prove sufficient conditions for the existence of a nonpositive solution of this inequality for differential-algebraic systems. Our conditions are given in terms of positivity of a modified Popov function in the right complex half-plane. Our results also apply to non-controllable systems. Consequences of our results are bounded real and positive real lemmas for differential-algebraic systems.

Exponential decay of measures and Tauberian theorems

We study behavior of a measure on [0,∞) by considering its Laplace transform. If it is possible to extend the Laplace transform to a complex half-plane containing the imaginary axis, then the exponential decay of the tail of the measure occurs and under certain assumptions we show that the rate of the decay is given by the so called abscissa of convergence and extend the result of Nakagawa (2005) [7]. Under stronger assumptions we give behavior of density of the measure by considering its Laplace transform. In situations when there is no exponential decay we study occurrence of heavy tails and give an application in the theory of non-local equations.

Modularity of a Ramanujan–Selberg continued fraction

We study a Ramanujan–Selberg continued fraction S(τ) by employing the modular function theory. We first find modular equations of S(τ) of level n for every positive integer n by using affine models of modular curves. This is an extension of Baruah–Saikia's results for level n=3,5 and 7. We further show that the ray class field modulo 4 over an imaginary quadratic field K is obtained by the value of S2(τ), and we prove the integrality of 1/S(τ) to find its class polynomial for K with τ∈K∩H, where H is the complex upper half plane.

Note on stability conditions for structured population dynamics models

We consider a characteristic equation to analyze asymptotic stability of a scalar renewal equation, motivated by structured population dynamics models. The characteristic equation is given by 1 = Z ∞ 0 k(a)e −λa da, where k : R+ → R can be decomposed into positive and negative parts. It is shown that if delayed negative feedback is characterized by a convex function, then all roots of the characteristic equation locate in the left half complex plane.

Symbolic Dynamics, Modular Curves, and Bianchi IX Cosmologies

It is well known that the so called Bianchi IX spacetimes with SO(3)-symmetry in a neighbourhood of the Big Bang exhibit a chaotic behaviour of typical trajectories in the backward movement of time. This behaviour (Mixmaster Model of the Universe) can be encoded by the shift of two-sided continued fractions. Exactly the same shift encodes the sequences of intersections of hyperbolic geodesics with purely imaginary axis in the upper complex half-plane, that is geodesic flow on an appropriate modular surface. A physical interpretation of this coincidence was suggested in arXiv:1402.2158: namely, that Mixmaster chaos is an approximate description of the passage from a hot quantum Universe at the Big Bang moment to the cooling classical Universe. Here we discuss and elaborate this suggestion, looking at the Mixmaster Model from the perspective of the second class of Bianchi IX spacetimes: those with SU(2)-symmetry (self-dual Einstein metrics). We also extend it to the more general context related to Painleve VI equations.

A rigorous version of R.P. Brent's model for the binary Euclidean algorithm

The binary Euclidean algorithm is a modification of the classical Euclidean algorithm for computation of greatest common divisors which avoids ordinary integer division in favour of division by powers of two only. The expectation of the number of steps taken by the binary Euclidean algorithm when applied to pairs of integers of bounded size was first investigated by R.P. Brent in 1976 via a heuristic model of the algorithm as a random dynamical system. Based on numerical investigations of the expectation of the associated Ruelle transfer operator, Brent obtained a conjectural asymptotic expression for the mean number of steps performed by the algorithm when processing pairs of odd integers whose size is bounded by a large integer. In 1998 B. Vallée modified Brent's model via an induction scheme to rigorously prove an asymptotic formula for the average number of steps performed by the algorithm; however, the relationship of this result with Brent's heuristics remains conjectural. In this article we establish previously conjectural properties of Brent's transfer operator, showing directly that it possesses a spectral gap and preserves a unique continuous density. This density is shown to extend holomorphically to the complex right half-plane and to have a logarithmic singularity at zero. By combining these results with methods from classical analytic number theory we prove the correctness of three conjectured formulae for the expected number of steps, resolving several open questions promoted by D.E. Knuth in The Art of Computer Programming.

A blow-up problem for a nonlinear heat equation in the complex plane of time

Solutions of a nonlinear heat equation are numerically computed in the time variable t lying in the complex plane, and possible singularities are sought. It turns out that in the complex half plane \( \{ \mathfrak {R}[t] \ge 0 \}\), where \(\mathfrak {R}\) denotes the real part of a complex number, there is no singularity other than that which exists on the real line. However, if we compute further in the Riemann surface, new singularities are found. A certain nonlinear Schrodinger equation which is associated with our problem is also computed numerically and we propose a conjecture that it is well-posed globally in time.

Disk scattering of open and closed strings (I)

At the tree level, the scattering processes involving open and closed strings are described by a disk world-sheet with vertex operator insertions at the boundary and in the bulk. Such amplitudes can be decomposed as certain linear combinations of pure open string amplitudes. While previous relations have been established on the double cover (complex sphere) in this letter we derive them on the disk (upper complex half plane) allowing for different momenta of the left- and right-movers of the closed string. Formally, the computation of disk amplitudes involving both open and closed strings is reduced to considering the monodromies on the underlying string world-sheet.

Consensus rate regulation for general linear multi-agent systems under directed topology

Recently, optimization in distributed multi-agent coordination has been studied concerning convergence speed. The optimal convergence speed of consensus for multi-agent systems consisting of general linear node dynamics is still an open problem. This paper aims to design optimal distributed consensus protocols for general identical linear continuous time cooperative systems which not only minimize some local quadric performances, but also regulate the consensus rate (including convergence rate and damping rate) for the multi-agent systems. The graph topology is assumed to be fixed and directed. The inverse optimal design method is utilized and the resulting optimal distributed protocols place part of close-loop poles of the global disagreement systems at specified locations asymptotically, while the remains far from the imaginary axis enough. It turns out that for the identical linear continuous time multi-agent systems, the convergence speed has no upper bound. The main advantages of the developed method over the LQR design method are that the resulting multi-agent systems can achieve specified consensus rate asymptotically and the resulting protocols have the whole right half complex plane as its asymptotical consensus region. Numerical examples are given to illustrate the effectiveness of the proposed design procedures.