Entire Complex Plane Research Articles

On L p -Resolvent Estimates and the Density of Eigenvalues for Compact Riemannian Manifolds

We address an interesting question raised by Dos Santos Ferreira, Kenig and Salo about regions ${\mathcal R}_g\subset {\mathbb C}$ for which there can be uniform $L^{\frac{2n}{n+2}}\to L^{\frac{2n}{n-2}}$ resolvent estimates for $\Delta_g+\zeta$, $\zeta \in {\mathcal R}_g$, where $\Delta_g$ is the Laplace-Beltrami operator with metric $g$ on a given compact boundaryless Riemannian manifold of dimension $n\ge3$. This is related to earlier work of Kenig, Ruiz and the third author for the Euclidean Laplacian, in which case the region is the entire complex plane minus any disc centered at the origin. Presently, we show that for the round metric on the sphere, $S^n$, the resolvent estimates in Ferreira et al, involving a much smaller region, are essentially optimal. We do this by establishing sharp bounds based on the distance from $\zeta$ to the spectrum of $\Delta_{S^n}$. In the other direction, we also show that the bounds in \cite{Kenig} can be sharpened logarithmically for manifolds with nonpositive curvature, and by powers in the case of the torus, ${\mathbb T}^n={\mathbb R}^n/{\mathbb Z}^n$, with the flat metric. The latter improves earlier bounds of Shen. Further improvements for the torus are obtained using recent techniques of the first author and his work with Guth based on the multilinear estimates of Bennett, Carbery and Tao. Our approach also allows us to give a natural necessary condition for favorable resolvent estimates that is based on a measurement of the density of the spectrum of $\sqrt{-\Delta_g}$, and, moreover, a necessary and sufficient condition based on natural improved spectral projection estimates for shrinking intervals.

One dimensional acoustic direct nonlinear inversion using the Volterra inverse scattering series

Direct inversion of acoustic scattering problems is nonlinear. One way to treat the inverse scattering problem is based on the reversion of the Born–Neumann series solution of the Lippmann–Schwinger equation. An important issue for this approach is the radius of convergence of the Born–Neumann series for the forward problem. However, this issue can be tackled by employing a renormalization technique to transform the Lippmann–Schwinger equation from a Fredholm to a Volterra integral form. The Born series of a Volterra integral equation converges absolutely and uniformly in the entire complex plane. We present a further study of this new mathematical framework. A Volterra inverse scattering series (VISS) using both reflection and transmission data is derived and tested for several acoustic velocity models. For large velocity contrast, series summation techniques (e.g., Cesàro summation, Euler transform, etc) are employed to improve the rate of convergence of VISS. It is shown that the VISS method with summation techniques can provide a relatively good estimation of the velocity profile. The method is fully data-driven in the respect that no prior information of the model is required. Besides, no internal multiple removal is needed. This one dimensional VISS approach is useful for inverse scattering and serves as an important step for studying more complicated and realistic inversions.

PHASE SHIFTING HOLOGRAPHY FOR THZ NEAR-FIELD/FAR-FIELD PREDICTION

With a view to extending techniques for THz antenna near-fleld/far-fleld prediction, this communication derives general analytic expressions for calibrated phase shifting holography (PSH) and introduces a new 120 - three-step PSH method that avoids switching ofi the reference fleld and has symmetrical performance over the entire complex plane, providing spurious free far-fleld prediction. Numerical tests with simulated near-fleld patterns at 372GHz conflrm the convenience of the method and give an indication of the precision required for the phase shifts.

Analytic continuation of the doubly-periodic Barnes zeta function

The aim of this work is to study the analytic continuation of the doubly-periodic Barnes zeta function. By using a suitable complex integral representation as a starting point we find the meromorphic extension of the doubly periodic Barnes zeta function to the entire complex plane in terms of a real integral containing the Hurwitz zeta function and the first Jacobi theta function. These allow us to explicitly give expressions for the derivative at all non-positive integer points.

Absence of wandering domains for some real entire functions with bounded singular sets

Let f be a real entire function whose set S(f) of singular values is real and bounded. We show that, if f satisfies a certain function-theoretic condition (the "sector condition"), then $f$ has no wandering domains. Our result includes all maps of the form f(z)=\lambda sinh(z)/z + a, where a is a real constant and {\lambda} is positive. We also show the absence of wandering domains for certain non-real entire functions for which S(f) is bounded and the iterates of f tend to infinity uniformly on S(f). As a special case of our theorem, we give a short, elementary and non-technical proof that the Julia set of the complex exponential map f(z)=e^z is the entire complex plane. Furthermore, we apply similar methods to extend a result of Bergweiler, concerning Baker domains of entire functions and their relation to the postsingular set, to the case of meromorphic functions.

Anosov flows and dynamical zeta functions

We study the Ruelle and Selberg zeta functions for C r Anosov ows, r > 2, on a compact smooth manifold. We prove several results, the most remarkable being (a) for C 1 ows the zeta function is meromorphic on the entire complex plane; (b) for contact ows satisfying a bunching condition (e.g., geodesic ows on manifolds of negative curvature better than 1 9 -pinched), the zeta function has a pole at the topological entropy and is analytic in a strip to its left; (c) under the same hypotheses as in (b) we obtain sharp results on the number of periodic orbits. Our arguments are based on the study of the spectral properties of a transfer operator acting on suitable Banach spaces of anisotropic currents.

On the Roots of Domination Polynomials

The domination polynomial of a graph G of order n is the polynomial $${D(G, x) = \sum_{i=\gamma(G)}^{n} d(G, i)x^i}$$ where d(G, i) is the number of dominating sets of G of size i, and ?(G) is the domination number of G. We investigate here domination roots, the roots of domination polynomials. We provide an explicit family of graphs for which the domination roots are in the right half-plane. We also determine the limiting curves for domination roots of complete bipartite graphs. Finally, we prove that the closure of the roots is the entire complex plane.

Graphing the Complex Roots of Quadratic Functions on a Three Dimensional Coordinate Space

We are familiar with the geometric definition of Real roots of a Quadratic function as the x-intercept of the Quadratic functions graph; however such a geometric definition is not given for the Complex roots of a Quadratic function. This paper geometrically defines complex numbers as solution of Quadratic Equations. Inorder to achieve this a new coordinate space is defined where given a quadratic function with complex solution we can geometrically plot complex solutions to it, in this new coordinate frame. Thus the solution to any Quadratic equation with complex solutions can be derived geometrically. Also this coordinate frame will define the entire Complex Plane as a Solution Space of Quadratic functions with complex roots. The paper further provides ideas and conjures to extend the research into polynomials of degree higher than two i.e. the cubic polynomials and higher. The author hopes that further research could develop the method to geometrically derive complex solutions to cubic and higher degree polynomials, and the complex numbers will no more be fictitious quantities but quantities with rigorous geometric definitions.

The meromorphic continuation of the zeta function of a product of Hilbert and Picard modular surfaces over CM-fields

In this paper we prove in particular the meromorphic continuation to the entire complex plane of the zeta function of a product of a Hilbert modular surface and a Picard modular surface regarded over arbitrary CM-fields (see the end of the Introduction for more results regarding the meromorphic continuation of the zeta functions associated to products of Shimura curves, quaternionic Shimura surfaces and Picard modular surfaces). In order to obtain the meromorphic continuation we show the simultaneous potential modularity for an arbitrary finite number of l -adic representations associated to Hilbert modular forms and Picard modular surfaces.

Dynamical rigidity of transcendental meromorphic functions

We prove the form of dynamical rigidity of transcendental meromorphic functions which asserts that if two tame transcendental meromorphic functions restricted to their Julia sets are topologically conjugate via a locally bi-Lipschitz homeomorphism, then they, treated as functions defined on the entire complex plane , are topologically conjugate via an affine map, i.e. a map from to of the form z ↦ az + b. As an intermediate step we show that no tame transcendental meromorphic function is essentially affine.

The Green’s function in a channel with a sound-absorbing cover in the case of a uniform flow

We study the modal structure of an acoustic field of a point source as function of channel wall admittance in the case of a two-dimensional channel. The characteristic equation for determining the eigen-values corresponding to the boundary problem is studied in the form of this equation’s dependence on the admittance, which varies in the entire complex plane. All modes, without exception, existing in the channel and forming the source field are classified based on the obtained topography of the characteristic equation. The expressions that describe the amplitudes and spatial distribution of the hydrodynamic modes, attenuation rate (for stable modes), or increment (for unstable modes) were obtained as functions of the wall admittance and flow velocity. It is shown that in addition to the hydrodynamic unstable modes existing downstream from the source, hydrodynamic unstable modes exist upstream from the source at any admittance. They appear only when the admittance has an elastic character. It is shown that hydrodynamic modes are induced only in the case when the source is located close to the wall or on the wall. The amplitude of these modes decreases exponentially with distance from the wall.

Efficient algorithmic implementation of the Voigt/complex error function based on exponential series approximation

We show that a Fourier expansion of the exponential multiplier yields an exponential series that can compute high-accuracy values of the complex error function in a rapid algorithm. Numerical error analysis and computational test reveal that with essentially higher accuracy it is as fast as FFT-based Weideman’s algorithm at a regular size of the input array and considerably faster at an extended size of the input array. As this exponential series approximation is based only on elementary functions, the algorithm can be implemented utilizing freely available functions from the standard libraries of most programming languages. Due to its simplicity, rapidness, high-accuracy and coverage of the entire complex plane, the algorithm is efficient and practically convenient in numerical methods related to the spectral line broadening and other applications requiring error-function evaluation over extended input arrays.

Linear twists of L-functions of degree 2 from the Selberg class

In 1983, S. M. Voronin obtain an analytic continuation to the entire complex plane for some twists of L-functions associated with holomorphic modular forms on SL(2, Z). In this paper, we extend Voronin’s result to L-functions of degree 2 from the extended Selberg class.

New series for the cosine lemniscate function and the polynomialization of the lemniscate integral

We discuss the numerical computation of the cosine lemniscate function and its inverse, the lemniscate integral. These were previously studied by Bernoulli, Euler, Gauss, Abel, Jacobi and Ramanujan. We review general elliptic formulas for this special case and provide some novelties. We show that a Fourier series by Ramanujan converges twice as fast as the standard elliptic cosine Fourier series specialized to this case. The so-called imbricate series, however, converges geometrically fast over the entire complex plane. We derive two new expansions. The rational-plus-Fourier series converges much faster than Ramanujan’s: for real z : each term is asymptotically 12,400 times smaller than its immediate predecessor: coslem ( z ) = 4 B { q ( 1 − q ) cos ( B z ) / [ ( 1 + q ) 2 − 4 q cos 2 ( B z ) ] + ∑ n = 1 ∞ q n − 1 / 2 { 1 / ( 1 + q 2 n − 1 ) − 1 } cos ( ( 2 n − 1 ) B z ) } where q = exp ( − π ) is the elliptic nome, K ≈ 1.85 … is the complete elliptic integral of the first kind for a modulus m = 1 / 2 and B = π / ( K 2 ) . The rational imbricate series is uniformly valid over the complex plane, but converges twice as fast as the sech-imbricate series: coslem ( z ) = 4 B q ( 1 − q ) ∑ j = − ∞ ∞ q ( 1 − q ) cos ( B z ) / { ( 1 + q ) 2 − 4 q cos 2 ( B [ z − j P i ] ) } where P = ( 4 / 2 ) K is the period in both the real and imaginary directions. We devise a new approximation for the lemniscate integral for real argument as the arccosine of a Chebyshev series and show that 17 terms yield about 15 digits of accuracy. For complex argument, we show that the lemniscate integral can be found to near machine precision (assumed as sixteen decimal digits) by computing the roots of a polynomial of degree thirteen. Alternatively, Newton’s iteration converges in three iterations with an initialization, accurate to four decimal places, that is the chosen root of a cubic equation.

ON MAZUR'S CONJECTURE FOR TWISTED L-FUNCTIONS OF ELLIPTIC CURVES OVER TOTALLY REAL OR CM FIELDS

Let E be an elliptic curve defined over a number field F, and let Σ be a finite set of finite places of F. Let L(s, E, ψ) be the L-function of E twisted by a finite-order Hecke character ψ of F. It is conjectured that L(s, E, ψ) has a meromorphic continuation to the entire complex plane and satisfies a functional equation s ↔ 2 − s. Then one can define the so called minimal order of vanishing ats = 1 of L(s, E, ψ), denoted by m(E, ψ) (see Section 2 for the definition).

Analytic continuation of weighted Bergman kernels

We show that the Bergman kernel Kα(x,y) on a smoothly bounded strictly pseudoconvex domain with respect to the weight ρα, where −ρ is a defining function and α>−1, extends meromorphically in α to the entire complex plane. This is somewhat reminiscent of scattering poles or resonances in scattering theory. With a small change, the assertion remains valid also for functions ρ of slightly more general type.

Probability and complex quantum trajectories: Finding the missing links

It is shown that a normalisable probability density can be defined for the entire complex plane in the modified de Broglie–Bohm quantum mechanics, which gives complex quantum trajectories. This work is in continuation of a previous one that defined a conserved probability for most of the regions in the complex space in terms of a trajectory integral, indicating a dynamical origin of quantum probability. There it was also shown that the quantum trajectories obtained are the same characteristic curves that propagate information about the conserved probability density. Though the probability density we now adopt for those regions left out in the previous work is not conserved locally, the net source of probability for such regions is seen to be zero in the example considered, allowing to make the total probability conserved. The new combined probability density agrees with the Born’s probability everywhere on the real line, as required. A major fall out of the present scheme is that it explains why in the classical limit the imaginary parts of trajectories are not observed even indirectly and particles are confined close to the real line.

Analytic Continuation of Dirichlet Series with Almost Periodic Coefficients

We consider Dirichlet series \({\zeta_{g,\alpha}(s)=\sum_{n=1}^\infty g(n\alpha) e^{-\lambda_n s}}\) for fixed irrational α and periodic functions g. We demonstrate that for Diophantine α and smooth g, the line Re(s) = 0 is a natural boundary in the Taylor series case λn = n, so that the unit circle is the maximal domain of holomorphy for the almost periodic Taylor series \({\sum_{n=1}^{\infty} g(n\alpha) z^n}\). We prove that a Dirichlet series \({\zeta_{g,\alpha}(s) = \sum_{n=1}^{\infty} g(n \alpha)/n^s}\) has an abscissa of convergence σ0 = 0 if g is odd and real analytic and α is Diophantine. We show that if g is odd and has bounded variation and α is of bounded Diophantine type r, the abscissa of convergence σ0 satisfies σ0 ≤ 1 − 1/r. Using a polylogarithm expansion, we prove that if g is odd and real analytic and α is Diophantine, then the Dirichlet series ζg,α(s) has an analytic continuation to the entire complex plane.

On L-functions of twisted 4-dimensional quaternionic Shimura varieties

In this article we prove the meromorphic continuation to the entire complex plane and also the functional equation of the zeta functions of twisted 4-dimensional quaternionic Shimura varieties in quite general cases.

Non-solvable base change for Hilbert modular representations and zeta functions of twisted quaternionic Shimura varieties

In this paper we prove some non-solvable base change for Hilbert modular representations, and we use this result to show the meromorphic continuation to the entire complex plane of the zeta functions of some twisted quaternionic Shimura varieties. The zeta functions of the twisted quaternionic Shimura varieties are computed at all places.