Simple Pole Research Articles

Eigenvalues of the Thurston operator

Let f : C ̂ → C ̂ be a postcritically finite rational map, and let Q ( C ̂ ) be the space of meromorphic quadratic differentials on C ̂ with simple poles. We study the set of eigenvalues of the pushforward operator f ∗ : Q ( C ̂ ) → Q ( C ̂ ) . In particular, we show that when f : C → C is a unicritical polynomial of degree D with periodic critical point, the eigenvalues of f ∗ : Q ( C ̂ ) → Q ( C ̂ ) are contained in the annulus 1 4 D < | λ | < 1 and belong to 1 D U where U is the group of algebraic units.

Propagation of time-resolved fluorescence in a diffuse medium: complex analytical derivation.

The spatiotemporal evolution of fluorescence in an optically diffusive medium following ultrashort laser pulse excitation is evaluated using complex analytical methods. When expressed as a Fourier integral, the integrand of the time-resolved diffuse fluorescence with embedded fluorophores is shown to exhibit branch points and simple pole singularities in the lower-half complex-frequency plane. Applying Cauchy's integral theorem to solve the Fourier integral, we calculate the time-resolved signal for fluorescence lifetimes that are both shorter and longer compared to the intrinsic absorption timescale of the medium. These expressions are derived for sources and detectors that are in the form of localized points and wide-field harmonic spatial patterns. The accuracy of the expressions derived from complex analysis is validated against the numerically computed, full time-resolved fluorescence signal. The complex analysis shows that the branch points and simple poles contribute to two physically distinct terms in the net fluorescence signal. While the branch points result in a diffusive term that exhibits spatial broadening (corresponding to a narrowing with time in the spatial Fourier domain), the simple poles lead to fluorescence decay terms with spatial/spatial-frequency distributions that are independent of time. This distinct spatiotemporal behavior between the diffuse and fluorescence signals forms the basis for direct measurement of lifetimes shorter than the intrinsic optical diffusion timescales in a turbid medium.

Focusing and defocusing mKdV equations with nonzero boundary conditions: Inverse scattering transforms and soliton interactions

We explore the inverse scattering transforms with matrix Riemann–Hilbert problems for both focusing and defocusing modified Korteweg–de Vries (mKdV) equations with non-zero boundary conditions (NZBCs) at infinity systematically. Using a suitable uniformization variable, the direct and inverse scattering problems are proposed on a complex plane instead of a two-sheeted Riemann surface. For the direct scattering problem, the analyticities, symmetries and asymptotic behaviors of the Jost solutions and scattering matrix, and discrete spectra are established. The inverse problems are formulated and solved with the aid of the matrix Riemann-Hilbert problems, and the reconstruction formulas, trace formulas, and theta conditions are also posed. In particular, we present the general solutions for the focusing mKdV equation with NZBCs and both simple and double poles, and for the defocusing mKdV equation with NZBCs and simple poles. Finally, some representative reflectionless potentials are in detail studied to illustrate distinct nonlinear wave structures containing solitons and breathers for both focusing and defocusing mKdV equations with NZBCs.

Renormalons in integrated spectral function moments and αs extractions

Precise extractions of $\alpha_s$ from $\tau\to {\rm (hadrons)}+\nu_\tau$ and from $e^+e^-\to {\rm (hadrons)}$ below the charm threshold rely on finite energy sum rules (FESRs) where the experimental side is given by integrated spectral function moments. Here we study the renormalons that appear in the Borel transform of polynomial moments in the large-$\beta_0$ limit and in full QCD. In large-$\beta_0$, we establish a direct connection between the renormalons and the perturbative behaviour of moments often employed in the literature. The leading IR singularity is particularly prominent and is behind the fate of moments whose perturbative series are unstable, while those with good perturbative behaviour benefit from partial cancellations of renormalon singularities. The conclusions can be extended to QCD through a convenient scheme transformation to the $C$-scheme together with the use of a modified Borel transform which make the results particularly simple; the leading IR singularity becomes a simple pole, as in large-$\beta_0$. Finally, for the moments that display good perturbative behaviour, we discuss an optimized truncation based on renormalisation scheme (or scale) variation. Our results allow for a deeper understanding of the perturbative behaviour of integrated spectral function moments and provide theoretical support for low-$Q^2$ $\alpha_s$ determinations.

Laplace domain method for computing the transient response of floating structures to irregular waves

Traditional methods for evaluating the dynamic response of floating structures to irregular waves have been performed in either the frequency or the time domain. Whereas the computed response from the frequency-domain method is merely the steady-state response, implementing the time-domain method is usually time-consuming. In this article, an efficient pole–residue method operated in the Laplace domain to compute transient response of floating structures to irregular waves is developed, given that the frequency-domain data has been obtained from running a standard hydrodynamic package. The proposed method requires both the irregular waves and the transfer functions associated with the wave-structure system be in their pole–residue forms. From those, pole–residue form of the response can be obtained by simple pole–residue operations. Once the poles and residues of the response are known, the response time history is readily obtained. In the numerical studies, this article considers a SDOF model for the heave motion of a spar platform excited by regular and irregular waves, respectively. Numerical studies demonstrated that the proposed method is more efficient and accurate than the traditional time-domain method.

The cosmological bootstrap: inflationary correlators from symmetries and singularities

Scattering amplitudes at weak coupling are highly constrained by Lorentz invariance, locality and unitarity, and depend on model details only through coupling constants and the particle content of the theory. For example, four-particle amplitudes are analytic for contact interactions and have simple poles with appropriately positive residues for tree-level exchange. In this paper, we develop an understanding of inflationary correlators which parallels that of flat-space scattering amplitudes. Specifically, we study slow-roll inflation with weak couplings to extra massive particles, for which all correlation functions are controlled by an approximate conformal symmetry on the boundary of the spacetime. After systematically classifying all possible contact terms in de Sitter space, we derive an analytic expression for the four-point function of conformally coupled scalars mediated by the tree-level exchange of massive scalars. Conformal symmetry implies that the correlator satisfies a pair of differential equations with respect to spatial momenta, encoding bulk time evolution in purely boundary terms. The absence of unphysical singularities (and the correct normalization of physical ones) completely fixes this correlator. Moreover, a “spin-raising” operator relates it to the correlators associated with the exchange of particles with spin, while “weight-shifting” operators map it to the four-point function of massless scalars. We explain how these de Sitter four-point functions can be perturbed to obtain inflationary three-point functions. Using our formalism, we reproduce many classic results in the literature, such as the three-point function of slow-roll inflation, and provide a complete classification of all inflationary three- and four-point functions arising from weakly broken conformal symmetry. Remarkably, the inflationary bispectrum associated with the exchange of particles with arbitrary spin is completely characterized by the soft limit of the simplest scalar-exchange four-point function of conformally coupled scalars and a series of contact terms. Finally, we demonstrate that the inflationary correlators contain flat-space scattering amplitudes via a suitable analytic continuation of the external momenta, which can also be directly connected with the signals for particle production seen in the squeezed limit.

BMS symmetry of celestial OPE

In this paper we study the BMS symmetry of the celestial OPE of two positive helicity gravitons in Einstein theory in four dimensions. The celestial OPE is obtained by Mellin transforming the scattering amplitude in the (holomorphic) collinear limit. The collinear limit at leading order gives the singular term of the celestial OPE. We compute the first subleading correction to the OPE by analysing the four graviton scattering amplitude directly in Mellin space. The subleading term can be written as a linear combination of BMS descendants with the OPE coefficients determined by BMS algebra and the coefficient of the leading term in the OPE. This can be done by defining a suitable BMS primary state. We find that among the descendants, which appear at the first subleading order, there is one which is created by holomorphic supertranslation with simple pole on the celestial sphere.

Dubrovin equation for periodic Dirac operator on the half-line

ABSTRACT We consider the Dirac operator with a periodic potential on the half-line with the Dirichlet boundary condition at zero. Its spectrum consists of an absolutely continuous part plus at most one eigenvalue in each open gap. The resolvent admits a meromorphic continuation onto a two-sheeted Riemann surface with a unique simple pole on each open gap: on the first sheet (an eigenvalue) or on the second sheet (a resonance). These poles are called levels and there are no other poles. If the potential is shifted by real parameter t, then the continuous spectrum does not change but the levels can change their positions. We prove that each level is smooth and in general, non-monotonic function of t. We prove that a level is a strictly monotone function of t for a specific potential. Using these results, we obtain formulas to recover potentials of special forms.

Solitons and rogue waves of the quartic nonlinear Schrödinger equation by Riemann–Hilbert approach

We systematically develop a Riemann–Hilbert approach for the quartic nonlinear Schrodinger equation on the line with both zero boundary condition and nonzero boundary conditions at infinity. For zero boundary condition, the associated Riemann–Hilbert problem is related to two cases of scattering data: N simple poles and one Nth-order pole, which allows us to find the exact formulae of soliton solutions. In the case of nonzero boundary conditions and initial data that allow for the presence of discrete spectrum, the pure one-soliton solution and rogue waves are presented. The important advantage of this method is that one can study the long-time asymptotic behavior of the solutions and the infinite order rogue waves based on the associated Riemann–Hilbert problems.

Masur–Veech volumes, frequencies of simple closed geodesics, and intersection numbers of moduli spaces of curves

We express the Masur–Veech volume and the area Siegel–Veech constant of the moduli space Qg,n of genus g meromorphic quadratic differentials with at most n simple poles and no other poles as polynomials in the intersection numbers ∫M‾ g′,n′ψ1d1⋯ψ n′dn′ with explicit rational coefficients, where g′<g and n′<2g+n. The formulas obtained in this article are derived from lattice point counts involving the Kontsevich volume polynomials N g′,n′(b1,…,bn′) that also appear in Mirzakhani’s recursion for the Weil–Petersson volumes of the moduli spaces Mg′,n′(b1,…,bn′) of bordered hyperbolic surfaces with geodesic boundaries of lengths b1,…,bn′. A similar formula for the Masur–Veech volume (but without explicit evaluation) was obtained earlier by Mirzakhani through a completely different approach. We prove a further result: the density of the mapping class group orbit Modg,n⋅γ of any simple closed multicurve γ inside the ambient set MLg,n(Z) of integral measured laminations, computed by Mirzakhani, coincides with the density of square-tiled surfaces having horizontal cylinder decomposition associated to γ among all square-tiled surfaces in Qg,n. We study the resulting densities (or, equivalently, volume contributions) in more detail in the special case when n=0. In particular, we compute explicitly the asymptotic frequencies of separating and nonseparating simple closed geodesics on a closed hyperbolic surface of genus g for all small genera g, and we show that in large genera the separating closed geodesics are 2 3πg⋅1 4g times less frequent.

Zeta-funkcija i neke njene osobine

Introduction/purpose: Some properties of the zeta function will be shown as well as its applications in calculus, in particular the "golden nugget formula" for the value of the infinite sum 1 + 2 + 3 + · · · . Some applications in physics will also be mentioned. Methods: Complex plane integrations and properties of the Gamma function will be used from the definition of the function to its analytic extension. Results: From the original definition of the z(s) function valid for s &gt; 1 a meromorphic function is obtained on the whole complex plane with a simple pole in s = 1. Conclusion: The relevance of the zeta function cannot be overstated, ranging from the infinite series to the number theory, regularization in theoretical physics, the Casimir force, and many other fields.

Spin Effect and Stokes Phenomenon for Fermion Production in Electric Fields

One of the intriguing features of (Sauter-)Schwinger pair production is the Stokes phenomenon of no-pair production in some bipolar electric fields. The spin effect is studied on fermion pair production in the same electric fields of New Physics (Sae Mulli) 68, 1225 (2018), in which pair production of bosons is explained in terms of polons, simple poles with complex frequency. The Stokes phenomenon is showed to occur for specific parameters when the electric field is bipolar and changes the direction of antisymmetry in time. We compare the pair production of bosons and fermions and discuss its physical implications.

Looking for a compact semi empirical equation of state for hard spheres and the possibility of a glassy transition

The equation of state of a hard sphere fluid at high density should exhibit a simple pole at the random close packing limit. Here we show that trying to obtain a compact semi-empirical equation of state simultaneously compatible with that asymptotic behaviour and with the known virial coefficients raises an analytical difficulty which can be solved if a glassy transition occurs in the disordered metastable phase at a density intermediate between the freezing point and the random close packing limit. The comparison of the estimated value for the transition point with earlier estimations suggests a more subtle situation.

On the cohomology of the moduli space of parabolic connections

We study the moduli space of logarithmic connections of rank 2 on $${\mathbb {P}}^1 {\setminus } \{ t_1, \dots , t_5 \}$$ with fixed spectral data. The aim of this paper is to compute the cohomology of this space, a computation that will be used to extend the results of the Geometric Langlands Correspondence due to D. Arinkin to the case where these types of connections have five simple poles on $${\mathbb {P}}^1$$ .

Principle of Torque Ripple Reduction in Synchronous Reluctance Motors With Shifted Asymmetrical Poles

In this paper, the principle of torque ripple reduction in synchronous reluctance motors (SynRMs) with shifted asymmetrical poles is revealed. The key of this principle is that the asymmetrical poles and shifted pole pairs are combined to reduce the torque ripple while their functions are different. The adjacent poles are designed as asymmetrical to suppress the second-order harmonic of torque ripple. Nevertheless, two asymmetrical poles are selected as a repeating unit, and then, the half of repeating units of the SynRM are shifted to eliminate the main order harmonic of torque ripple. In order to obtain the principle of torque ripple reduction, a simple five-phase SynRM with salient poles is proposed and the analytical model is built. Then, the asymmetrical angle and shifted angle to reduce the second-order and main order harmonics of torque ripple are revealed by the analytical method, respectively. Furthermore, the revealed principle can be extended into SynRMs with multilayer flux barriers or other combinations of stator slots and rotor poles. In order to evaluate the revealed torque ripple reduction principle, a prototype with simple salient poles is manufactured for experiments. Through the experimental test, it is verified that the revealed principle is feasible and effective.

On harmonic univalent mappings with nonzero pole

In this article we consider the class AH(p) of all sense preserving harmonic functions f in the open unit disc D having a simple pole at z=p∈(0,1) with the normalizations f(0)=fz(0)−1=0. We first derive a sufficient condition for univalence of such functions. Next we consider the class SH0(p) which consists of all functions belonging to AH(p) and are univalent with the additional normalization fz¯(0)=0. We study the class SH0(p) in the geometric function theoretic viewpoint. As a byproduct of our investigation, we see that consideration of nonzero pole yields nontrivial lower bounds for the Taylor coefficients of such functions. We also prove a growth result for a non-vanishing subclass associated with SH0(p).

Details of Krakow’s Modernist townhouses

The aim of this paper is to present the key features of façade details typical for townhouses erected in Krakow in the period 1918-1939. It emphasizes the importance of the townhouse details for Krakow’s cultural heritage and the pressing need to document them. Since the funds of the Social Committee for the Renovation of Krakow’s Monuments are allocated in the first priority to overhauls of public utility buildings and registered monuments, the modernist buildings have little, if any, chances for renovation. Meanwhile, they continue to fall into disrepair, which makes the documentation efforts even more urgent. Only a few most precious buildings from the interwar period are listed in the register of monuments. The remaining ones are not awarded by appropriate legal protection. Based on the analysis of the influences of a variety of trends on the development of Krakow’s townhouse façades and their details in the period 1918-1939, three basic groups of façades were identified. The first one relied on the historic style, the second, and most numerous one, combined historic and functionalist forms, while the third one emerged under explicit influence of functionalism. Key elements of the historic stylistics include: symmetry and division of the façade into three parts (bossage on the wall base, vertically articulated main part, finial), strong emphasis on the entrance portal, conspicuous cornices and decorations. In townhouses combining historic and functional forms, the prevailing stylistic elements include simple poles, lesenes (often worked in the plaster texture), enhanced with decorations (coats of arms, friezes, etc.). The ornamental-stylistic current typical of the ‘Krakow school’ comes in several different iterations. Few townhouses were explicitly influenced by the European functionalism. Designers aimed at simplifying the detail forms and their geometrization, producing an effect that revealed more deeply the functional content of the mass. The functionalist current in Krakow differed artistically from the avant-garde European projects. Nonetheless, it shared similar rationalist objectives – usability, economy and simplicity. Façade details often served as a link between contemporaneity and previous periods; they took account of features typical of the Krakow’s historic architecture.

Study of vector charmonium electromagnetic Dalitz decays

Using the largest [Formula: see text] and [Formula: see text] data samples in the world, BESIII has measured some charmonium electromagnetic Dalitz decays. Stimulated by the recent experimental results from BESIII, we study the electromagnetic Dalitz decays of [Formula: see text] and [Formula: see text], based on the pointlike QED theory and the simple pole approximation theory, and we present the numerical values of the branching fractions. With the available large samples of [Formula: see text] and [Formula: see text] events collected with the BESIII detector, the expected events for these electromagnetic Dalitz decays are estimated, which offers a unique opportunity to check the theoretical predictions and to measure the transition form factor (TFF). The TFF could characterize the electromagnetic structure arising at the vertex of the transition, and serve as a sensitive probe to the internal structures. These results could be useful for the data taking plan in the future at BESIII.

WKB asymptotics of meromorphic solutions to difference equations

We consider the difference Schrödinger equation where z is a complex variable, E is a spectral parameter, and h is a small positive parameter. If the potential v is an analytic function, then, for h sufficiently small, the analytic solutions to this equation have standard semi-classical behavior that can be described by means of an analog of the complex WKB method for differential equations. We assume that v has a simple pole and, in its neighborhood, we study the asymptotics of meromorphic solutions to the difference Schrödinger equation.

Study of the electromagnetic Dalitz decays ψ(ϒ)→ηc(ηb)l+l−

We study the electromagnetic Dalitz (EM) decays, $\psi \to \eta_{c} l^{+} l^{-}$ and $\Upsilon \to \eta_{b} l^{+} l^{-}$ ($l = e$ or $\mu$), in which the lepton pair comes from the virtual photon emitted by the M1 transition from $c\bar{c}$ ($b\bar{b}$) spin triplet state to the spin singlet state. We estimate the partial width of $\psi(\Upsilon) \to \eta_{c}(\eta_{b}) l^{+} l^{-}$, based on the simple pole approximation. Besides, based on different QCD models, the partial width of $\psi(\Upsilon) \to \eta_{c}(\eta_{b}) \gamma$ is determined.