Essential Singularity Research Articles

A Lehto–Virtanen-type theorem and a rescaling principle for an isolated essential singularity of a holomorphic curve in a complex space

We establish a Lehto–Virtanen-type theorem and a rescaling principle for an isolated essential singularity of a holomorphic curve in a complex space, which are useful for establishing a big Picard-type theorem and a big Brody-type one for holomorphic curves.

Generalizations of the abstract boundary singularity theorem

The abstract boundary singularity theorem was first proven by Ashley and Scott. It links the existence of incomplete causal geodesics in strongly causal, maximally extended spacetimes to the existence of abstract boundary essential singularities, i.e., non-removable singular boundary points. We give two generalizations of this theorem: the first to continuous causal curves and the distinguishing condition, the second to locally Lipschitz curves in manifolds such that no inextendible locally Lipschitz curve is totally imprisoned. To do this we extend generalized affine parameters from C1 curves to locally Lipschitz curves.

Quasi-one-dimensional pair density wave superconducting state

We provide a quasi-one-dimensional model which can support a pair-density-wave (PDW) state, in which the superconducting (SC) order parameter modulates periodically in space, with gapless Bogoliubov quasiparticle excitations. The model consists of an array of strongly-interacting one-dimensional systems, where the one-dimensional systems are coupled to each other by local interactions and tunneling of the electrons and Cooper pairs between them. Within the interchain mean-field theory, we find several SC states from the model, including a conventional uniform SC state, PDW SC state, and a coexisting phase of the uniform SC and PDW states. In this quasi-1D regime we can treat the strong correlation physics essentially exactly using bosonization methods and the crossover to the 2D system by means of interchain mean field theory. The resulting critical temperatures of the SC phases generically exhibit a power-law scaling with the coupling constants of the array, instead of the essential singularity found in weak-coupling BCS-type theories. Electronic excitations with an open Fermi surface, which emerge from the electronic Luttinger liquid systems below their crossover temperature to the Fermi liquid, are then coupled to the SC order parameters via the proximity effect. From the Fermi surface thus coupled to the SC order parameters, we calculate the quasiparticle spectrum in detail. We show that the quasiparticle spectrum can be fully gapped or nodal in the uniform SC phase and also in the coexisting phase of the uniform SC and PDW parameters. In the pure PDW state, the excitation spectrum has a reconstructed Fermi surface in the form of Fermi pockets of Bogoliubov quasiparticles.

Zeros of combinations of the Riemann ξ-function on bounded vertical shifts

In this paper we consider a series of bounded vertical shifts of the Riemann ξ-function. Interestingly, although such functions have essential singularities, infinitely many of their zeros lie on the critical line. We also generalize some integral identities associated with the theta transformation formula and some formulae of G.H. Hardy and W.L. Ferrar in the context of a pair of functions reciprocal in Fourier cosine transform.

Rescaling principle for isolated essential singularities of quasiregular mappings

We establish a rescaling theorem for isolated essential singularities of quasiregular mappings. As a consequence we show that the class of closed manifolds receiving a quasiregular mapping from a punctured unit ball with an essential singularity at the origin is exactly the class of closed quasiregularly elliptic manifolds, that is, closed manifolds receiving a non-constant quasiregular mapping from a Euclidean space.

Reexamination of the nonperturbative renormalization-group approach to the Kosterlitz-Thouless transition.

We reexamine the two-dimensional linear O(2) model (φ4 theory) in the framework of the nonperturbative renormalization-group. From the flow equations obtained in the derivative expansion to second order and with optimization of the infrared regulator, we find a transition between a high-temperature (disordered) phase and a low-temperature phase displaying a line of fixed points and algebraic order. We obtain a picture in agreement with the standard theory of the Kosterlitz-Thouless (KT) transition and reproduce the universal features of the transition. In particular, we find the anomalous dimension η(T(KT))≃0.24 and the stiffness jump ρ(s)(T(KT)(-))≃0.64 at the transition temperature T(KT), in very good agreement with the exact results η(T(KT))=1/4 and ρ(s)(T(KT)(-))=2/π, as well as an essential singularity of the correlation length in the high-temperature phase as T→T(KT).

A Numerical Test of Padé Approximation for Some Functions with Singularity

The aim of this study is to examine some numerical tests of Padé approximation for some typical functions with singularities such as simple pole, essential singularity, brunch cut, and natural boundary. As pointed out by Baker, it was shown that the simple pole and the essential singularity can be characterized by the poles of the Padé approximation. However, it was not fully clear how the Padé approximation works for the functions with the branch cut or the natural boundary. In the present paper, it is shown that the poles and zeros of the Padé approximated functions are alternately lined along the branch cut if the test function has branch cut, and poles are also distributed around the natural boundary for some lacunary power series and random power series which rigorously have a natural boundary on the unit circle. On the other hand, Froissart doublets due to numerical errors and/or external noise also appear around the unit circle in the Padé approximation. It is also shown that the residue calculus for the Padé approximated functions can be used to confirm the numerical accuracy of the Padé approximation and quasianalyticity of the random power series.

DMRG study of the Berezinskii–Kosterlitz–Thouless transitions of the 2D five-state clock model

The two Berezinskii–Kosterlitz–Thouless phase transitions of the two-dimensional 5-state clock model are studied on infinite strips using the DMRG algorithm. Because of the open boundary conditions, the helicity modulus ϒ2 is computed by imposing twisted magnetic fields at the two boundaries. Its scaling behavior is in good agreement with the existence of essential singularities with σ = 1/2 at the two transitions. The predicted universal values of ϒ2 are shown to be reached in the thermodynamic limit. The fourth-order helicity modulus is observed to display a dip at the high-temperature BKT transition, like the XY model, and shown to take a new universal value at the low-temperature BKT transition. Finally, the scaling behavior of magnetization at the low-temperature transition is compatible with η = 1/4.

Prestack exploding reflector modelling and migration for anisotropic media

ABSTRACTThe double‐square‐root equation is commonly used to image data by downward continuation using one‐way depth extrapolation methods. A two‐way time extrapolation of the double‐square‐root‐derived phase operator allows for up and downgoing wavefields but suffers from an essential singularity for horizontally travelling waves. This singularity is also associated with an anisotropic version of the double‐square‐root extrapolator. Perturbation theory allows us to separate the isotropic contribution, as well as the singularity, from the anisotropic contribution to the operator. As a result, the anisotropic residual operator is free from such singularities and can be applied as a stand alone operator to correct for anisotropy. We can apply the residual anisotropy operator even if the original prestack wavefield was obtained using, for example, reverse‐time migration. The residual correction is also useful for anisotropic parameter estimation. Applications to synthetic data demonstrate the accuracy of the new prestack modelling and migration approach. It also proves useful in approximately imaging the Vertical Transverse Isotropic Marmousi model.

Source–receiver two‐way wave extrapolation for prestack exploding‐reflector modelling and migration

ABSTRACTMost modern seismic imaging methods separate input data into parts (shot gathers). We develop a formulation that is able to incorporate all available data at once while numerically propagating the recorded multidimensional wavefield forward or backward in time. This approach has the potential for generating accurate images free of artiefacts associated with conventional approaches. We derive novel high‐order partial differential equations in the source–receiver time domain. The fourth‐order nature of the extrapolation in time leads to four solutions, two of which correspond to the incoming and outgoing P‐waves and reduce to the zero‐offset exploding‐reflector solutions when the source coincides with the receiver. A challenge for implementing two‐way time extrapolation is an essential singularity for horizontally travelling waves. This singularity can be avoided by limiting the range of wavenumbers treated in a spectral‐based extrapolation. Using spectral methods based on the low‐rank approximation of the propagation symbol, we extrapolate only the desired solutions in an accurate and efficient manner with reduced dispersion artiefacts. Applications to synthetic data demonstrate the accuracy of the new prestack modelling and migration approach.

The Baker–Akhiezer Function and Factorization of the Chebotarev–Khrapkov Matrix

A new technique is proposed for the solution of the Riemann-Hilbert problem with the Chebotarev-Khrapkov matrix coefficient $G(t)=\alpha_1(t)I+\alpha_2(t)Q(t)$, $\alpha_1(t), \alpha_2(t)\in H(L)$, $Q(t)$ is a $2\times 2$ zero-trace polynomial matrix, and $I$ is the unit matrix. This problem has numerous applications in elasticity and diffraction theory. The main feature of the method is the removal of the essential singularities of the solution to the associated homogeneous scalar Riemann-Hilbert problem on the hyperelliptic surface of an algebraic function by means of the Baker-Akhiezer function. The consequent application of this function for the derivation of the general solution to the vector Riemann-Hilbert problem requires finding of the $\rho$ zeros of the Baker-Akhiezer function ($\rho$ is the genus of the surface). These zeros are recovered through the solution to the associated Jacobi problem of inversion of abelian integrals or, equivalently, the determination of the zeros of the associated degree-$\rho$ polynomial and solution of a certain linear algebraic system of $\rho$ equations.

Capillary contact angle in a completely wet groove.

We consider the phase equilibria of a fluid confined in a deep capillary groove of width L with identical side walls and a bottom made of a different material. All walls are completely wet by the liquid. Using density functional theory and interfacial models, we show that the meniscus separating liquid and gas phases at two phase capillary coexistence meets the bottom capped end of the groove at a capillary contact angle θ(cap)(L) which depends on the difference between the Hamaker constants. If the bottom wall has a weaker wall-fluid attraction than the side walls, then θ(cap) > 0 even though all the isolated walls are themselves completely wet. This alters the capillary condensation transition which is now first order; this would be continuous in a capped capillary made wholly of either type of material. We show that the capillary contact angle θ(cap)(L) vanishes in two limits, corresponding to different capillary wetting transitions. These occur as the width (i) becomes macroscopically large, and (ii) is reduced to a microscopic value determined by the difference in Hamaker constants. This second wetting transition is characterized by large scale fluctuations and essential critical singularities arising from marginal interfacial interactions.

A Hecke correspondence theorem for automorphic integrals with infinite log-polynomial sum period functions

We generalize the correspondence between Dirichlet series with finitely many poles that satisfy a functional equation and automorphic integrals with log-polynomial sum period functions. In particular, we extend the correspondence to hold for Dirichlet series with finitely many essential singularities. We also study Dirichlet series with infinitely many poles in a vertical strip. For Hecke groups with λ ≥ 2 and some weights, we prove a similar correspondence for these Dirichlet series. For this case, we provide a way to estimate automorphic integrals with infinite log-polynomial periods by automorphic integrals with finite log-polynomial periods.

Experimental analysis of density and compressive strain of porous metal with the use of Taylor test

The article presents a new experimental method of analysis of density and longitudinal engineering compressive strain (LECS) in porous ductile rod, plastically deformed with a Taylor direct impact experiment (Taylor DIE). Two essential singularities in the distribution of the LECS are revealed, namely: maximum of the LECS e r max occurs near the rod striking end (Fig. 2), and at high impact velocities ( U > 100 m/s) e r max decreases along with an increase of the rod initial porosity, this behaviour is inverse to that under static loading. The dynamical density – LECS curves for porous copper are also presented in the paper. According to the authors’ best knowledge, these singularities have not been published in available literature so far.

Holomorphic functions associated with indeterminate rational moment problems

We consider indeterminate rational moment problems on the real line with their associated orthogonal rational functions. There exists a Nevanlinna type parametrization relating to the problem, with associated Nevanlinna matrices of functions having singularities in the closure of the set of poles of the rational functions belonging to the problem. We prove that the growth at the essential singularities of the four functions in the Nevanlinna matrix is the same.

A Matrix Model with a Singular Weight and Painlevé III

We investigate the matrix model with weight $$w(x) := {\rm exp}\left(-\frac{z^2}{2x^2}+\frac{t}{x}-\frac{x^2}{2}\right)$$ and unitary symmetry. In particular we study the double scaling limit as $${N \to \infty}$$ and $${(\sqrt{N} t, Nz^2 ) \to (u_1,u_2)}$$ , where N is the matrix dimension and the parameters (u 1, u 2) remain finite. Using the Deift-Zhou steepest descent method, we compute the asymptotics of the partition function when z and t are of order $${O\bigl(N^{-1/2} \bigr)}$$ . In this regime we discover a phase transition in the (z, N)-plane characterised by the Painlevé III equation. This is the first time that Painlevé III appears in studies of double scaling limits in Random Matrix Theory and is associated to the emergence of an essential singularity in the weighting function. The asymptotics of the partition function is expressed in terms of a particular solution of the Painlevé III equation. We derive explicitly the initial conditions in the limit $${Nz^2\rightarrow u_2}$$ of this solution.

On the generating function for consecutively weighted permutations

We show that the analytic continuation of the exponential generating function associated to consecutive weighted pattern enumeration of permutations only has poles and no essential singularities. The proof uses the connection between permutation enumeration and functional analysis, and as well as the Laurent expansion of the associated resolvent. As a consequence, we give a partial answer to a question of Elizalde and Noy: when is the multiplicative inverse of the exponential generating function for the number permutations avoiding a single pattern an entire function? Our work implies that it is enough to verify that this function has no zeros to conclude that the inverse function is entire.

Universal departure from Johnson-Nyquist relation caused by limited resolution

Exploiting the two-point measurement statistics, we propose a quantum measurement scheme of current with limited resolution of electron counting. Our scheme is equivalent to the full counting statistics in the long-time measurement with the ideal resolution, but is theoretically extended to take into account the resolution limit of actual measurement devices. Applying our scheme to a resonant level model, we show that the limited resolution of current measurement gives rise to a positive excess noise, which leads to a deviation from the Johnson-Nyquist relation. The deviation exhibits universal single-parameter scaling with the scaling variable $Q\ensuremath{\equiv}{S}_{\mathrm{M}}/{S}_{0}$, which represents the degree of the insufficiency of the resolution. Here, ${S}_{0}$ is the intrinsic noise, and ${S}_{\mathrm{M}}$ is the positive quantity that has the same dimension as ${S}_{0}$ and is defined solely by the measurement scheme. For the lack of the ideal resolution, the deviation emerges for $Q<1$ as $2exp[\ensuremath{-}{(2\ensuremath{\pi})}^{2}/Q]$ having an essential singularity at $Q=0$, which followed by the square root dependence $\sqrt{Q/4\ensuremath{\pi}}$ for $Q\ensuremath{\gg}1$. Our findings offer an explanation for the anomalous enhancement of noise temperature observed in Johnson noise thermometry.

Wilson loops and Riemann theta functions II

In this paper we extend and simplify previous results regarding the computation of Euclidean Wilson loops in the context of the AdS/CFT correspondence, or, equivalently, the problem of finding minimal area surfaces in hyperbolic space (Euclidean AdS3). If the Wilson loop is given by a boundary curve X(s) we define, using the integrable properties of the system, a family of curves X(lambda,s) depending on a complex parameter lambda known as the spectral parameter. This family has remarkable properties. As a function of lambda, X(lambda,s) has cuts and therefore is appropriately defined on a hyperelliptic Riemann surface, namely it determines the spectral curve of the problem. Moreover, X(lambda,s) has an essential singularity at the origin lambda=0. The coefficients of the expansion of X(lambda,s) around lambda=0, when appropriately integrated along the curve give the area of the corresponding minimal area surface. Furthermore we show that the same construction allows the computation of certain surfaces with one or more boundaries corresponding to Wilson loop correlators. We extend the area formula for that case and give some concrete examples. As the main example we consider a surface ending on two concentric circles and show how the boundary circles can be deformed by introducing extra cuts in the spectral curve.

Consciousness – Brain Gene Behavioral Psychology by Using Quantum Dots for DNA Sequences and for a Single DNA Sequence

Psychological / philosophical terms such as soul, self, Atman, ego, mind, consciousness are randomly used without clear cut meanings and definitions in philosophy and psychology. In this paper, we have scientifically interpreted and arrived at possible meanings and definitions for consciousness in terms of quantum dots which could be applied in interpreting the biological DNA base pair sequences and for DNA individual molecules. A theoretical detection of cognitive behavior of mind is attempted with individual molecules of DNA and with that of DNA base pairs. A rule governing the DNA base pairs with that of DNA single molecule is stated in the form of DNA theorem as “If left hemisphere of the brain is equal to DNA sequence information hemisphere and right hemisphere is equal to the individual DNA sequence hemisphere then it may be proved that the distinct part of Ramanujan numbers represent base pair DNA hemisphere and odd part of Ramanujan numbers represent the other non-base pair DNA individual molecules hemisphere are equal”. This theorem has multiple applications in tackling the social and health issues. It is possible to detect cognitive behavior of mind with quantum dots that bind to sequences of DNA which are associated with the individual DNA which shows cognitive behavior. If quantum dots are stimulated with light, they emit their unique bar codes making the critical consciousness associated DNA sequences visible and explain the personality development. And also, using number theory, consciousness may be defined as a line of essential singularity lying in a circle of brain having radius unity with an origin at zero point. This paper will not only has applications in tackling health and social issues but will also start a new chapter in bringing psychology nearer to the door steps of neurological sciences.