New Measure of Economic Development Based on the Four-Colour Theorem

The location quotient is one of the basic quantitative tools for identifying the regional poles and the turnpikes of economic growth in spatial economy. The disadvantage of this traditional measure is the limited scope of economic information contained in it. The new measure of economic development proposed in the article encompasses a complex spectrum of phenomena in one number, as it takes into account the influence of the public administration sector, as well as top technology in the form of ICT and its practical business models. It also takes into account the digital prosumption and the platforms for participation. The participation platforms in the public administration sector are the websites of municipal public administration offices. A cluster analysis was used to distinguish four quality classes of these websites. These classes were assigned four different colours, which were then used to draw up a map of the selected province. Each municipality is marked with a colour that corresponds to the quality class of the website of the state administration office operating on its territory. The colour system resulting from the four-colour theorem and the corresponding dual graph play the role of a reference system in relation to each empirical colour distribution and another dual graph related to it. The measure of the economic development of a region is the degree of reduction of the dual graph corresponding to the empirical distribution of colours, which identifies the actual growth poles and determines the routes of growth. The presented indicator better and more precisely identifies poles and routes of economic growth than the traditional location quotient.     

Poles and zeros in ππ scattering amplitudes

We draw attention to the remarkable systematics exhibited in the ππ amplitudes provided by the CERN-Munich data. It is shown that the existence of resonance poles with specific spin and mass values and the behaviours of the amplitude zeros in the Mandelstam plane give strong support to the ideas associated with dual resonance models. It is emphasized that some gauge-like conditions of dual resonance models play an essential role in these considerations.     

NOETHER AND TOPOLOGICAL CHARGE OF GAUGE FIELD

"Electric" chaise currents with gauge-dependent isotopic axis are gauge invariant Noether currents. These currnnts include themselves the contribution of gauge field. "Magrnetic" poles are related to the topologically nontrivial orientation of isotopic axis. The dual symmetry trans-forms a known conserved current into infinitely many ved currents. The induced Noether charges of vacuum different chiral boundary condition and the topological background charges always compensate each other.     

Duality,scaling and fixed poles in virtual compton scattering

J-plane fixed poles are considered in the lepton-hadron scattering amplitudes. A relation between the scale-invariant behaviour of the virtual Compton scattering structure W₂ and the J = 1 pole is established using the wrong signature FESR. A dual mechanism for the generation of fixed poles, leading to a number of interesting implications for a singular behaviour of the pole residue in the variable q² (photon mass square), is indicated.     

Isotopic spin splitting of trajectories in dual multiparticle models

A dual model is considered with trajectories which depend linearly on the square of the total isotopic spin of the channel. It is shown that in spite of the noncommutativity of the operators of isotopic spin of different channels such a model leads to parent trajectories with normal properties: the absence of exotic states and bootstrap-type factorization. On the daughter level nonfactorizeable singularities inevitably appear. Being dependent on the details of the model they are either branch points or non-factorizeable poles.     

Off-mass shell dual amplitudes (II)

A new method for continuing dual model amplitudes off the mass shell was recently proposed. This paper explores some of the properties of the resulting amplitudes. It is demonstrated that one-current amplitudes contain fixed poles in the J-plane at positions that are correlated with the asymptotic power behavior of form factors. The two-point function is explicitly calculated and shown to fall asymptotically as a power provided that a certain condition involving the dimension of space-time and a parameter (that is believed to be correlated with the leading Regge intercept) is satisfied. Certain formulas required for future investigation of more complicated amplitudes are also derived.     

Quantum damped oscillator I: Dissipation and resonances

Quantization of a damped harmonic oscillator leads to so called Bateman’s dual system. The corresponding Bateman’s Hamiltonian, being a self-adjoint operator, displays the discrete family of complex eigenvalues. We show that they correspond to the poles of energy eigenvectors and the corresponding resolvent operator when continued to the complex energy plane. Therefore, the corresponding generalized eigenvectors may be interpreted as resonant states which are responsible for the irreversible quantum dynamics of a damped harmonic oscillator.     

A fresh look at generalized Veneziano amplitudes

The dual resonance model, which was a precursor of string theory was based upon the idea that two-particle scattering amplitudes should be expressible equivalently as a sum of contributions of an infinite number of s channel poles each corresponding to a finite number of particles with definite spin, or as a similar sum of t channel poles. The famous example of Veneziano [Nuovo Cimento A 57 (1968) 190] satisfies all these requirements, and is additionally ghost free. We recall other trajectories which provide solutions to the duality constraints, e.g. the general Mobiüs trajectories and the logarithmic trajectories, which were thought to be lacking this last feature. We however demonstrate, partly empirically, the existence of a regime within a particular deformation of the Veneziano amplitude for logarithmic trajectories for which the 4-point amplitude remains ghost free.     

Conformal field theory interpretation of black hole quasinormal modes

We obtain exact expressions for the quasinormal modes of various spin for the Ba?ados-Teitelboim-Zanelli black hole. These modes determine the relaxation time of black hole perturbations. Exact agreement is found between the quasinormal frequencies and the location of the poles of the retarded correlation function of the corresponding perturbations in the dual conformal field theory. This then provides a new quantitative test of the anti-de Sitter/conformal field theory correspondence.     

Duality and multi-gluon scattering

For the six-gluon scattering process we give explicit and simple expressions for the amplitude and its square. To achieve this we use an analogy with string theories to identify a unique procedure for writing the multi-gluon scattering amplitudes in terms of a sum of gauge invariant dual sub-amplitudes multiplied by an appropriate color (Chan-Paton) factor. The sub-amplitudes defined in this way are invariant under cyclic permutations, satisfy powerful identities which relate different non-cyclic permutations and factorize in the soft gluon limit, the two-gluon collinear limit and on multi-gluon poles. Also, to leading order in the number of colors these sub-amplitudes sum incoherently in the square of the full matrix element. The results contained here are important for Monte Carlo studies of multi-processes at hadron colliders as well as for understanding the general structure of QCD.     

Characteristics of multipole lenses with the ideal central part of the poles

A procedure for calculating the potential and field of multipole lenses in which the central part of the poles has the form of an ideal curve is presented. The procedure is based on the method of conformal mapping. The properties of these lenses are compared with those whose poles have a polygonal profile. It is shown that sufficiently wide ideal-center poles provide a better field quality than polygonal poles.     

Elastic scattering and the K matrix: (II). Phase shifts,resonances and few-level approximations

The K-matrix expansions given in part I are applied to the scattering by a square-well potential. Their convergence proves very satisfactory from a physical point of view since few-level approximations allow very good approximations to the phase shifts and cross sections. It also appears that all the complex poles of K_l and the real ones with positive residue should undoubtedly be associated with physical resonances. As for the real ones with negative residue, i.e. the echo poles, they are obviously unrelated to resonances, but they provide a very good parametrization of the background part of the scattering. The time delay is given a major role in the argument. The possibility of having double poles is also discussed and sum rules are given for the energies and residues of the poles.     

李模型中极点的运动

本文指出,在李模型中,不论有否左方割线和非弹性割线,只要不引入CDD极点,第二叶不具有代表共振态的极点。当CDD极点被引入后,第二叶极点可能在偶合常数增加时穿过正常割线而走入第一叶,但在这以前,另有一极点自左方(或穿过左方割线,或在-∞)走入第一叶。文中画出了一种情形下的二叶的全部极点的运动。     

Pole Dynamics for Elliptic Solutions of the Korteweg-deVries Equation

The real, nonsingular elliptic solutions of the Korteweg-de Vries equation are studied through the time dynamics of their poles in the complex plane. The dynamics of these poles is governed by a dynamical system with a constraint. This constraint is solvable for any finite number of poles located in the fundamental domain of the elliptic function, often in many different ways. Special consideration is given to those elliptic solutions that have a real nonsingular soliton limit. This revised version was published online in July 2006 with corrections to the Cover Date.     

Point-like structure and off-shell dual strings

We argue that in a consistent off-shell dual formalism the amplitude for the emission of a scalar off-shell state by a string consists of two components. One of these contains the particle poles in the off-shell leg and the other is intimately related to the insertion of a point-like energy density on the string. As a result, the amplitude for a string to emit a zero momentum scalar state into the vacuum (which may be relevant for spontaneous symmetry breaking) is described by the amplitude for a finite fraction of the energy in the string to collapse to a spatial point at some time (this fraction and its space-time position being integrated over). The off-shell amplitudes have an elegant formulation in terms of a set of “confined modes” which can be assigned quark flavour quantum numbers to reproduce the Chan-Paton scheme.We suggest the dual model be modified by allowing for the coupling of scalar closed strings to the vacuum and the resulting effect on the space-time structure of dual Green functions is described.We find that even the emission of a single zero-momentum closed string modifies the elastic amplitude in a significant manner, leading to a power-behaved fixed-angle cross section in contrast to the usual exponential decrease of the dual model. This arises from point-like scattering between energy densities accumulating in the colliding strings. The relationship between the fixed angle and Regge limits is discussed. The fixed angle behaviour is found to be the asymptotic limit in momentum transfer of a fixed pole that arises in the Regge limit.     

费米子Regge轨迹的解析性和阈行为

从π介子核子散射振幅出发,根据Mandelstam表示和么正条件,对于奇异量子数为零、重子数为1的费米子Regge轨迹,求得了下列结果:(1)位置参数α(s)的解析性,只有右方物理割而无左方动力割,但是左方有从s=0到-∞的运动学割。(2)当s<0时α(s)取复数值,并且还有一条同它成复共轭(在s<0范围内)的Regge轨迹。这一对轨迹它们的J-宇称相同,但是空间宇称相反。02时,α(s)为实数。不同宇称的轨迹对s的依赖不同。(3)讨论了费米子Regge迹在阈能(W₀=M+μ)附近的行为。凡是与共振态、束缚态有关的Regge轨迹,α(W₀)≠0。求得了在阈附近的表示,定性方面同玻色子轨迹相同。(4)还有一大类Regge轨迹,它们同共振态、束缚态无关,α(W₀)=0。这类轨迹在能量趋近阈能时,有无穷多个极点趋于ReJ=0(即我们在费米子情况下得到Gribov-Pomeranchuk极点凝聚现象),这类轨迹当能量由阈下趋于阈时,从J平面左半平面共轭成对趋于ReJ=0;当能量从阈上趋于阈时,成对地从第一和第三象限趋于ReJ=0。所有这些极点实数部分Reα→0比虚数部分Imα→0快一或二个数量级。J平面原点是一个凝聚点,这类“非动力”的Regge轨迹阈行为,无论对玻色子、费米子,以至非相对论势散射,定性都相似(只是凝聚的轴线和点不同)。它们实际上是s矩阵在阈的普遍性质的反映。     

Regge-Pole bei geschwindigkeitsabhängigen Potentialen

The analytical behaviour of the scattering amplitude with respect to energy and angular momentum in considered for a group of soluble velocity-dependent potentials. Generalized Jost functions as well as trajectories for the poles are given explicitly. In certain cases the positions of the poles have been determined numerically.     

An automatic frequency domain modal parameter estimation algorithm

In this article, a frequency-domain modal parameter estimation method is proposed. The algorithm automatically separates physical poles from mathematical ones. An important issue in the automatization of the algorithm is the inclusion of noise information to estimate the standard deviations of the poles. These standard deviations are used (together with other features) as the inputs of a fuzzy clustering algorithm. The clustering algorithm then classifies the poles into the mathematical and physical ones. The method requires no user interaction, and a parameter is available quantifying the success of the classification.     

Pseudo-gap phase and duality in a holographic fermionic system with dipole coupling on Q-lattice

We classify the different phases by the "pole-zero mechanism" for a holographic fermionic system which contains a dipole coupling with strength p on a Q-lattice background. A complete phase structure in p space can be depicted in terms of Fermi liquid, non-Fermi liquid, Mott phase and pseudo-gap phase. In particular, we find that in general the region of the pseudo-gap phase in p space is suppressed when the Q-lattice background is dual to a deep insulating phase, while for an anisotropic background, we have an anisotropic region for the pseudo-gap phase in p space as well. In addition, we find that the duality between zeros and poles always exists regardless of whether or not the model is isotropic.     

Toward bias minimization in acoustic feedback cancellation systems

A novel technique for bias suppression within acoustic feedback cancellation systems is proposed. This is achieved based on the use of all-pass filters in the forward part of the hearing aid. The poles of these filters are made time-varying, which results in a frequency response with constant magnitude and varying phase. This is a desired feature of the proposed approach, since the results from human psychoacoustics show that the human ear is not sensitive to moderate phase perturbations. The derivation of the proposed algorithms for the time variation of the location of the poles of all pass filters is based on a rigorous analysis of the phenomenon of bias in acoustic systems. Practical issues, such as the dependence of the steady-state error on the order of the all-pass filter, the number of varying poles, and their standard deviation are examined and strategies for the variation of the poles are introduced. Results obtained from a simulated hearing aid are provided to support the analysis. The quality of the processed audio signals is evaluated through subjective tests.