Taylor Expansion Coefficients Research Articles

QCD critical point, Lee-Yang edge singularities, and Padé resummations

I analyze the trajectory of the Lee-Yang edge singularities of the QCD equation of state in the complex baryon chemical potential (μB) plane for different values of the temperature by using the recent lattice results for the Taylor expansion coefficients up to eighth order in μB and various resummation techniques that blend in Padé expansions and conformal maps. By extrapolating from this information, I estimate for the location of the QCD critical point: Tc≈100 MeV, μc≈580 MeV. I also estimate the crossover slope at the critical point to be α1≈9∘ and further constrain the nonuniversal mapping parameters between the three-dimensional Ising model and QCD equations of state. Published by the American Physical Society 2024

Positivity, low twist dominance and CSDR for CFTs

We consider a crossing symmetric dispersion relation (CSDR) for CFT four point correlation with identical scalar operators, which is manifestly symmetric under the cross-ratios u,vu,v interchange. This representation has several features in common with the CSDR for quantum field theories. It enables a study of the expansion of the correlation function around u=v=1/4u=v=1/4, which is used in the numerical conformal bootstrap program. We elucidate several remarkable features of the dispersive representation using the four point correlation function of \Phi_{1,2}Φ1,2 operators in 2d minimal models as a test-bed. When the dimension of the external scalar operator (\Delta_\sigmaΔσ) is less than \frac{1}{2}12, the CSDR gets contribution from only a single tower of global primary operators with the second tower being projected out. We find that there is a notion of low twist dominance (LTD) which, as a function of \Delta_\sigmaΔσ, is maximized near the 2d Ising model as well as the non-unitary Yang-Lee model. The CSDR and LTD further explain positivity of the Taylor expansion coefficients of the correlation function around the crossing symmetric point and lead to universal predictions for specific ratios of these coefficients. These results carry over to the epsilon expansion in 4-\epsilon4−ϵ dimensions. We also conduct a preliminary investigation of geometric function theory ideas, namely the Bieberbach-Rogosinski bounds.

Taylor‐based adaptive sliding mode control method for robot manipulators

AbstractThis paper develops a novel Taylor‐based adaptive sliding mode control method (ASMC) for robot manipulators. In the first new scheme, sliding mode control (SMC) is effectively enhanced using the Taylor expansion for achieving a less conservative sign‐function gain that enables chattering attenuation. After that, a new Taylor‐based adaptive SMC scheme is proposed without prior knowledge of the upper bound of uncertainty and the Taylor expansion coefficients. Then, a new Taylor‐based boundary layer ASMC scheme is proposed for chattering attenuation. As a fact, no chattering is expected to be observed as long as the sliding surface remains an invariant set. However, the sliding surface will not be an invariant set unless the sign‐function is precisely determined on the sliding surface. It is verified that the sign‐function cannot be precisely determined on the sliding surface due to the presence of uncertainty, hence, the chattering phenomenon. In the new Taylor‐based boundary layer ASMC scheme, the conventional ASMC is modified by removing the sign‐function in the vicinity of the sliding surface and using the Taylor expansion to estimate and compensate for the uncertainty. Compared to a time‐delay ASMC applied to a robot manipulator, the Taylor‐based ASMC scheme exhibits a less conservative sign‐function gain resulting in chattering attenuation.

Constructing continuum models of acoustic metamaterials via the symbiotic organisms search (SOS) optimization

Based on the symbiotic organisms search (SOS) optimization algorithm, a robust strain gradient (SG) continuum model has been proposed to accurately capture the broadband dispersion relations of one-dimensional acoustic metamaterials. Under the continuous assumption, an unavoidable key step is the Taylor expansion of displacements, which directly influences the accuracy of the corresponding continuum theory. When the wavelength becomes comparable to the periodic characteristic size, the coefficients of Taylor expansions need necessary adjustments due to the discreteness of the microstructure. Thus, the continuum theories still face critical challenges in predicting the broadband dispersion feature. This remains widely open so far. In this study, we attempt to adopt the SOS optimization to determine the optimal Taylor expansion coefficients to guarantee the dispersion diagrams causing the minimal error throughout the first Brillouin zone. The robustness of the SOS-based SG continuum model is demonstrated with three benchmark examples, i.e., the monoatomic, diatomic, and mass-in-mass lattices. Such an attempt of constructing continuum models with the help of optimization tools may shed some new light on continuum mechanics of structure media.

Thawing k-essence dark energy in the PAge space

A broad class of dark energy models can be written in the form of k-essence, whose Lagrangian density is a two-variable function of a scalar field ϕ and its kinetic energy . In the thawing scenario, the scalar field becomes dynamic only when the Hubble friction drops below its mass scale in the late Universe. Thawing k-essence dark energy models can be randomly sampled by generating the Taylor expansion coefficients of its Lagrangian density from random matrices [Huang Z 2021 Phys. Rev. D 104 103533]. Reference [Huang Z 2021 Phys. Rev. D 104 103533] points out that the non-uniform distribution of the effective equation of state parameters (w 0, w a ) of the thawing k-essence model can be used to improve the statistics of model selection. The present work studies the statistics of thawing k-essence in a more general framework that is Parameterized by the Age of the Universe (PAge) [Huang Z 2020 Astrophys. J. Lett. 892 L28]. For fixed matter fraction Ω m , the random thawing k-essence models cluster in a narrow band in the PAge parameter space, providing a strong theoretical prior. We simulate cosmic shear power spectrum data for the Chinese Space Station Telescope optical survey, and compare the fisher forecast with and without the theoretical prior of thawing k-essence. For an optimal tomography binning scheme, the theoretical prior improves the figure of merit in PAge space by a factor of 3.3.

Statistics of thawing k -essence dark energy models

K-essence is a minimally-coupled scalar field whose Lagrangian density $\mathcal{L}$ is a function of the field value $\phi$ and the kinetic energy $X=\frac{1}{2}\partial_\mu\phi\partial^\mu\phi$. In the thawing scenario, the scalar field is frozen by the large Hubble friction in the early universe, and therefore initial conditions are specified. We construct thawing k-essence models by generating Taylor expansion coefficients of $\mathcal{L}(\phi, X)$ from random matrices. From the ensemble of randomly generated thawing k-essence models, we select dark energy candidates by assuming negative pressure and non-growth of sub-horizon inhomogeneities. For each candidate model the dark energy equation of state function is fit to the Chevallier-Polarski-Linder parameterization $w(a) \approx w_0+w_a(1-a)$, where $a$ is the scale factor. The thawing k-essence dark models distribute very non-uniformly in the $(w_0, w_a)$ space. About 90\% models cluster in a narrow band in the proximity of a slow-roll line $w_a\approx -1.42 \left(\frac{\Omega_m}{0.3}\right)^{0.64}(1+w_0)$, where $\Omega_m$ is the present matter density fraction. This work is a proof of concept that for a certain class of models very non-uniform theoretical prior on $(w_0, w_a)$ can be obtained to improve the statistics of model selection.

Net-baryon Number Fluctuations

The appearance of large, none-Gaussian cumulants of the baryon number distribution is commonly discussed as a signal for the QCD critical point. We review the status of the Taylor expansion of cumulant ratios of baryon number fluctuations along the freeze-out line and also compare QCD results with the corresponding proton number fluctuations as measured by the STAR Collaboration at RHIC. To further constrain the location of a possible QCD critical point we discuss poles of the baryon number fluctuations in the complex plane. Here we use not only the Taylor coefficients obtained at zero chemical potential but perform also calculations of Taylor expansion coefficients of the pressure at purely imaginary chemical potentials.

New Paley–Wiener Theorems

In this paper, for an arbitrary fixed compact set K, we find necessary and sufficient conditions on the Taylor expansion coefficients of entire functions of exponential type so that these functions are the Fourier image of distributions supported in K. In other words, we state the Paley–Wiener theorem in the language of Taylor expansion coefficients.

Insights on Skyrme parameters from GW170817

The binary neutron-star merger event, GW170817, has cast a new light on nuclear physics research. Using a neutron-star model that includes a crust equation of state (EoS), we calculate the properties of a 1.4 solar-mass neutron star. The model incorporates more than 200 Skyrme energy density functionals, which describe nuclear matter properties, in the outer liquid core region of the neutron star. We find a power-law relation between the neutron-star tidal deformability, Λ, and the neutron-star radius, R. Without an explicit crust EoS, the model predicts a smaller R and the difference becomes significant for stars with large radii. To connect the neutron star properties with nuclear matter properties, we confront the predicted values for Λ against the Taylor expansion coefficients of the Skyrme interactions. There is no pronounced correlation between Skyrme parameters relevant to symmetric nuclear matter and neutron star properties. However, we find the strongest correlation between Λ and Ksym, the curvature of the density dependence of the symmetry energy at saturation density. At twice the saturation density, our calculations show a strong correlation between Λ and total pressure, providing guidance to laboratory nucleus-nucleus collision experiments.

Reliable estimation of the radius of convergence in finite density QCD

We study different estimators of the radius of convergence of the Taylor series of the pressure in finite density QCD. We adopt the approach in which the radius of convergence is estimated first in a finite volume, and the infinite-volume limit is taken later. This requires an estimator for the radius of convergence that is reliable in a finite volume. Based on general arguments about the analytic structure of the partition function in a finite volume, we demonstrate that the ratio estimator cannot work in this approach, and propose three new estimators, capable of extracting reliably the radius of convergence, which coincides with the distance from the origin of the closest Lee-Yang zero. We also provide an estimator for the phase of the closest Lee-Yang zero, necessary to assess whether the leading singularity is a true critical point. We demonstrate the usage of these estimators on a toy model, namely 4 flavors of unimproved staggered fermions on a small $6^3 \times 4$ lattice, where both the radius of convergence and the Taylor coefficients to any order can be obtained by a direct determination of the Lee-Yang zeros. Interestingly, while the relative statistical error of the Taylor expansion coefficients steadily grows with order, that of our estimators stabilizes, allowing for an accurate estimate of the radius of convergence. In particular, we show that despite the more than 100\% error bars on high-order Taylor coefficients, the given ensemble contains enough information about the radius of convergence.

The construction of high order convergent look-ahead finite difference formulas for Zhang Neural Networks

ABSTRACTZhang Neural Networks rely on convergent 1-step ahead finite difference formulas of which very few are known. Those which are known have been constructed in ad-hoc ways and suffer from low truncation error orders. This paper develops a constructive method to find convergent look-ahead finite difference schemes of higher truncation error orders. The method consists of seeding the free variables of a linear system comprised of Taylor expansion coefficients followed by a minimization algorithm for the maximal magnitude root of the formula's characteristic polynomial. This helps us find new convergent 1-step ahead finite difference formulas of any truncation error order. Once a polynomial has been found with roots inside the complex unit circle and no repeated roots on it, the associated look-ahead ZNN discretization formula is convergent and can be used for solving any discretized ZNN based model. Our method recreates and validates the few known convergent formulas, all of which have truncation error orders at most 4. It also creates new convergent 1-step ahead difference formulas with truncation error orders 5 through 8.

Relation between exponential behavior and energy denominators — weak coupling limit

We show some interesting properties of tridiagonal and pentadiagonal matrices in the weak coupling limits. In the former case of this limit, the ground state wave function amplitudes are identical to the Taylor expansion coefficients of the exponential function e[Formula: see text]. With regards to transition rates, a dip in the pentadiagonal case which is not present in the tridiagonal case is explained. An intimate connection between energy denominators and exponential behavior is demonstrated.

Robust Reconstruction of Continuously Time-Varying Topologies of Weighted Networks

Methodological advance in reconstructing the structure of a complex dynamical network have enabled increasingly intricate comprehension of its evolutionary mechanisms and functional behaviors. However, we are often involved in such a situation that only a limited number of observations can be collected and the network structures are completely unknown and continuously time-varying and state-dependent. Few studies have been involved in reconstructing the continuously varying topologies of networks in time intervals via limited discrete observations. We develop a new way to reconstruct the structures of continuously time-varying and state-dependent networks by reconstructing the Taylor expansion coefficients of couplings. The alternating direction method of multipliers algorithm is applied to solve the reconstruction problem by integrating each component’s information of a high-dimensional node. The robustness analysis of our method shows that the structure reconstruction under noisy observations and outliers approximates the one under accurate observations as nodal dimension increases. Different from the existing works, the advantage of our method lies in two aspects. First, the ability to reconstruct network structures via one-time dynamical evolution. Second, instead of reconstructing structures at discrete time points, we obtain the continuously varying structures in continuous time intervals. Numerical simulations are provided to illustrate the feasibility, effectiveness, and robustness of the reconstruction scheme.

Vibration measurement with nonlinear converter in the presence of noise

Conventional vibration measurement methods use the linear properties of physical converters. These methods are strongly influenced by nonlinear distortions, because ideal linear converters are not available. Practically, any converter can be considered as a linear one, when an output signal is very small. However, the influence of noise increases significantly and signal-to-noise ratio decreases at lower signals. When the output signal is increasing, the nonlinear distortions are also augmenting. If the wide spectrum vibration is measured, conventional methods face a harmonic distortion as well as intermodulation effects. Purpose of this research is to develop a measurement method of wide spectrum vibration by using a converter described by a nonlinear function of type f(x), where x =x(t) denotes the dependence of coordinate x on time t due to the vibration. Parameter x(t) describing the vibration is expressed as Fourier series. The spectral components of the converter output f(x(t)) are determined by using Fourier transform. The obtained system of nonlinear equations is solved using the least squares technique that permits to find x(t) in the presence of noise. This method allows one to carry out the absolute or relative vibration measurements. High resistance to noise is typical for the absolute vibration measurement, but it is necessary to know the Taylor expansion coefficients of the function f(x). If the Taylor expansion is not known, the relative measurement of vibration parameters is also possible, but with lower resistance to noise. This method allows one to eliminate the influence of nonlinear distortions to the measurement results, and consequently to eliminate harmonic distortion and intermodulation effects. The use of nonlinear properties of the converter for measurement gives some advantages related to an increased frequency range of the output signal (consequently increasing the number of equations) that allows one to decrease the noise influence on the measurement results. The greater is the nonlinearity the lower is noise. This method enables the use of the converters that are normally not suitable due to the high nonlinearity.

QCD equation of state to O(μB6) from lattice QCD

We calculated the QCD equation of state using Taylor expansions that include contributions from up to sixth order in the baryon, strangeness and electric charge chemical potentials. Calculations have been performed with the Highly Improved Staggered Quark action in the temperature range $T\in [135~{\rm MeV}, 330~{\rm MeV}]$ using up to four different sets of lattice cut-offs corresponding to lattices of size $N_\sigma^3\times N_\tau$ with aspect ratio $N_\sigma/N_\tau=4$ and $N_\tau =6-16$. The strange quark mass is tuned to its physical value and we use two strange to light quark mass ratios $m_s/m_l=20$ and $27$, which in the continuum limit correspond to a pion mass of about $160$ MeV and $140$ MeV espectively. Sixth-order results for Taylor expansion coefficients are used to estimate truncation errors of the fourth-order expansion. We show that truncation errors are small for baryon chemical potentials less then twice the temperature ($\mu_B\le 2T$). The fourth-order equation of state thus is suitable for the modeling of dense matter created in heavy ion collisions with center-of-mass energies down to $\sqrt{s_{NN}}\sim 12$ GeV. We provide a parametrization of basic thermodynamic quantities that can be readily used in hydrodynamic simulation codes. The results on up to sixth order expansion coefficients of bulk thermodynamics are used for the calculation of lines of constant pressure, energy and entropy densities in the $T$-$\mu_B$ plane and are compared with the crossover line for the QCD chiral transition as well as with experimental results on freeze-out parameters in heavy ion collisions. These coefficients also provide estimates for the location of a possible critical point. We argue that results on sixth order expansion coefficients disfavor the existence of a critical point in the QCD phase diagram for $\mu_B/T\le 2$ and $T/T_c(\mu_B=0) > 0.9$.

Momentum space orthogonal polynomial projection quantization

The orthogonal polynomial projection quantization (OPPQ) is an algebraic method for solving Schrödinger’s equation by representing the wave function as an expansion in terms of polynomials orthogonal with respect to a suitable reference function R(x), which decays asymptotically not faster than the bound state wave function. The expansion coefficients are obtained as linear combinations of power moments . In turn, the 's are generated by a linear recursion relation derived from Schrödinger’s equation from an initial set of low order moments. It can be readily argued that for square integrable wave functions representing physical states . Rapidly converging discrete energies are obtained by setting Ω coefficients to zero at arbitrarily high order. This paper introduces an extention of OPPQ in momentum space by using the representation , where Qn(k) are polynomials orthogonal with respect to a suitable reference function T(k). The advantage of this new representation is that it can help solving problems for which there is no coordinate space moment equation. This is because the power moments in momentum space are the Taylor expansion coefficients, which are recursively calculated via Schrödinger’s equation. We show the convergence of this new method for the sextic anharmonic oscillator and an algebraic treatment of Gross–Pitaevskii nonlinear equation.

Reconstructing inflationary paradigm within Effective Field Theory framework

In this paper my prime objective is to analyze the constraints on a sub-Planckian excursion of a single inflaton field within Effective Field Theory framework in a model independent fashion. For a generic single field inflationary potential, using the various parameterization of the primordial power spectrum I have derived the most general expression for the field excursion in terms of various inflationary observables, applying the observational constraints obtained from recent Planck 2015 and Planck 2015 +BICEP2/Keck Array data. By explicit computation I have reconstructed the structural form of the inflationary potential by constraining the Taylor expansion coefficients appearing in the generic expansion of the potential within the Effective Field Theory. Next I have explicitly derived, a set of higher order inflationary consistency relationships, which would help us to break the degeneracy between various class of inflationary models by differentiating them. I also provided two simple examples of Effective Theory of inflation- inflection-point model and saddle-point model to check the compatibility of the prescribed methodology in the light of Planck 2015 and Planck 2015 +BICEP2/Keck Array data. Finally, I have also checked the validity of the prescription by estimating the cosmological parameters and fitting the theoretical CMB TT, TE and EE angular power spectra with the observed data within the multipole range $2<l<2500$.

On Taylor coefficients of Cauchy-type integrals of finite complex measures

Boundary values of Cauchy-type integrals of finite complex measures given on a unit circle, generally speaking, are not Lebesgue integrable, and therefore at expansion of Cauchy-type integrals in Taylor series, the expansion coefficients cannot be expressed by boundary values using the Lebesgue integral. In this paper, using the notion of A-integration and N-integration, we get a formula for calculating the Taylor expansion coefficients of Cauchy-type integrals of finite complex measures.

Conjectures on the evaluation of certain functions with algebraic properties

In this article we consider functions with Moebius-periodic Taylor expansion coefficients. These functions under some conditions take algebraic values and can be evaluated in terms of theta functions and the Dedekind eta function. Special cases are the elliptic singular moduli, the Rogers–Ramanujan continued fraction, Eisenstein series and functions associated with Jacobi symbol coefficients.

Equation of state at finite net-baryon density using Taylor coefficients up to sixth order

We employ the lattice QCD data on Taylor expansion coefficients up to sixth order to construct an equation of state at finite net-baryon density. When we take into account how hadron masses depend on lattice spacing and quark mass, the coefficients evaluated using the p4 action are equal to those of hadron resonance gas at low temperature. Thus the parametrised equation of state can be smoothly connected to the hadron resonance gas equation of state. We see that the equation of state using Taylor coefficients up to second order is realistic only at low densities, and that at densities corresponding to s/n_B > 40, the expansion converges by the sixth order term.