Kernel Formalism Research Articles

Late-time correlators and complex geodesics in de Sitter space

We study two-point correlation functions of a massive free scalar field in de Sitter space using the heat kernel formalism. Focusing on two operators in conjugate static patches we derive a geodesic approximation to the two-point correlator valid for large mass and at late times. This expression involves a sum over two complex conjugate geodesics that correctly reproduces the large-mass, late-time limit of the exact two-point function in the Bunch-Davies vacuum. The exponential decay of the late-time correlator is associated to the timelike part of the complex geodesics. We emphasize that the late-time exponential decay is in tension with the finite maximal entropy of empty de Sitter space, and we briefly discuss how non-perturbative corrections might resolve this paradox.

Memory kernel formalism with fractional exponents and its application to dielectric relaxation

A fractal Fokker–Planck formalism applied to the dielectric relaxation in glass forming liquids is proposed. This formalism is a modality of the generalized equation of Langevin on the use of fractional-time derivatives simultaneously with the memory function to describe the dynamics of the dipolar-moment autocorrelation function. The goal is to get the description of the complex autocorrelation function numerically, and the real and imaginary parts of the second-order memory function, related to the kernel of the integral hierarchy representation of this autocorrelation function. The results exhibit the memory effect associates with α-dielectric relaxation mode. From the analysis, it is shown the existence of a maximum and the appropriated frequency limit in the imaginary and real parts, respectively, of the second-order memory function. That is required to describe experimental well the complex shear viscosity of the material.The theoretical model was tested with experimental data from dielectric spectroscopy of biphenyl-2-yl isobutyrate (OBPI), a glass forming liquid, in measurements in the frequency domain from 10−2 to 106 Hz. This ester has been selected in this study for its predominant α-relaxation when compared to the β relaxation. The real and imaginary parts of the second-order memory function were evaluated by means of the proposed model and the aid of the parameters of the Havriliak–Negami relationship.

Kernel formalism applied to Fourier-based wave-front sensing in presence of residual phases

In this paper, we describe Fourier-based wave-front sensors (WFSs) as linear integral operators, characterized by their kernel. In the first part, we derive the dependency of this quantity with respect to the WFS's optical parameters: pupil geometry, filtering mask, and tip/tilt modulation. In the second part, we focus the study on the special case of convolutional kernels. The assumptions required to be in such a regime are described. We then show that these convolutional kernels allow to drastically simplify the WFS model by summarizing its behavior in a concise and comprehensive quantity called the WFS impulse response. We explain in particular how it allows to compute the sensor's sensitivity with respect to spatial frequencies. Such an approach therefore provides a fast diagnostic tool to compare and optimize Fourier-based WFSs. In the third part, we develop the impact of the residual phases on the sensor's impulse response, and show that the convolutional model remains valid. Finally, a section dedicated to the pyramid WFS concludes this work and illustrates how the slope maps are easily handled by the convolutional model.

A vertex operator algebra construction of the colour-kinematics dual numerator

We derive a vertex operator based expression for the kinematic numerators of Yang-Mills amplitudes by applying the momentum kernel formalism to open string amplitudes. The expression involves an α′-weighted commutator induced by the monodromy relations between the colour ordered Yang-Mills amplitudes, which mirrors the α′ deformed colour structure observed in open string and semi-abelian Z-theory. The kinematic algebra given by this construction contains the Lie algebra of diffeomorphism as an obvious sub-algebra.

Quantum gravity effects on the thermodynamic stability of 4D Schwarzschild black hole

Based on the Euclidean approach, we consider the effects of quantum gravity and mass-less matter on the thermodynamic properties of Schwarzschild black hole. The techniques of effective field theory are utilized to analytically construct the partition function at the one-loop level. Using the non-local heat kernel formalism, the partition function is expressed as a curvature expansion. We extensively discuss the effect of the corrections on the thermodynamic stability. The one-loop free energy shows, remarkably, that a large number of gauge fields is able to render Schwarzschild black hole thermodynamically stable. The black hole mass at which stability is achieved scales as sqrt{N} in Planck units, where N is the number of gauge fields, and is independent of any UV completion of quantum gravity.

CONSTRUCTING A FLEXIBLE LIKELIHOOD FUNCTION FOR SPECTROSCOPIC INFERENCE

We present a modular, extensible likelihood framework for spectroscopic inference based on synthetic model spectra. The subtraction of an imperfect model from a continuously sampled spectrum introduces covariance between adjacent datapoints (pixels) into the residual spectrum. For the high signal-to-noise data with large spectral range that is commonly employed in stellar astrophysics, that covariant structure can lead to dramatically underestimated parameter uncertainties (and, in some cases, biases). We construct a likelihood function that accounts for the structure of the covariance matrix, utilizing the machinery of Gaussian process kernels. This framework specifically address the common problem of mismatches in model spectral line strengths (with respect to data) due to intrinsic model imperfections (e.g., in the atomic/molecular databases or opacity prescriptions) by developing a novel local covariance kernel formalism that identifies and self-consistently downweights pathological spectral line "outliers." By fitting many spectra in a hierarchical manner, these local kernels provide a mechanism to learn about and build data-driven corrections to synthetic spectral libraries. An open-source software implementation of this approach is available at http://iancze.github.io/Starfish, including a sophisticated probabilistic scheme for spectral interpolation when using model libraries that are sparsely sampled in the stellar parameters. We demonstrate some salient features of the framework by fitting the high resolution $V$-band spectrum of WASP-14, an F5 dwarf with a transiting exoplanet, and the moderate resolution $K$-band spectrum of Gliese 51, an M5 field dwarf.

Analytic representations with theta functions for systems on ℤ(d) and on 𝕊

An analytic representation with theta functions on a torus, for systems with variables in ℤ(d), is considered. Another analytic representation with theta functions on a strip, for systems with positions in a circle 𝕊 and momenta in ℤ, is also considered. The reproducing kernel formalism for these two systems is studied. Wigner and Weyl functions in this language are also studied.

Experimental and theoretical dosimetry of the RIC-100phosphorus-32 brachytherapy source for implant geometries encountered in the intraoperative setting.

Experimental and theoretical dosimetry of the RIC-100phosphorus-32 brachytherapy source is presented for implant geometries that may occur in an intraoperative setting during treatment of localized spinal tumors with temporary superficial radiation. Dose variation, due to source shape and size, is evaluated, and nonideal implant conditions are simulated. Calibration, depth dose, and dose profiles were evaluated for several implant geometries and source sizes. Experimental measurements were performed using EBT3 gafchromic film. Theoretical calculations were performed using dose point kernel (DPK) formalism, which simulates isotropic, monoenergetic point sources distributed uniformly throughout the source and emitting electrons radially outward. Calibration and depth dose for RIC-100 are independent of source size for diameters >1cm. Sources should be ordered with physical dimensions ∼0.2cm larger than the target size, in all dimensions, to deliver >90% prescription dose to target edges. Relative dose profile shape is approximately constant as a function of target depth. Air gaps between the source and target cause narrower dose profile widths and shallower depth dose in the therapeutic range. DPK for RIC-100 agrees with published P-32 kernels, and DPK calculations agree with measurement (within 5%) for many depths and geometries. Intraoperative placement and measurement dosimetry of RIC-100 require careful setup due to steep dose gradients. Physical source dimensions should be chosen carefully based on treatment site dimensions, and air gaps between source and target should be minimized, to prevent underdosing the target in the lateral extent. Radiological scaling should be used to calculate expected dose when nonwater materials are used in experimental measurements, such as calibration or depth dose.

Development graphs—Proof management for structured specifications

Development graphs are a tool for dealing with structured specifications in a formal program development in order to ease the management of change and reusing proofs. In this work, we extend development graphs with hiding (e.g. hidden operations). Hiding is a particularly difficult to realize operation, since it does not admit such a good decomposition of the involved specifications as other structuring operations do. We develop both a semantics and proof rules for development graphs with hiding. The rules are proven to be sound, and also complete relative to an oracle for conservative extensions. We also show that an absolutely complete set of rules cannot exist. The whole framework is developed in a way independent of the underlying logical system (and thus also does not prescribe the nature of the parts of a specification that may be hidden). We also show how various other logic independent specification formalisms can be mapped into development graphs; thus, development graphs can serve as a kernel formalism for management of proofs and of change.

Excluded volume effect on diffusion-influenced reactions in one dimension

The excluded volume (EV) effect between nonreactive like-particles of diffusion-influenced pseudo-first-order reaction A+B→C is investigated by the hierarchical Smoluchowski approach of Kuzovkov and Kotomin [Rep. Prog. Phys. 51, 1479 (1988)] and the many-particle kernel formalism of Lee et al. [J. Chem. Phys. 113, 8686 (2000)] in one dimension. Contrary to the three-dimensional analysis, the latter theory can be formulated without additional approximations in one dimension so that more accurate results are obtained. Although formulations and resulting expressions are different, these two theories show almost identical results numerically. The EV effect becomes significant at higher concentrations of B molecules as in three dimensions. However, we found that the EV effect in one dimension is more pronounced than in three dimensions. A similar trend appears as the size of the B molecule increases. Theoretical results are compared with Monte Carlo simulations. The simulation results reveal much larger EV effect than that predicted by both theories. This behavior may be attributed to the “cage” effect which is not considered in both theories.

Motional smearing of electrically recovered couplings measured from multipulse transients

AbstractThe measurement of residual dipolar and quadrupolar coupling constants in the liquid phase by using an electric field to destroy the isotropic nature of molecular tumbling is complicated by charge‐induced turbulent motion. In many cases this motion is due to charge injection at electrode surfaces, an effect that leads to an apparent removal of electrically recovered anisotropic spectral splittings when measured from a spin‐echo envelope modulation produced by a train of radio frequency (rf) pulses. To understand this averaging, the effect of quadrupolar couplings and enhanced molecular diffusion on free‐induction, spin‐echo, and Carr–Purcell signals is analytically determined in the special case of homogeneous rf pulses. Additional signal damping due to rf inhomogeneity and coupling constant heterogeneity is determined by numerically extending the kernel formalism introduced by Herzog and Hahn to understand spin diffusion in solids. Finally, the merit of the numerical approach is tested by comparison with analytical results for homogeneous rf pulses and experimental results for perdeuterated nitrobenzene involving inhomogeneous rf pulses and coupling heterogeneity. © 2001 John Wiley & Sons, Inc. Concepts Magn Reson 13: 171–189, 2001

Excluded volume effects on the diffusion-influenced reaction: The many-particle kernel approach

In most theories of diffusion-influenced reactions, the reaction system is assumed to consist of a central reactant molecule surrounded by the other reactant molecules that pass each other freely. That is, excluded volumes among the like reactant molecules are neglected. We use the many-particle kernel formalism to investigate the effect of excluded volumes on the diffusion-influenced reaction. We obtain approximate analytic expressions for the many-particle kernel and the time profile of the survival probability of reactant molecules. The result is shown to be in good agreement with the Brownian dynamics simulation.

Networked Enterprises as Multi-Agent Systems: Concept-Oriented Logic Approach

Nowadays it is popular to treat networked enterprises as multi-agent systems. Efficient engineering of multi-agent systems requires the ability to reuse formal descriptions of system components, their organization, coordination and behavior. So, the need in expressive and suitable languages for supporting such engineering becomes crucial. We propose to construct such languages on the basis of concept-oriented approach. Concept-oriented logic languages are cognitively adequate thanks to the fact that experts usually describe their knowledge in terms of concepts. In particular, we have developed an advanced knowledge representation language called CONCEPT. In this paper we show, in outline, how CONCEPT can be used as a kernel formalism in building languages for describing agents knowledge and communication, information distribution in multi-agent systems, conflict management etc.

Consistency checking in complex object database schemata with integrity constraints

Integrity constraints are rules that should guarantee the integrity of a database. Provided an adequate mechanism to express them is available, the following question arises: is there any way to populate a database which satisfies the constraints supplied by a database designer? That is, does the database schema, including constraints, admit at least a nonempty model? This work answers the above question in a complex object database environment, providing a theoretical framework, including the following ingredients: (1) two alternative formalisms, able to express a relevant set of state integrity constraints with a declarative style; (2) two specialized reasoners, based on the tableaux calculus, able to check the consistency of complex objects database schemata expressed with the two formalisms. The proposed formalisms share a common kernel, which supports complex objects and object identifiers, and which allow the expression of acyclic descriptions of: classes, nested relations and views, built up by means of the recursive use of record, quantified set, and object type constructors and by the intersection, union, and complement operators. Furthermore, the kernel formalism allows the declarative formulation of typing constraints and integrity rules. In order to improve the expressiveness and maintain the decidability of the reasoning activities, we extend the kernel formalism into two alternative directions. The first formalism, OLCP, introduces the capability of expressing path relations. Because cyclic schemas are extremely useful, we introduce a second formalism, OLCD, with the capability of expressing cyclic descriptions but disallowing the expression of path relations. In fact, we show that the reasoning activity in OLCDP (i.e., OLCP with cycles) is undecidable.

A model for irreversible scattering of gas atoms on a metal surface

Abstract Within the framework of the scattering kernel formalism, the temperature dependent isothermal energy accommodation coefficient (IEAC) is deduced from the truncated hard-cube model. This model, which introduces a threshold velocity ν ∗ for trapping, can be justified by hard-cube scattering within a rigid potential well together with electronic dissipation of energy. The irreversible mechanism is modelled as a relaxing image force.

A model for irreversible scattering of gas atoms on a metal surface

Within the framework of the scattering kernel formalism, the temperature dependent isothermal energy accommodation coefficient (IEAC) is deduced from the truncated hard-cube model. This model, which introduces a threshold velocity ν ∗ for trapping, can be justified by hard-cube scattering within a rigid potential well together with electronic dissipation of energy. The irreversible mechanism is modelled as a relaxing image force.

Accommodation in Ar(Kr, Xe) and metal-surface systems as a test for electronic dissipation

A reinterpretation of experimental results on energy accommodation for the gas-surface systems Ar W, Kr W, and Xe W is provided in the frame of the scattering kernel formalism. A threshold velocity for trapping can be drawn from the analysis. This threshold is explained on the basis of charge transfer and electronic relaxation during scattering.

A New Connected Kernel Formalism for Nuclear Reactions

A new connected kernel formalism is proposed for multichannel nuclear reactions. The basic set of equations is derived from a set of Lippmann-Schwinger equations by transforming all divergent disconnected subseries into rearrangement channel components inductively. This set of equations determines the wave function in such a way that the system is resonating among all possible groupings of the constituent nucleons just as in the resonating group theory. In the approximation where only two-body channels are retained, our connected kernel equations are exactly the same as the conventional coupled-channel equations.

A New Connected Kernel Formalism for Nuclear Reactions

A New Connected Kernel Formalism for Nuclear Reactions

A comparison of projection operator formalisms for the study of self-diffusion

Received April 18, 1973 The utility of projection operator formalisms for describing the dynamics of many-body systems is studied, and the compatibility of these formalisms with certain approximation schemes is evaluated in the light of known behavior of such systems. For simplicity the investigation is limited to the study of Brownian motion. Specifically, a memory kernel formalism and a kinetic equation formalism are compared for the calculation of the time evolution of the momentum autocorrelation function. Both perturbation expansions and averaged propagator approximations are investigated. The results from these studies suggest that the long-time behavior of the momentum autocorrelation function is sensitive to the long-range nature of the interparticle potential.