About the chiral symmetry breaking in QED<Subscript>3</Subscript>

The problem of the chiral symmetry breaking in QED₃ is considered by solving the Schwinger–Dyson equation for the fermion propagator in the ladder approximation using the Landau gauge for the photon propagator. Within the framework of the indicated approximation, different simplifications that allow expressions for the fermion mass function to be retrieved in an explicit form are analyzed. The results obtained are compared with the data of numerical analysis. It appears that the neglect of higher Gegenbauer harmonics in the kernel of the initial integral equation for the fermion mass function influences the dynamic mass value and the asymptotics of the mass function only weakly. On the other hand, it is established that the conclusion about a complicated structure of the fermion vacuum of the massive phase is an artifact of linearization of the Schwinger–Dyson equation kernel: consideration of the kernel nonlinearity yields a simple massive phase structure of the fermion vacuum.     

Path integration of the harmonic oscillator with a two-time quadratic action

Path integration of the harmonic oscillator with a two-time quadratic action characterizing memory effects is given in terms of the solutions of some integrodifferential equations. The exact propagator in closed form is then obtained for the specific kernel introduced by Feynman in the polaron problem.     

Solving the Bethe-Salpeter equation for two fermions in Minkowski space

The method of solving the Bethe-Salpeter equation in Minkowski space, developed previously for spinless particles (V.A. Karmanov, J. Carbonell, Eur. Phys. J. A 27, 1 (2006)), is extended to a system of two fermions. The method is based on the Nakanishi integral representation of the amplitude and on projecting the equation on the light-front plane. The singularities in the projected two-fermion kernel are regularized without modifying the original BS amplitudes. The numerical solutions for the J = 0 bound state with the scalar, pseudoscalar and massless vector exchange kernels are found. The stability of the scalar and positronium states without vertex form factor is discussed. Binding energies are in close agreement with the Euclidean results. Corresponding amplitudes in Minkowski space are obtained.     

Null-plane dynamics for few-quark systems

A Lorentz-invariant formulation of the 3-dimensional harmonic oscillator is proposed for serving as the confining part of the kernel of a Bethe-Salpeter equation for aqq system, and likewise for aqqq system. Such a kernel is amenable to an effective 3-dimensional reduction in the null-plane approximation (NPA), so as to maintain explicit NPA-covariance for the (reduced) BS equations as well as the corresponding wave functions. This rectifies the mathematical limitation of an earlier BS formulation (by Mitra and coworkers) forqq andqqq systems with non-covariant h. o. kernels which was restricted to the instantaneous approximation and was therefore valid only for slowly moving hadrons. The present (covariant) formulation agrees exactly with the earlier (non-covariant) treatment for hadrons at rest, thus preserving all the physical results on mass spectra, etc., obtained with the latter, but is now formally applicable to processes involving hadrons in motion.     

Chebyshev polynomials and Fourier transform of <Emphasis Type="Italic">SU</Emphasis>(2) irreducible representation character as spin tomographic star-product kernel

Spin tomographic symbols of qudit states and spin observables are studied. Spin observables are associated with the functions on a manifold whose points are labeled by the spin projections and sphere S ² coordinates. The star-product kernel for such functions is obtained in an explicit form and connected with the Fourier transform of characters of the SU(2) irreducible representation. The kernels are shown to be in close relation to the Chebyshev polynomials. Using specific properties of these polynomials, we establish the recurrence relation between the kernels for different spins. Employing the explicit form of the star-product kernel, a sum rule for Clebsch–Gordan and Racah coefficients is derived. Explicit formulas are obtained for the dual tomographic star-product kernel as well as for intertwining kernels which relate spin tomographic symbols and dual tomographic symbols.     

Propositional Kernels

The pervasive presence of artificial intelligence (AI) in our everyday life has nourished the pursuit of explainable AI. Since the dawn of AI, logic has been widely used to express, in a human-friendly fashion, the internal process that led an (intelligent) system to deliver a specific output. In this paper, we take a step forward in this direction by introducing a novel family of kernels, called Propositional kernels, that construct feature spaces that are easy to interpret. Specifically, Propositional Kernel functions compute the similarity between two binary vectors in a feature space composed of logical propositions of a fixed form. The Propositional kernel framework improves upon the recent Boolean kernel framework by providing more expressive kernels. In addition to the theoretical definitions, we also provide an algorithm (and the source code) to efficiently construct any propositional kernel. An extensive empirical evaluation shows the effectiveness of Propositional kernels on several artificial and benchmark categorical data sets.     

Study of the gluon propagator in the axial gauge

It is shown that a non-linear integral equation for the gluon propagator in the axial gauge (Baker et al.) can be simplified considerably. A comparison is made with an approximate equation for the gluon propagator in the Landau gauge (Mandelstam). Both equations have polynomial kernels where the argument is the divisor of the internal and external momenta. A solution which behaves as a double pole for low momenta remains consistent.     

Phase space propagators for quantum operators

The Heisenberg evolution of a given unitary operator corresponds classically to a fixed canonical transformation that is viewed through a moving coordinate system. The operators that form the bases of the Weyl representation and its Fourier transform, the chord representation are, respectively, unitary reflection and translation operators. Thus, the general semiclassical study of unitary operators allows us to propagate arbitrary operators, including density operators, i.e., the Wigner function. The various propagation kernels are different representations of the super-operators which act on the space of operators of a closed quantum system. We here present the mixed semiclassical propagator, that takes translation chords to reflection centres, or vice versa. In contrast to the centre-centre propagator that directly evolves Wigner functions, they are guaranteed to be caustic free, having a simple WKB-like universal form for a finite time, whatever the number of degrees of freedom. Special attention is given to the near-classical region of small chords, since this dominates the averages of observables evaluated through the Wigner function.     

Difference between standard and quasi-conformal BFKL kernels

As it was recently shown, the colour singlet BFKL kernel, taken in Möbius representation in the space of impact parameters, can be written in quasi-conformal shape, which is unbelievably simple compared with the conventional form of the BFKL kernel in momentum space. It was also proved that the total kernel is completely defined by its Möbius representation. In this paper we calculated the difference between standard and quasi-conformal BFKL kernels in momentum space and discovered that it is rather simple. Therefore we come to the conclusion that the simplicity of the quasi-conformal kernel is caused mainly by using the impact parameter space.     

Low-x evolution equations in M?bius representation

The M?bius form of the BFKL kernel in the next-to-leading order (NLO) in theories containing fermions and scalars in arbitrary representations of the colour group is presented. The ambiguity of the NLO kernels permits to get agreement between the BFKL approach and the colour dipole model and to find the quasi-conformal representation of the BFKL kernel.     

The Bethe-Salpeter equation with fermions

The Bethe-Salpeter (BS) equation in the ladder approximation is studied within a fermion theory: two fermion fields (constituents) with mass m interacting via an exchange of a scalar field with mass μ. The BS equation can be written in the form of an integral equation in the configuration Euclidean x-space with the symmetric kernel K for which Tr K ² = ∞ due to the singular character of the fermion propagator. This kernel is represented in the form K = K ₀ + K _I . The operator K ₀ with Tr K ₀ ² = ∞ is of the “fall at the center” potential type and describes a continuous spectrum only. Besides the presence of this operator leads to a restriction on the value of the coupling constant. The kernel K _I with Tr K _I ² < ∞ is responsible for bound fermion-fermion states. Our approach is that the eigenvalue problem of the equation $\Lambda\Psi = g^2(K_0 + K_I)\Psi \qquad {\rm with}\qquad \Lambda = 1The Bethe-Salpeter (BS) equation in the ladder approximation is studied within a fermion theory: two fermion fields (constituents) with mass m interacting via an exchange of a scalar field with mass μ. The BS equation can be written in the form of an integral equation in the configuration Euclidean x-space with the symmetric kernel K for which Tr K ² = ∞ due to the singular character of the fermion propagator. This kernel is represented in the form K = K ₀ + K _I . The operator K ₀ with Tr K ₀ ² = ∞ is of the “fall at the center” potential type and describes a continuous spectrum only. Besides the presence of this operator leads to a restriction on the value of the coupling constant. The kernel K _I with Tr K _I ² < ∞ is responsible for bound fermion-fermion states. Our approach is that the eigenvalue problem of the equation can be rewritten in the form The kernel of the last equation is finite for g ² < g _c ² and the variational procedure of calculations of eigenvalues and eigenfunctions can be applied. The quantum pseudoscalar and scalar mesodynamics is considered. The binding energy of the state 1⁺ (deuteron) as a function of the coupling constant is calculated in the framework of the procedure formulated above. It is shown that this bound state is absent in the pseudoscalar mesodynamics and does exist in the scalar mesodynamics. A comparison with the non-relativistic Schr?dinger picture is made. Correspondence: G. V. Efimov, Bogoliubov Laboratory of Theoretical Physics, Joint Institute for Nuclear Research, 141980 Dubna, Russia     

Cusp kernels for velocity-changing collisions

We introduce an analytical kernel, the "cusp" kernel, to model the effects of velocity-changing collisions on optically pumped atoms in low-pressure buffer gases. Like the widely used Keilson-Storer kernel [J. Keilson and J.?E. Storer, Q. Appl. Math. 10, 243 (1952)], cusp kernels are characterized by a single parameter and preserve a Maxwellian velocity distribution. Cusp kernels and their superpositions are more useful than Keilson-Storer kernels, because they are more similar to real kernels inferred from measurements or theory and are easier to invert to find steady-state velocity distributions.     

The PCAC relation in the NJL model in the BS/Faddeev approach

Evaluations of bound state expectation values of quantized one-body operators in the canonical operator formalism are reviewed, based on the Bethe-Salpeter(BS)/Faddeev approach in the NJL model. The PCAC relation in the BS/Faddeev approach is derived without assuming any particular form of the Bethe-Salpeter kernel. These results are applied to ladder truncation schemes and the truncation scheme in which the qq interactions induced by the q exchange in the pion and sigma mesonic channel are included.     

基于可见光光谱高效鉴别玉米单倍体籽粒

单倍体技术已发展成为玉米遗传研究及现代玉米育种的重要技术之一,单倍体籽粒的鉴别筛选是其中的重要环节。目前单倍体籽粒主要是依赖于籽粒的R1-nj遗传标记通过人工肉眼观察颜色的有或无进行鉴别,费时费工。而且部分材料由于标记颜色很难从籽粒外部观察到,导致人工筛选准确率较低。基于可见光光谱分析建立玉米单倍体籽粒鉴别方法,探索利用可见光光谱鉴别玉米单倍体籽粒的可行性。同时,由于每季用于诱导单倍体的育种材料不尽相同,模型须能够鉴别未参加建模的材料的单倍体。本研究以9个遗传背景的单倍体和杂交籽粒共284粒作为试验材料,利用便携式紫外-可见光光纤光谱仪采集单个玉米籽粒的可见光漫透射光谱。光谱数据经平滑、矢量归一化预处理和主成分分析,基于支持向量机方法建立单倍体和杂交籽粒判别模型。每次选择1个背景的样本作为测试集,其余背景的样本作为建模集对模型进行交叉验证。模型交叉验证平均正确判别率达到92.06%。其中8次测试正确判别率在85%以上。结果表明利用可见光光谱分析建立玉米单倍体籽粒鉴别方法,并使模型可鉴别未参与建模材料的单倍体具有可行性。并且基于该方法有望建立玉米单倍体籽粒的自动化快速筛选系统,提高玉米单倍体育种效率。     

Nucleon properties in the covariant quark-diquark model

In the covariant quark-diquark model the effective Bethe-Salpeter (BS) equations for the nucleon and the Δ are solved including scalar and axial vector diquark correlations. Their quark substructure is effectively taken into account in both, the interaction kernel of the BS equations and the currents employed to calculate nucleon observables. Electromagnetic current conservation is maintained. The electric form factors of proton and neutron match the data. Their magnetic moments improve considerably by including axial vector diquarks and photon induced scalar-axial vector transitions. The isoscalar magnetic moment can be reproduced, the isovector contribution is about 15% too small. The ratio μG _E/G _M and the axial and strong couplings g _A, g _NN, provide an upper bound on the relative importance of axial vector diquarks confirming that scalar diquarks nevertheless describe the dominant 2-quark correlations inside nucleons. Received: 3 July 2000 / Accepted: 15 August 2000     

Classical spin and quantum propagation

We consider a classical Brownian motion model of diffusion in two spatial dimensions, where the Brownian particle moves on spiral paths. The classical spin does not change the propagator for the probability density for the position of the particle. However, the subdominant eigenvalues of the classical kernel are simply related to the dominant eigenvalues of the Feynman kernel for an analogous quantum system. The Feynman kernel can be extracted from the classical kernel by coupling to a spin angular momentum of the particle.     

Autoignition-controlled flame initiation and flame stabilization in a reacting jet in crossflow

This paper describes an analysis of the mechanisms of autoignition-controlled flame initiation and flame stabilization in a nonpremixed jet in crossflows, using simultaneous high-speed (10 kHz) tomographic particle image velocimetry, OH-PLIF and line-of-sight flame emissions. Measurements are conducted on a turbulent, transverse, reacting propane jet issued into a crossflow generated by combustion of natural gas at an equivalence ratio of 0.4 with the crossflow velocity of 10 m/s, the crossflow temperature of 1350 K and the jet momentum flux ratio of 41. While several prior studies have analyzed the lifted character of the flame in similar configurations, we show that several dynamic processes precede the leading edge of the lifted diffusion flame, including formation and evolution of “autoignition kernels”, “flame kernels” and “flame fragments”. “Autoignition kernels”, i.e., discrete compact reaction zones with the peak hydroxyl (OH) fluorescence intensity below that of the diffusion flame, initiate preferably at bulges along the jet periphery where the strain rates and the scalar dissipation rates are lower. The autoignition kernel grows in both size and the OH-fluorescence intensity as it convects downstream. An autoignition kernel transitions into a propagating flame kernel, which quickly gets distorted and elongated in the direction of the principal expansion strain rate to form a flame fragment. Neighboring flame fragments merge with each other and with the downstream diffusion flame via edge-flame propagation. Merging of upstream flame fragments with the downstream diffusion flame results in an upstream advancement of the diffusion-flame front. The diffusion flame front is intrinsically unsteady because of the rather random formation and evolution of autoignition kernels, flame kernels and flame fragments, presumably due to the stochastic velocity, the strain rate and mixture-fraction oscillations.     

Acceleration-induced nonlocality: uniqueness of the kernel

We consider the problem of uniqueness of the kernel in the nonlocal theory of accelerated observers. In a recent work [Ann. Phys. (Leipzig) 11 (2002) 309], we showed that the convolution kernel is ruled out as it can lead to divergences for nonuniform accelerated motion. Here we determine the general form of bounded continuous kernels and use observational data regarding spin-rotation coupling to argue that the kinetic kernel given by K(τ,τ′)=k(τ′) is the only physically acceptable solution.     

Evolution kernel for the dirac field

The evolution kernel for the free Dirac field is calculated using Wilson lattice fermions. We discuss the difficulties due to which this calculation has not been previously performed in continuum theory. The continuum limit is taken, and the complete energy eigenfunctions as well as the propagator are then evaluated in a new manner using the kernel.     

A variational principle for the Landau equation

The Landau equation describing the collective modes of an infinite many-fermion system is cast into a form which displays explicitly its structure as a homogeneous Fredholm equation of the second kind with non-symmetric kernel. A variational principle appropriate to this equation is constructed by noting the analogy to the integral form of the Schrödinger equation. The Landau equation is also re-expressed in terms of two coupled equations with symmetric kernels.