Irreducible Kernels Research Articles

Information Geometry of Reversible Markov Chains

We analyze the information geometric structure of time reversibility for parametric families of irreducible transition kernels of Markov chains. We define and characterize reversible exponential families of Markov kernels, and show that irreducible and reversible Markov kernels form both a mixture family and, perhaps surprisingly, an exponential family in the set of all stochastic kernels. We propose a parametrization of the entire manifold of reversible kernels, and inspect reversible geodesics. We define information projections onto the reversible manifold, and derive closed-form expressions for the e-projection and m-projection, along with Pythagorean identities with respect to information divergence, leading to some new notion of reversiblization of Markov kernels. We show the family of edge measures pertaining to irreducible and reversible kernels also forms an exponential family among distributions over pairs. We further explore geometric properties of the reversible family, by comparing them with other remarkable families of stochastic matrices. Finally, we show that reversible kernels are, in a sense we define, the minimal exponential family generated by the m-family of symmetric kernels, and the smallest mixture family that comprises the e-family of memoryless kernels.

Mode-coupling theory of the glass transition for colloidal liquids in slit geometry

ABSTRACT We provide a detailed derivation of the mode-coupling equations for a colloidal liquid confined by two parallel smooth walls. We introduce irreducible memory kernels for the different relaxation channels thereby extending the projection operator technique to colloidal liquids in slit geometry. Investigating both the collective dynamics as well as the tagged-particle motion, we prove that the mode-coupling functional assumes the same form as in the Newtonian case corroborating the universality of the glass-transition singularity with respect to the microscopic dynamics.

The diameter of inhomogeneous random graphs

AbstractIn this paper, we study the diameter of inhomogeneous random graphs that are induced by irreducible kernels . The kernels we consider act on separable metric spaces and are almost everywhere continuous. We generalize results known for the Erdős–Rényi model G(n, p) for several ranges of p. We find upper and lower bounds for the diameter of in terms of the expansion factor and two explicit constants that depend on the behavior of the kernel over partitions of the metric space.

Spinor structures of the Bethe-Salpeter irreducible kernels.

In this paper the possible types of the Bethe-Salpeter (BS) irreducible kernels are determined from a phenomenologically reasonable requirement, and the spinor structures of BS wave functions and equations for all the three categories of natural ${\mathit{J}}^{\mathit{P}\mathit{C}}$ mesons are obtained.

Asymptotic completeness and multiparticle structure in field theories

The ideas developed in Part I (ref. (1)) are applied to the recently constructed massive Gross-Neveu model. We define in this case an irreducible kernel satisfying a regularized Bethe-Salpeter equation which is convenient to derive asymptotic completeness in the 2-particle region. As in Part I, the method aIlows direct graphical definition of general irreducible kernels and is well suited to the analysis of asymptotic completeness and related results in more general energy regions. A large part of the paper is devoted to a new self-contained construction (via phase space expansion) of the Gross-Neveu model. The presentation is somewhat simpler than previous ones, is more complete on some points and is best suited to our purposes.

Asymptotic completeness and multiparticle structure in field theories

Previous proofs of asymptotic completeness and related results on scattering in field theories are restricted toP(ϕ)2 models in the 2- and 3-particle regions. In this paper, new cluster expansions that are well adapted to more direct proofs and generalizations of these results are presented. In contrast to previous ones, they are designed to provide direct graphical definitions of general irreducible kernels satisfying structure equations recently proposed and shown to be closely linked with asymptotic completeness and with the multiparticle structure of Green functions and collision amplitudes in general energy regions. The method can be applied as previously toP(ϕ)2 and can also be extended to theories involving renormalization which are controlled by phase-space analysis. It is here illustrated in detail for the Bethe-Salpeter kernel in ϕ 42 , in which case a new proof of its 4-particle decay (which yields asymptotic completeness in the 2-particle region) is given.

On the running coupling constant in relativistic bound-state problems

By inspection of the relation between one-particle irreducible functions and Bethe-Salpeter kernels and making use of Weinberg's theorem the running coupling constant for a given bound state is shown to have the usual form in the short-distance region of an asymptotically free theory. This is done explicitly for a two-body bound state in the absence ofs-channel contributions which could induce difficulties. Two different choices for the boundary condition on the running coupling are pointed out, the more unconventional one is found preferable for several reasons. This difference would manifest itself unambiguously in strongly relativistic systems.

General Irreducible Markov Chains and Non-Negative Operators.

Preface 1. Preliminaries 2. Irreducible kernels 3. Transience and recurrence 4. Embedded renewal processes 5. Positive and null recurrence 6. Total variation limit theorems 7. Miscellaneous limit theorems for Harris recurrent Markov chains Notes and comments List of symbols and notation Bibliography Index.

R-particle irreducible kernels, asymptotic completeness and analyticity properties of several particle collision amplitudes

r-particle irreducible kernels, asymptotic completeness and analyticity properties of several particle collision amplitudes

Irreducible kernels and nonperturbative expansions in a theory with purem ?m interaction

Recent results on the structure of theS matrix at them-particle threshold (m≧2) in a simplifiedm→m scattering theory with no subchannel interaction are extended to the Green functionF on the basis of off-shell unitarity, through an adequate mathematical extension of some results of Fredholm theory: local two-sheeted or infinite-sheeted structure ofF arounds=(mμ)2 depending on the parity of (m−1)(ν−1) (where μ>0 is the mass and ν is the dimension of space-time), off-shell definition of the irreducible kernelU which is the analogue of theK matrix in the two different parity cases (m−1)(ν−1) odd or even, and related local expansion ofF, for (m−1)(ν−1) even, in powers of σβ ln σ(σ=(mμ)2−s). It is shown that each term in this expansion is the dominant contribution to a Feynman-type integral in which each vertex is a kernelU. The links between the kernelU and Bethe-Salpeter type kernelsG of the theory are exhibited in both parity cases, as also the links between the above expansion ofF and local expansions, in the Bethe-Salpeter type framework, ofFλ in terms of Feynman-type integrals in which each vertex is a kernelG and which include both dominant and subdominant contributions.

Mass spectrum of the two dimensional λφ4−1/4φ2−μφ quantum field model

It is shown thatr-particle irreducible kernels in the two-dimensional λφ4−1/4φ2−μφ quantum field theory have (r+1)-particle decay for |μ|≦λ2≪1. As a consequence there is an upper mass gap and, in the subspace of two-particle states, a bound state. The proof extends Spencer's expansion [20] to handle fluctuations between the two wells of the classical potential. A new method for resumming the low temperature cluster expansion is introduced.

The decay of the Bethe-Salpeter kernel inP(ϕ)2 quantum field models

We extend methods of high temperature expansions to show that for even weakly coupledP(ϕ)2 quantum field models the Bethe-Salpeter kernel has 4 particle decay. More precisely ifK denotes the Euclidean Bethe-Salpeter kernel $$|K(x_1 ,x_2 ,x_3 ,x_4 )| \leqq Oe^{ - m_0 (1 - \varepsilon )d_2 } ,$$ wherex=(x0,x1),d2=2|x10+x20−x30−x40|+|x10−x20|+|x30−x40| and e(λ)→0 as λ→0. Our methods apply to otherr particle irreducible kernels.

Spectral properties of integral operators with nonnegative kernels

This paper is concerned with spectral properties of an integral operator with a nonnegative continous kernel. Various conditions are given for the existence of a positive eigenvalue of maximum modulus, its simplicity, and the existence of a corresponding positive eigenfunction. Other results describe the distribution of secondary eigenvalues. The operators considered include those with stochastic kernels and those with irreducible kernels. In addition to new results, simple new proofs are given for fundamental properties of operators with positive kernels. The analysis involves collectively compact operator approximation theory, transpose operators, the spectral mapping theorem, and an extremal characterization of the spectral radius.

Split and minimal abelian extensions of finite groups

Criteria for an abelian extension of a group to split are given in terms of a Sylow decomposition of the kernel and of normal series for the Sylow subgroups. An extension is minimal if only the entire extension is carried onto the given group by the canonical homomorphism. Various basic results on minimal extensions are given, and the structure question is related to the case of irreducible kernels of prime exponent. It is proved that an irreducible modular representation of SL(2, p) or PSL(2, p) for p prime and > 5 afford a minimal extension with kernel of exponent p only when the representation has degree 3, 3 i.e., when the kernel has order p

Laplace Transforms and the Diagonalization of Bethe-Salpeter Equations for Absorptive Parts

A Laplace transform is developed for the crossed-channel partial-wave analysis of Bethe-Salpeter-like equations for the absorptive part of scattering matrix elements. The transform requires no assumption about rotating contours to a Euclidian region and allows from the outset power growth in energy of the transformed absorptive part. This eliminates the need for any analytic continuation after the transformation. The diagonalization of the forward and nonforward equations with arbitrary irreducible kernels is explicitly carried out.

Three-Dimensional Covariant Integral Equations for Low-Energy Systems

Integral equations suitable for the dynamical treatment of strongly interacting particles are derived. The equations can be described as Bethe-Salpeter equations with one particle restricted to the mass shell, resulting in a three-dimensional covariant equation which can be easily interpreted physically. To restore the dynamical terms omitted in the process of restricting one particle to the mass shell, additional kernels are added to the irreducible kernels from the original Bethe-Salpeter equation. The addition of these extra terms leads to a resulting simplification in the kernels themselves, since the new kernels have the same structure as the original ones, with some partial cancellations. Estimates as to the convergence of the procedure and the sizes of the various potentials are given. The special case of the hydrogen atom is discussed briefly, and comments are made on the application of these equations to the nuclear-force problem. Connections between scattering equations and bound-state equations are discussed, and the relativistic normalization condition for bound-state wave functions is derived.