Interaction of Elastic and Scalar Fields

The three-dimensional mathematical problems of the interaction of an elastic and some scalar fields are investigated. It is assumed that the elastic structure under consideration is a bounded homogeneous anisotropic body occupying domain Ω¯⁺⊂ℝ³ and the physical scalar field is defined in the exterior domain Ω⁻ = ℝ³\Ω⁺. These two fields satisfy the governing equations in the corresponding domains together with the transmission conditions on the interface ∂Ω⁺. The problems are studied by the potential method and the existence and uniqueness theorems are proved.     

Lax Representation for a Triplet of Scalar Fields

We construct a 3×3 matrix zero-curvature representation for the system of three two-dimensional relativistically invariant scalar fields. This system belongs to the class described by the Lagrangian L = [g _ij(u)u _x ⁱ u _t ^j]/2+f(u), where g _ij is the metric tensor of a three-dimensional reducible Riemannian space. We previously found all systems of this class that have higher polynomial symmetries of the orders 2, 3, 4, or 5. In this paper, we find a zero-curvature representation for one of these systems. The calculation is based on the analysis of an evolutionary system u _t = S(u), where S is one of the higher symmetries. This approach can also be applied to other hyperbolic systems. We also find recursion relations for a sequence of conserved currents of the triplet of scalar fields under consideration.     

Spacelike Localization of Long-Range Fields in a Model of Asymptotic Electrodynamics

A previously proposed algebra of asymptotic fields in quantum electrodynamics is formulated as a net of algebras localized in regions which in general have unbounded spacelike extension. Electromagnetic fields may be localized in ‘symmetrical spacelike cones’, but there are strong indications this is not possible in the present model for charged fields, which have tails extending in all space directions. Nevertheless, products of appropriately ‘dressed’ fermion fields (with compensating charges) yield bi-localized observables.     

Quantum Backreaction (Casimir) Effect II. Scalar and Electromagnetic Fields

Casimir effect in most general terms may be understood as a backreaction of a quantum system causing an adiabatic change of the external conditions under which it is placed. This paper is the second installment of a work scrutinizing this effect with the use of algebraic methods in quantum theory. The general scheme worked out in the first part is applied here to the discussion of particular models. We consider models of the quantum scalar field subject to external interaction with “softened” Dirichlet or Neumann boundary conditions on two parallel planes. We show that the case of electromagnetic field with softened perfect conductor conditions on the planes may be reduced to the other two. The “softening” is implemented on the level of the dynamics, and is not imposed ad hoc, as is usual in most treatments, on the level of observables. We calculate formulas for the backreaction energy in these models. We find that the common belief that for electromagnetic field the backreaction force tends to the strict Casimir formula in the limit of “removed cutoff” is not confirmed by our strict analysis. The formula is model dependent and the Casimir value is merely a term in the asymptotic expansion of the formula in inverse powers of the distance of the planes. Typical behaviour of the energy for large separation of the plates in the class of models considered is a quadratic fall-of. Depending on the details of the “softening” of the boundary conditions the backreaction force may become repulsive for large separations. Communicated by Klaus Fredenhagen submitted 9/09/04, accepted 1/07/05     

Dimensional Regularization and the n-Wave Procedure for Scalar Fields in Many-Dimensional Quasi-Euclidean Spaces

We obtain the vacuum expectation values of the energy–momentum tensor for a scalar field arbitrarily coupled to a curvature in the case of an N-dimensional quasi-Euclidean space–time; the vacuum is defined in accordance with the Hamiltonian diagonalization method. We extend the n-wave procedure to the many-dimensional case. We find all the counterterms in the case N=5 and the counterterms for the conformal scalar field in the cases N=6,7. We determine the geometric structure of the first three counterterms in the N-dimensional case. We show that all the subtractions in the four-dimensional case and the first three subtractions in the many-dimensional case correspond to the renormalization of the parameters in the bare gravitational Lagrangian. We discuss the geometric structure of the other counterterms in the many-dimensional case and the problem of eliminating the conformal anomaly in the four-dimensional case.     

Scalar Curvature, Killing Vector Fields and Harmonic One-Forms on Compact Riemannian Manifolds

It is well known that no non-trivial Killing vector field existson a compact Riemannian manifold of negative Ricci curvature;analogously, no non-trivial harmonic one-form exists on a compactmanifold of positive Ricci curvature. One can consider the following,more general, problem. By reducing the assumption on the Riccicurvature to one on the scalar curvature, such vanishing theoremscannot hold in general. This raises the question: "What informationcan we obtain from the existence of non-trivial Killing vectorfields (or, respectively, harmonic one-forms)?" This paper givesanswers to this problem; the results obtained are optimal. 2000Mathematics Subject Classification 53C20 (primary), 53C24 (secondary).     

Constrained Extremum Problems,Regularity Conditions and Image Space Analysis. Part I: The Scalar Finite-Dimensional Case

Image space analysis has proved to be instrumental in unifying several theories, apparently disjoint from each other. With reference to constraint qualifications/regularity conditions in optimization, such an analysis has been recently introduced by Moldovan and Pellegrini. Based on this result, the present paper is a preliminary part of a work, which aims at exploiting the image space analysis to establish a general regularity condition for constrained extremum problems. The present part deals with scalar constrained extremum problems in a Euclidean space. The vector case as well as the case of infinite-dimensional image will be the subject of a subsequent part.     

Scattering of Dirac Particles by Electromagnetic Fields with Small Support in Two Dimensions and Effect from Scalar Potentials

We study the asymptotic behavior of scattering amplitudes for the scattering of Dirac particles in two dimensions when electromagnetic fields with small support shrink to point-like fields. The result is strongly affected by perturbations of scalar potentials and the asymptotic form changes discontinuously at half-integer fluxes of magnetic fields even for small perturbations. The analysis relies on the behavior at low energy of resolvents of magnetic Schrödinger operators with resonance at zero energy. The magnetic scattering of relativistic particles appears in the interaction of cosmic string with matter. We discuss this closely related subject as an application of the obtained results. Communicated by Bernard Helffersubmitted 05/05/03, accepted 31/07/03     

Stability of Quantum Electrodynamics and Quantum Chromodynamics

We consider the vacuum energy in QED viewed as in a system of charged fermions and bosons and in QCD viewed as in a system of quarks (fermions) and gluons (bosons) in a self-dual field with a constant strength. We show that the cause of instability is the instability of bosons in the self-dual vacuum field. For the global stability of a system consisting of fermions and bosons, the number of fermions should be sufficiently large. The nonzero self-dual field leading to the confinement of fermions realizes the minimum of the vacuum energy in the case where the boson has the smallest mass in the system. Confinement therefore does not arise in QED, where the fermion (electron) has the smallest mass, and does arise in QCD, where the boson (gluon) has the smallest mass.     

The Discrete Logarithm Problem Over Prime Fields: The Safe Prime Case. The Smart Attack,Non-Canonical Lifts and Logarithmic Derivatives

We connect the discrete logarithm problem over prime fields in the safe prime case to the logarithmic derivative.     

Parametrization of the Solution of the Kepler Problem and New Adaptive Numerical Methods Based on This Parametrization

We propose a one-parameter family of adaptive numerical methods for solving the Kepler problem. The methods preserve the global properties of the exact solution of the problem and approximate the time dependence of the phase variables with the second or fourth approximation order. The variable time increment is determined automatically from the properties of the solution.     

Prescribing Morse Scalar Curvatures: Pinching and Morse Theory

We consider the problem of prescribing conformally the scalar curvature on compact manifolds of positive Yamabe class in dimension . We prove new existence results using Morse theory and some analysis on blowing-up solutions under suitable pinching conditions on the curvature function. We also provide new nonexistence results showing the sharpness of some of our assumptions, both in terms of the dimension and of the Morse structure of the prescribed function. © 2021 Wiley Periodicals, Inc.     

Matrices of Scalar Differential Operators: Divisibility and Spaces of Solutions

Computational Mathematics and Mathematical Physics - We investigate the connection between divisibility of full-rank square matrices of linear scalar differential operators over some differential...     

Clifford Algebras and Euclid’s Parametrization of Pythagorean Triples

We show that the space of Euclid’s parameters for Pythagorean triples is endowed with a natural symplectic structure and that it emerges as a spinor space of the Clifford algebra R₂₁, whose minimal version may be conceptualized as a 4-dimensional real algebra of “kwaternions.” We observe that this makes Euclid’s parametrization the earliest appearance of the concept of spinors. We present an analogue of the “magic correspondence” for the spinor representation of Minkowski space and show how the Hall matrices fit into the scheme. The latter obtain an interesting and perhaps unexpected geometric meaning as certain symmetries of an Apollonian gasket. An extension to more variables is proposed and explicit formulae for generating all Pythagorean quadruples, hexads, and decuples are provided.     

Imprimitive Parametrization of Analytic Curves and Factorizations of Entire Functions

Let f(z) and g(z) be transcendental entire functions. Fuchsand Song proved that if (f(z); g(z)) parametrizes some complexalgebraic curve, then f and g must have a transcendental commonright factor. The paper proves this result by a different methodthat also allows a similar result to be proved for some transcendentalcurves. This result is then used to solve some factorizationproblems of entire functions.