Bogdanov–Takens bifurcation in a predator–prey model

In this paper, we investigate a class of predator–prey model with age structure and discuss whether the model can undergo Bogdanov–Takens bifurcation. The analysis is based on the normal form theory and the center manifold theory for semilinear equations with non-dense domain combined with integrated semigroup theory. Qualitative analysis indicates that there exist some parameter values such that this predator–prey model has an unique positive equilibrium which is Bogdanov–Takens singularity. Moreover, it is shown that under suitable small perturbation, the system undergoes the Bogdanov–Takens bifurcation in a small neighborhood of this positive equilibrium.     

Meromorphic solutions in the FitzHugh–Nagumo model

The FitzHugh–Nagumo model is studied in the framework of analytic theory of differential equations. The Nevanlinna theory is used to find all meromorphic solutions of a second-order ordinary differential equation related to the FitzHugh–Nagumo model. As a consequence new exact solutions of the FitzHugh–Nagumo system are obtained in explicit form.     

SU(3) Color Gauge Invariance and the Jaffe–Witten Mass Gap in QCD

We discuss the role of group theory, the theory of distributions, and some theorems of the theory of functions of complex variable in connection with the so-called Jaffe–Witten mass gap in QCD, which is responsible for the large-scale structure of the QCD ground state and hence plays a dominant role in QCD as a theory of strong interactions at low energies. We show how the mass gap may appear without violating the SU(3) color gauge invariance of QCD. The theory of generalized functions (distributions) and the Weierstrass–Sochocki–Casorati theorem are used in order to perform the renormalization of the regularized mass gap.     

Comparison of linear beam theories

In this paper we consider three models for a cantilever beam based on three different linear theories: Euler–Bernoulli, Timoshenko and two-dimensional elasticity. Using the natural frequencies and modes as a yardstick, we conclude that the Timoshenko theory is close to the two-dimensional theory for modes of practical importance, but that the applicability of the Euler–Bernoulli theory is limited.     

Topological degree theory and fixed point theorems in fuzzy normed space

In this paper, the Leray–Schauder topological degree theory is developed in a fuzzy normed space. Since the linear topology on this fuzzy normed space is not necessarily locally convex, and since each Menger probabilistic normed space can be considered as a special fuzzy normed space, the degree theory in this paper is different from the degree theory in locally convex linear topological space presented by Nagumo (Amer. J. Math. 73 (1951) 497–511), and it also is an extension of the degree theory in Menger probabilistic normed space studied by Zhang and Chen (Appl. Math. Mech. 10(6) (1989) 477–486). Applying this degree theory, some fixed point theorems for operators are given in fuzzy normed spaces, and some former corresponding results are extended and improved.     

From Riemann and Kodaira to Modern Developments on Complex Manifolds

We survey the theory of complex manifolds that are related to Riemann surface, Hodge theory, Chern class, Kodaira embedding and Hirzebruch–Riemann–Roch, and some modern development of uniformization theorems, Kähler–Einstein metric and the theory of Donaldson–Uhlenbeck–Yau on Hermitian Yang–Mills connections. We emphasize mathematical ideas related to physics. At the end, we identify possible future research directions and raise some important open questions.     

Tetrad-Gauge Theory of Gravity

We present a tetrad–gauge theory of gravity based on the local Lorentz group in a four-dimensional Riemann–Cartan space–time. Using the tetrad formalism allows avoiding problems connected with the noncompactness of the group and includes the possibility of choosing the local inertial reference frame arbitrarily at any point in the space–time. The initial quantities of the theory are the tetrad and gauge fields in terms of which we express the metric, connection, torsion, and curvature tensor. The gauge fields of the theory are coupled only to the gravitational field described by the tetrad fields. The equations in the theory can be solved both for a many-body system like the Solar System and in the general case of a static centrally symmetric field. The metric thus found coincides with the metric obtained in general relativity using the same approximations, but the interpretation of gravity is quite different. Here, the space–time torsion is responsible for gravity, and there is no curvature because the curvature tensor is a linear combination of the gauge field tensors, which are absent in the case of pure gravity. The gauge fields of the theory, which (together with the tetrad fields) define the structure of space–time, are not directly coupled to ordinary matter and can be interpreted as the fields describing dark energy and dark matter.     

Dirac reduction for nonholonomic mechanical systems and semidirect products

This paper develops the theory of Dirac reduction by symmetry for nonholonomic systems on Lie groups with broken symmetry. The reduction is carried out for the Dirac structures, as well as for the associated Lagrange–Dirac and Hamilton–Dirac dynamical systems. This reduction procedure is accompanied by reduction of the associated variational structures on both Lagrangian and Hamiltonian sides. The reduced dynamical systems obtained are called the implicit Euler–Poincaré–Suslov equations with advected parameters and the implicit Lie–Poisson–Suslov equations with advected parameters. The theory is illustrated with the help of finite and infinite dimensional examples. It is shown that equations of motion for second order Rivlin–Ericksen fluids can be formulated as an infinite dimensional nonholonomic system in the framework of the present paper.     

Sturm intersection theory for periodic Jacobi matrices and linear Hamiltonian systems

Sturm–Liouville oscillation theory for periodic Jacobi operators with matrix entries is discussed and illustrated. The proof simplifies and clarifies the use of intersection theory of Bott, Maslov and Conley–Zehnder. It is shown that the eigenvalue problem for linear Hamiltonian systems can be dealt with by the same approach.     

Vertex maps on graphs – Perron–Frobenius theory

The goal of this paper is to describe the connections between Perron–Frobenius theory and vertex maps on graphs. In particular, it is shown how Perron–Frobenius theory gives results about the sets of integers that can arise as periods of periodic orbits, about the concepts of transitivity and topological mixing and about horseshoes and topological entropy.     

Mathematical foundations for a theory of confidence structures

This paper introduces a new mathematical object: the confidence structure. A confidence structure represents inferential uncertainty in an unknown parameter by defining a belief function whose output is commensurate with Neyman–Pearson confidence. Confidence structures on a group of input variables can be propagated through a function to obtain a valid confidence structure on the output of that function. The theory of confidence structures is created by enhancing the extant theory of confidence distributions with the mathematical generality of Dempster–Shafer evidence theory. Mathematical proofs grounded in random set theory demonstrate the operative properties of confidence structures. The result is a new theory which achieves the holistic goals of Bayesian inference while maintaining the empirical rigor of frequentist inference.     

Lie Superalgebras and Calogero–Moser–Sutherland Systems

We review recent results obtained at the intersection of the theory of quantum deformed Calogero–Moser–Sutherland systems and the theory of Lie superalgebras. We begin with a definition of admissible deformations of root systems of basic classical Lie superalgebras. For classical series, we prove the existence of Lax pairs. Connections between infinite-dimensional Calogero–Moser–Sutherland systems, deformed quantum CMS systems, and representation theory of Lie superalgebras are discussed.     

Microstructural behaviour of transversal isotropic material

We discuss and simulate transversal isotropic material under tension loading. The preferential direction of the material is inclined under 45 degrees to the direction of the tensile resultant. In this configuration the deformation of a rectangular test specimen differ from the behaviour of isotropic material in the way, that beside Poissons effect additional displacement appear perpendicular to the tension direction. In classical continuum theories, this transverse deformations describe a typical S–shape. By using a non–local continuum theory, the effect of microstructural orientation is incorporated into the numerical model. Then, it depends on a phenomenological parameter of inner structure whether the energetically favoured configuration is classical or contains microstructural behaviour. In the second case, the transverse deformation is not described by the typical S–shape, but with higher forms of it. A simple experimental model will show the connection between the inner structure of the material and the rotational parameters within the non–local continuum theory. It is evident, that these parameters are responsible for the non–classical behaviour and give the possibility to find energetically favoured solutions. The results of the finite–element–analyses can help to understand constitutive parameters for the non–local continuum theory and to apply it to other specimens. (© 2008 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim)     

A dual geometric theory for bundle shifts

Inspired by analytic model theory for Hilbert space operators and some recent developments in Cowen–Douglas operators, we formulate in this paper a geometric theory for bundle shifts and relate it in duality to the geometric theory of Cowen–Douglas operators.     

Stable Projective Homotopy Theory of Modules,Tails, and Koszul Duality

A contravariant functor is constructed from the stable projective homotopy theory of finitely generated graded modules over a finite-dimensional algebra to the derived category of its Yoneda algebra modulo finite complexes of modules of finite length. If the algebra is Koszul with a noetherian Yoneda algebra, then the constructed functor is a duality between triangulated categories. If the algebra is self-injective, then stable homotopy theory specializes trivially to stable module theory. In particular, for an exterior algebra the constructed duality specializes to (a contravariant analog of) the Bernstein–Gelfand–Gelfand correspondence.     

The Lie theory of certain modular form and arithmetic identities

Modular form identities lying in the framework of Shimura’s theory of nearly holomorphic modular forms are obtained by Lie theoretic means as consequences of identities relating the Maass–Shimura operator and the Rankin–Cohen brackets, which in turn follow from change-of-basis formulae in the theory of Verma modules. The Lie theoretic origin of known van der Pol and Lahiri-type arithmetic identities is thus unveiled, and similar new ones are derived in a systematic way. These identities relate divisor functions, Ramanujan’s τ-function and functions defined by the Fourier coefficients of other cusp forms and involve hybrid coefficients, drawn from Lie theory and number theory, given explicitly by formulae combining the arithmetic Clebsch–Gordan coefficients and the Bernoulli numbers. A few side results, interesting in their own right, such as Leibniz-type rules satisfied by the Rankin–Cohen brackets, are also obtained.     

Bohr–Sommerfeld–Heisenberg theory in geometric quantization

In the framework of geometric quantization we extend the Bohr–Sommerfeld rules to a full quantum theory which resembles the Heisenberg matrix theory. This extension is possible because Bohr–Sommerfeld rules not only provide an orthogonal basis in the space of quantum states, but also give a lattice structure to this basis. This permits the definition of appropriate shifting operators. As examples, we discuss the 1–dimensional harmonic oscillator and the coadjoint orbits of the rotation group.     

Material Forces in elasto-plastic Materials

In this contribution the concept of configurational forces, also called material forces, is applied to rate–independent, elasto–plastic materials. The theory of configurational forces is briefly recast. Zones of plastic deformation can be interpreted as distributed inhomogeneities. With this background the theory of configurational forces can be applied in many situations, including plastic zones at crack tips, elastic inclusions in elasto–plastic materials and localized deformation. The numerical evaluation is done with the Finite Element Method. (© 2008 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Global weak solutions for the initial–boundary-value problems Vlasov–Poisson–Fokker–Planck System

This work is devoted to prove the existence of weak solutions of the kinetic Vlasov–Poisson–Fokker–Planck system in bounded domains for attractive or repulsive forces. Absorbing and reflection-type boundary conditions are considered for the kinetic equation and zero values for the potential on the boundary. The existence of weak solutions is proved for bounded and integrable initial and boundary data with finite energy. The main difficulty of this problem is to obtain an existence theory for the linear equation. This fact is analysed using a variational technique and the theory of elliptic–parabolic equations of second order. The proof of existence for the initial–boundary value problems is carried out following a procedure of regularization and linearization of the problem. © 1998 B. G. Teubner Stuttgart—John Wiley & Sons, Ltd.