Morse theory, Higgs fields, and Yang–Mills–Higgs functionals

In this mostly expository paper we describe applications of Morse theory to moduli spaces of Higgs bundles. The moduli spaces are finite-dimensional analytic varieties but they arise as quotients of infinite-dimensional spaces. There are natural functions for Morse theory on both the infinite-dimensional spaces and the finite-dimensional quotients. The first comes from the Yang?CMills?CHiggs energy, while the second is provided by the Hitchin function. After describing what Higgs bundles are, we explore these functions and how they may be used to extract topological information about the moduli spaces.     

Grothendieck’s comparison theorem and multivariable Fredholm theory

We use a variant of Grothendieck’s comparison theorem to show that, for a Fredholm tuple T ∈ L(X)ⁿ on a complex Banach space, there are isomorphisms . We conclude that a Fredholm tuple T ∈ L(X)ⁿ satisfies Bishop’s property (β) at z = 0 if and only if the vanishing conditions hold for . We apply these observations and results from commutative algebra to show that a graded tuple on a Hilbert space is Fredholm if and only if it satisfies Bishop’s property (β) at z = 0 and that, in this case, its cohomology groups can grow at most like k^p. Received: 14 January 2009     

Rohlin’s invariant and gauge theory,I. Homology 3-tori

This is the first in a series of papers exploring the relationship between the Rohlin invariant and gauge theory. We discuss a Casson-type invariant of a 3-manifold Y with the integral homology of the 3-torus, given by counting projectively flat U(2)-connections. We show that its mod 2 evaluation is given by the triple cup product in cohomology, and so it coincides with a certain sum of Rohlin invariants of Y. Our counting argument makes use of a natural action of H ¹ (Y;₂) on the moduli space of projectively flat connections; along the way we construct perturbations that are equivariant with respect to this action. Combined with the Floer exact triangle, this gives a purely gauge-theoretic proof that Cassons homology sphere invariant reduces mod 2 to the Rohlin invariant.     

On vector variational-like inequalities and vector optimization problems with (G,α)-invexity

The aim of this paper is to study the relationship among Minty vector variationallike inequality problem, Stampacchia vector variational-like inequality problem and vector optimization problem involving(G, α)-invex functions. Furthermore, we establish equivalence among the solutions of weak formulations of Minty vector variational-like inequality problem,Stampacchia vector variational-like inequality problem and weak efficient solution of vector optimization problem under the assumption of(G, α)-invex functions. Examples are provided to elucidate our results.     

Stochastic investment models—theory and applications

The paper discusses a stochastic model for investment variables, involving four series: the Retail Prices Index, an index of share dividend yields, an index of share yields, and the yield on ‘consols’. Section 2 describes the model and explains its derivation on the basis of historical data. Section 3 shows how the model can be used for forecasting the distributions of the variables. Section 4 discusses possible applications, and describes two in detail, relating to the expense charges of unit trusts and to guarantees incorporated in index linked life annuities.     

From microscopic theory to macroscopic theory——symmetries and order parameters of rigid molecules

Density functional theory is used to describe the phase behaviors of rigid molecules. The construction of the kernel function is discussed. Excluded-volume potential is calculated for two types of molecules with C _2v symmetry. Molecular symmetries lead to the symmetries of the kernel function and the density function, enabling a reduction of configuration space. By approximating the kernel function with a polynomial, the system can be fully characterized by some moments corresponding to the form of the kernel function. The symmetries of the kernel function determine the form of the polynomial, while the coefficients are determined by the temperature and molecular parameters. The analysis of the impact of coefficients helps us to choose independent variables in the moments as order parameters. Combining the analysis and some simulation results, we propose a minimal set of order parameters for bent-core molecules.     

Pseudo-killing and pseudoharmonic vector fields on a Riemann—Cartan manifold

Six classes of Riemann—Cartan manifolds are distinguished in an invariant way. Geometric characteristics of some of the distinguished classes of Riemann—Cartan manifolds are found, and also conditions hindering the existence, are determined. The local geometry of Riemann—Cartan manifolds carrying pseudo-Killing and pseudoharmonic vector fields is studied. Conditions hindering the existence “in the large” of pseudo-Killing and pseudoharmonic vector fields on Riemann—Cartan manifolds are obtained.     

Progress in physical concepts of string and superstring theory—thirty years of string theory

In this paper we discuss the evolution of physical concepts which led to the generation and development of string theories. The paper is conceived with the intention of summarizing and extending with new aspects the specific characteristics of strings which refer to the physical intuition and experiment. We hope to present new insights into the physics of strings and make it understandable from the point of view of a non-string theorist. Even if there exist some opinions that the (super)string theory appertains to the twenty-first or twenty-second century or that there are no concrete new predictions of string theory at low energies, we believe that string theory presents a rich field of research and a source of physical intuition not only for mathematicians but also for theoretical and experimental physicists. We offer as an example an atomic electron cloud which can also be interpreted in terms of a fixed point in a string theory We propose also an experiment to verify the fundamental hypotheses. Finally we deduce that the number of dimensions of spacetime must be infinite by virtue of the axiom of universality of motion.     

Local well-posedness for the (n + 1)-dimensional Yang–Mills and Yang–Mills–Higgs system in temporal gauge

The Yang–Mills and Yang–Mills–Higgs equations in temporal gauge are locally well-posed for small and rough initial data, which can be shown using the null structure of the critical bilinear terms. This carries over a similar result by Tao for the Yang–Mills equations in the (3+1)-dimensional case to the more general Yang–Mills–Higgs system and to general dimensions.     

Generalized magneto-thermoelasticity with two temperature and initial stress under Green–Naghdi theory

Magneto-thermoelastic interactions in an initially stressed isotropic homogeneous elastic half-space with two temperature are studied using mathematical methods under the purview of the Green–Naghdi theory with type II and III. The medium is considered to be permeated by a uniform magnetic field. The normal mode analysis is used to obtain the exact expressions for the displacement components, force stresses, temperature and couple stresses distribution. The variations of the considered variables through the horizontal distance are illustrated graphically. Comparisons are made with the results between type II and III. Numerical work is also performed for a suitable material with the aim of illustrating the results.     

Conley pairs in geometry—Lusternik–Schnirelmann theory and more

Firstly, we wish to motivate that Conley pairs, realized via Salamon’s definition (Salamon, 1990), are rather useful building blocks in geometry: Initially we met Conley pairs in an attempt to construct Morse filtrations of free loop spaces (Weber, 2017). From this fell off quite naturally, firstly, an alternative proof (Weber, 2016) of the cell attachment theorem in Morse theory (Milnor, 1963) and, secondly, some ideas (Majer and Weber, 2015) how to try to organize the closures of the unstable manifolds of a Morse–Smale gradient flow as a CW decomposition of the underlying manifold. Relaxing non-degeneracy of critical points to isolatedness we use these Conley pairs to implement the gradient flow proof of the Lusternik–Schnirelmann Theorem (Lusternik and Schnirelmann, 1934) proposed in Bott’s survey (Bott, 1982).Secondly, we shall use this opportunity to provide an exposition of Lusternik–Schnirelmann (LS) theory based on thickenings of unstable manifolds via Conley pairs. We shall cover the Lusternik–Schnirelmann Theorem (Lusternik and Schnirelmann, 1934), cuplength, subordination, the LS refined minimax principle, and a variant of the LS category called ambient category.     

Reliability theory and applications — Proceedings of the China-Japan Reliability Symposium

Book Reviews

Reliability theory and applications — Proceedings of the China-Japan Reliability SymposiumShunji Osaki and Jinhua Cao (eds): World Scientific, Singapore, New Jersey, Hong Kong, 1987, xii+433 pp, £47.15     

Rational Unfoldings of ± X k and their relation with geometric equilibrium theory

In this paper, two theorems determine the locus B (buoyancy locus) of the centre of gravity of the displaced water of a ship with any number of 2n sides, as a function of the arbitrary continuous curves that make up those sides, and that generalize the classical wall-sided case (n=1). Although the theorems may have interest from the pure geometrical point of view, they have been obtained with a view to simulate processes in biology that, in analogy to the self-righting of ships, are self-regulating, such as the neural control of body heat and the servo-control of the heart rate.     

Invariant submanifolds and properCR foliations on a para-coKählerian manifold with concircular structure vector field

After recalling the basic properties of para-coKählerian manifolds \(\tilde M\) with concircular structure vector field ξ, the infinitesimal auto morphismsX of the structure 1-form \(\tilde \eta \) are considered. One of the results is that the Lie derivative of all powers of the structure 2-form \(\tilde \Omega ,\) i.e. \(\mathcal{L}x\tilde \Omega ^p ;p = 1,...,m,\) is exterior recurrent. Further two types of horizontal distributionsD _n which are normal to ξ. IfD _t (resp.D _n ) is involutive, the corresponding leafM _t (resp.M _n ) is a minimal submanifold of \(\tilde M\) . FurtherM _n is a symplectic submanifold and ξ is an umbilical normal section ofM _n . Finally proper immersion \(M \to \tilde M\) are discussed, whereM is aCR-sub-manifold whose horizontal distribution isD _t . It is shown that the vertical distribution is involutive, and the restriction of ξ toM is an symptotic direction. Some interesting special cases are treated.     

Mellin analysis and its basic associated metric—applications to sampling theory

In this paper a notion of functional “distance” in the Mellin transform setting is introduced and a general representation formula is obtained for it. Also, a determination of the distance is given in terms of Lipschitz classes and Mellin–Sobolev spaces. Finally applications to approximate versions of certain basic relations valid for Mellin band-limited functions are studied in details.     

A generalized poincaré-invariant action with possible application in strings and E-infinity theory

Using the semi-inverse method [He. Variational principles for some nonlinear partial differential equations with variable coefficients. Chaos Solitons & Fractals 2004;19(4):847–851], a generalized Poincaré-invariant action using the tetrad in any number of dimensions is obtained. The so-obtained generalized action with a free parameter might find potential applications in bosonic strings and E-infinity theory.     

Algebraic games—playing with groups and rings

Two players alternate moves in the following impartial combinatorial game: Given a finitely generated abelian group A, a move consists of picking some \(0 \ne a \in A\). The game then continues with the quotient group \(A/\langle a \rangle \). We prove that under the normal play rule, the second player has a winning strategy if and only if A is a square, i.e. \(A \cong B \times B\) for some abelian group B. Under the misère play rule, only minor modifications concerning elementary abelian groups are necessary to describe the winning situations. We also compute the nimbers, i.e. Sprague–Grundy values of 2-generated abelian groups. An analogous game can be played with arbitrary algebraic structures. We study some examples of non-abelian groups and commutative rings such as R[X], where R is a principal ideal domain.