Linear structures on the collections of minimal surfaces inℝ 3 andℝ 4

The collection of minimal herissons in ³ is endowed with a vector space structure. The existence of this structure is related to the fact that null curves inC ³ are described by a single map from the étalé space of the sheaf of germs of holomorphic sections of the line bundle of degree 2 over ₁ to C³, which islinear on stalks. There is an analogous construction for null curves inC ⁴. This gives a similar class of minimal surfaces in ⁴.     

Isometric embeddings of locally Euclidean metrics in ℝ3 as conical surfaces

It is proved that if a domain with a locally Euclidean metric can be isometrically immersed in the Euclidean plane ?² with the standard metric, then it can be isometrically embedded in ?³ as a conical surface whose projection on a sphere centered at the vertex of the cone is a self-avoiding planar graph with sufficiently smooth edges of specially selected lengths.     

Stiefel–Whitney surfaces and the tri-genus of non-orientable 3-manifolds

Every non-orientable 3-manifold M can be expressed as a union of three orientable handlebodies V ₁,V ₂,V ₃ whose interiors are pairwise disjoint. If g _i denotes the genus of ∂V _i and g ₃≤g ₂≤g ₃, then the tri-genus of M is the minimum triple (g ₁,g ₂,g ₃), ordered lexicographically. If the Bockstein of the first Stiefel–Whitney class βw ₁(M)=0, then M has tri-genus (0,2g,g ₃), where g is the minimal genus of a 2-sided Stiefel Whitney surface of M. In this paper it is shown that, if βw ₁(M)≠0, then M has tri-genus (1,2g−1,g ₃), where g is the minimal genus of a (1-sided) Stiefel–Whitney surface. As an application the tri-genus of certain graph manifolds is computed. Received: 28 April 1999     

Singular surfaces and the rigidity of ℙ3

In this paper we give two elementary proofs for the rigidity of the complex projective space ₃ with respect to global deformations.     

On the Morse index of minimal surfaces in ℝp with polygonal boundaries

The minimal surfaces spanning a polygon in ^p (p2) correspond to the critical points of an analytic function in finitely many variables, namely Shiffman's function. We shall prove that the Morse index of the minimal surface coincides with the Morse index of at the corresponding critical point. Alternatively expressed, the Schwarz operator of the minimal surface and the Hessian of have the same number of negative eigenvalues. Finally we control the degeneration of the critical points.     

Intrinsic third derivatives for Plateau's problem and the Morse inequalities for disc minimal surfaces in ℝ3

Summary A simply branched minimal surface in ³ cannot be a non-degenerate critical point of Dirichlet's energy since the Hessian always has a kernel. However such minimal surface can be non-degenerate in another sense introduced earlier by R. Böhme and the author. Such surfaces arise as the zeros of a vector field on the space of all disc surfaces spanning a fixed contour. In this paper we show that the winding number of this vector field about such a surface is ±2 ^p , wherep is the number of branch points. As a consequence we derive the Morse inequalities for disc minimal surfaces in ³, thereby completing the program initiated by Morse, Tompkins, and Courant. Finally, this result implies that certain contours in ⁴ arbitrarily close to the given contour must span at least 2 ^p disc minimal surfaces.     

The Lyapunov–Movchan method in problems of the stability of flows and deformation processes

The extension of Lyapunov's method to continuous mechanical systems are discussed. An annotated bibliography of papers is given in which, based on the Lyapunov–Movchan method, with the construction of corresponding functionals, a direct analysis is carried out of the stability of motion (deformation) of continuous mechanical systems. The material is divided into sections, devoted to the following: (a) the extension of the mathematical apparatus as a whole to continuous and dynamic systems, (b) the stability of elastic, elastoplastic and viscoelastic deformable solids, (c) stability in aeroelasticity and hydroelasticity theory, (d) the linearized theory of hydrodynamic stability, and (e) the stability with reference to perturbations of material functions in the theory of constitutive relations.     

Simple Inductive Proofs of the Fishburn and Mirkin Theorem and the Scott–Suppes Theorem

This paper presents new proofs of two classic characterization theorems for families of ordered sets. The first is that any finite poset with no restriction isomorphic to has an interval representation. The second is that any finite poset with no restriction isomorphic to or to has a unit interval representation. Both proofs are straightforward and inductive.     

Degree theory on oriented infinite dimensional varieties and the Morse number of minimal surfaces spanning a curve in ℝn

The algebraic number of disc minimal surfaces spanning a wire in ₃ is defined and shown to be equal to one.The author wishes to acknowledge the support of the NSF.     

A characterisation of the Hoffman–Wohlgemuth surfaces in terms of their symmetries

For an embedded singly periodic minimal surface [(M)\tilde]{\tilde{M}} with genus r 3 4{\varrho\ge4} and annular ends, some weak symmetry hypotheses imply its congruence with one of the Hoffman–Wohlgemuth examples. We give a very geometrical proof of this fact, along which they come out many valuable clues for the understanding of these surfaces.     

On the L2-Discrepancy of the Sobol–Hammersley Net in Dimension 3

With the help of Walsh series analysis we show that the symmetrized Sobol–Hammersley net in base 2 and dimension 3 has almost best possible order of L₂ -discrepancy.     

Character Relations and Simple Modules in the Auslander–Reiten Graph of the Symmetric and Alternating Groups and their Covering Groups

From character relations for symmetric groups or Hecke algebras such as the Murnaghan–Nakayama formula and the Jantzen–Schaper formula, we obtain a lower bound for the diagonal entries of Cartan matrices. Moreover, we prove an analogous character relation for covering groups of symmetric groups and obtain a similar lower bound. As an application, we show in these situations that for wild blocks simple modules must lie at the end of the Auslander–Reiten quiver, which is equivalent to the fact that the hearts of projective indecomposable modules are indecomposable.     

On a Chain of Harmonic and Monogenic Potentials in Euclidean Half–space

In the framework of Clifford analysis, a chain of harmonic and monogenic potentials is constructed in the upper half of Euclidean space ? ^m+1, including a higher dimensional generalization of the complex logarithmic function. Their distributional limits at the boundary ? ^m turn out to be well-known distributions such as the Dirac distribution, the Hilbert kernel, the fundamental solution of the Laplace and Dirac operators, the square root of the negative Laplace operator, and the like. It is shown how each of those potentials may be recovered from an adjacent kernel in the chain by an appropriate convolution with such a distributional limit.     

Factorable surfaces in the three-dimensional Euclidean and Lorentzian spaces satisfying Δr i ?=?λ i r i

In this paper we classify the factorable surfaces in the three-dimensional Euclidean space ${\mathbb{E}^{3}}$ and Lorentzian ${\mathbb{E}_{1}^{3}}$ under the condition ??r _i ?=??? _i r _i , where ${\lambda_{i}\in\mathbb{R}}$ and ?? denotes the Laplace operator and we obtain the complete classification for those ones.     

Extension of the Barzilai–Borwein Method for Quadratic Forms in Finite Euclidean Spaces

In this paper, we present an extension of the Barzilai–Borwein method for the minimization of a quadratic form defined in a finite-dimensional real Euclidean space. We present a convergence analysis of the proposed method. We also present some applications in the resolution of linear matrix equations such as the Sylvester equation, which are of great interest in control theory and other engineering disciplines. Our numerical results indicate that the new method competes satisfactorily in the resolution of linear matrix equations with the function lyap of the Toolbox of Control of MATLAB.     

Periodic maximal surfaces in the Lorentz–Minkowski space $${\mathbb{L}^3}$$

A maximal surface with isolated singularities in a complete flat Lorentzian 3-manifold

Evolutions of the momentum density,deformation tensor and the nonlocal term of the Camassa–Holm equation

Methods originally developed to study the finite time blow-up problem of the regular solutions of the three dimensional incompressible Euler equations are used to investigate the regular solutions of the Camassa–Holm equation. We obtain results on the relative behaviors of the momentum density, the deformation tensor and the nonlocal term along the trajectories. In terms of these behaviors, we get new types of asymptotic properties of global solutions, blow-up criterion and blow-up time estimate for local solutions. More precisely, certain ratios of the quantities are shown to be vaguely monotonic along the trajectories of global solutions. Finite time blow-up of the accumulated momentum density is necessary and sufficient for the finite time blow-up of the solution. An upper estimate of the blow-up time and a blow-up criterion are given in terms of the initial short time trajectorial behaviors of the deformation tensor and the nonlocal term.