Peaked singular wave solutions associated with singular curves

We present new types of singular wave solutions with peaks in this paper. When a heteroclinic orbit connecting two saddle points intersects with the singular curve on the topological phase plane for a generalized KdV equation, it may be divided into segments. Different combinations of these segments may lead to different singular wave solutions, while at the intersection points, peaks on the waves can be observed. It is shown for the first time that there coexist different types of singular waves corresponding to one heteroclinic orbit.     

Two-weight estimates for singular and strongly singular integral operators

We give a Pontryagin-Thom type construction for Stein factorizations of fold maps of 3-manifolds into the plane. As an application, we compute the cobordism group of Stein factorizations of fold maps of oriented 3-manifolds into the plane and the oriented cobordism group of fold maps of 3-manifolds into the plane. It turns out that these two groups are isomorphic to Z ₂ ⊕ Z ₂. We have the analogous results about bordism groups as well.      

Existence of singular extremals and singular functionals in reachable spaces

Given a linear, infinite dimensional control system with point target and "full" control we show that singular extremals for the minimum norm problem exist except in certain exceptional cases ("singular" means "not satisfying Pontryagin's maximum principle"). Existence of singular extremals implies existence of certain functionals (also called singular) in the space of reachable states. Received March 5, 2001; accepted April 10, 2001.     

Generating singular transformations

We compute the singular rank and the idempotent rank of those subsemigroups of the full transformation semigroup that contain all singular transformations.     

Virasoro singular vectors

We present an explicit formula for the series S_2,p(t) of Virasoro singular vectors.     

On singular compactness

We present some applications of Shelah's singular compactness theorem to algebraic situations where the Shreier property fails. The principal application is to valuated vector spaces, where we make use of an alternate, unpublished, version of Shelah's theorem. Research partially supported by NSF Grant No. MCS 80-03591 and by the US-Israel BSF. Presented by L. Fuchs.     

The heat equation with singular nonlinearity and singular initial data

We study the existence, uniqueness and regularity of positive solutions of the parabolic equation ut−Δu=a(x)uq+b(x)up in a bounded domain and with Dirichlet's condition on the boundary. We consider here a∈Lα(Ω), b∈Lβ(Ω) and 0<q?1<p. The initial data u(0)=u₀ is considered in the space Lr(Ω), r?1. In the main result (0<q<1), we assume a,b?0 a.e. in Ω and we assume that u₀?γdΩ for some γ>0. We find a unique solution in the space .     

Coercive singular perturbations

Summary We consider general boundary value problems with small parameter ɛ in the operator and boundary conditions. Both the perturbed and reduced operators are supposed to be elliptic. We point outnecessary andsufficient conditions of Shapiro-Lopatinsky type for the singularly perturbed problem to be coercive, i.e. for a two-sided a priori estimate to hold for its solutions uniformly with respect to ɛ. Entrata in Redazione il 6 luglio 1977.     

非可换的奇异l-群

通过建立奇异元以及奇异l-群的刻划，研究了一般非可换的奇异l-群的性质及相关的结构．     

On the definition of singular bilinear forms and singular linear operators

We analyze various definitions of the concepts of a singular operator and a singular bilinear form and propose the most suitable ones. We also study the simplest properties of these objects.Published in Ukrainskii Matematicheskii Zhurnal, Vol. 45, No. 8, pp. 1084–1089, August, 1993.     

Generalized singular point quantity and integrability of degenerate resonant singular point

In this paper, integrability and generalized complex resonant center condition of degenerate resonant singular point for a class of complex polynomial differential system were studied. The concept of generalized singular point quantity of degenerate resonant singular point was proposed and the construction of that was studied. Two methods of computing generalized singular point quantities were given. Furthermore, the sufficient and necessary condition of integrability of degenerate resonant singular point was discussed for the first time.     

On a class of singular boundary value problems with singular perturbation

In this paper we investigate a class of singular second order differential equations with singular perturbation subject to three-point boundary value conditions, whose solution exhibits a couple of boundary layers at two endpoints. We first establish a lower–upper solutions theorem by using the Schauder fixed point theorem. By the asymptotic expansions and the lower–upper solutions theorem we obtain the existence, asymptotic estimates and uniqueness for the proposed problem. Several examples are given for illustrating our results.     

On the notions of singular domination and (multi-)singular hyperbolicity

Science China Mathematics - The properties of uniform hyperbolicity and dominated splitting have been introduced to study the stability of the dynamics of diffeomorphisms. One meets difficulties...     

Invariant singular minimal surfaces

We classify all singular minimal surfaces in Euclidean space that are invariant by a uniparametric group of translations and rotations.     

On singular formal deformations

We discuss the notion of singular formal deformation in algebraic setup. Such deformations show up in both finite and infinite dimensional structures. It turns out that there is a stronger version of singular deformation—called essentially singular—which arises from a singular curve in the base of the versal deformation.     

Maximally singular Sobolev functions

It is known that for any Sobolev function in the space Wm,p(RN), p?1, mp?N, where m is a nonnegative integer, the set of its singular points has Hausdorff dimension at most N−mp. We show that for p>1 this bound can be achieved. This is done by constructing a maximally singular Sobolev function in Wm,p(RN), that is, such that Hausdorff's dimension of its singular set is equal to N−mp. An analogous result holds also for Bessel potential spaces Lα,p(RN), provided αp<N, α>0, and p>1. The existence of maximally singular Sobolev functions has been announced in [Chaos Solitons Fractals 21 (2004), p. 1287].