The singular simplicial set Sing(X) of a space X completely captures its weak homotopy type. We introduce a category of controlled sets, yielding simplicial controlled sets, such that one can functorially produce a singular simplicial controlled set CSing(MaxCtl(X)) from a locally compact X. We then argue that this CSing(MaxCtl(X)) captures the (weak) proper homotopy type of X. Moreover, our te...

We define several notions of singular set for Type-I Ricci flows and show that they all coincide. In order to do this, we prove that blow-ups around singular points converge to nontrivial gradient shrinking solitons, thus extending work of Naber [15]. As a by-product we conclude that the volume of a finite-volume singular set vanishes at the singular time. We also define a notion of density for...

Dedicated to Professor WeiYue Ding on the occasion of his 60th birthday Abstract For k ≥ 3, we establish new estimate on Hausdorff dimensions of the singular set of stable-stationary harmonic maps to the sphere S. We show that the singular set of stable-stationary harmonic maps from B to S is the union of finitely many isolated singular points and finitely many Hölder continuous curves. We also...

We prove that a harmonic map with small energy and monotonicity property is smooth if its singular set is rectifiable and has a finite uniform density; moreover, the monotonicity property holds if the singular set has a lower dimension or its gradient has higher integrability. This work generalizes the results in [CL][DF][LG12], which were proved under the assumptions that the singular sets are...

We present a set of planar parallel manipulators of Stewart Gough type which are singular with respect to the Schönflies group X(a) without being architecturally singular. This set of so called Schönflies-singular planar parallel manipulators is characterized by the property that the carrier plane of the platform or of the base anchor points is orthogonal to the rotational axis a of the Schönfl...

We consider the case of an exponential map Eκ : z 7→ exp(z) + κ for which the singular value κ is accessible from the set of escaping points of Eκ. We show that there are dynamic rays of Eκ which do not land. In particular, there is no analog of Douady’s “pinched disk model” for exponential maps whose singular value belongs to the Julia set. We also prove that the boundary of a Siegel disk U fo...

This paper presents an improved technique for extracting singular streamlines that extends the capabilities of vector field topology. Singular streamlines are a unique set of tangent curves that connect singular or critical points. In prior work [Ken99], we demonstrated that vector field topology (a well-known technique for extracting singular streamlines) failed to identify a class of singular...

In this work, the convection-diffusion integro-differential equation with a weakly singular kernel is discussed. The  Legendre spectral tau method is introduced for finding the unknown function. The proposed method is based on expanding the approximate solution as the elements of a shifted Legendre polynomials. We reduce the problem to a set of algebraic equations by using operational matrices....

Parallel manipulators which are singular with respect to the Schönflies motion group X(a) are called Schönflies-singular, or more precisely X(a)-singular, where a denotes the rotary axis. A special class of such manipulators are architecturally singular ones because they are singular with respect to any Schönflies group. Another remarkable set of Schönflies-singular planar parallel manipulators...