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A topological space is called self-covering if it is a nontrivial cover of itself. We prove that a closed self-covering manifold M with free abelian fundamental group fibers over a circle under mild assumptions. In particular, we give a complete answer to the question whether a self-covering manifold with fundamental group \mathbb Z is a fiber bundle over S^1, except for the 4-dimensional smooth case. As an algebraic Hilfssatz, we develop a criterion for finite generation of modules over a commutative Noetherian ring. We also construct examples of self-covering manifolds with nonfree abelian fundamental group, which are not fiber bundles over S^1. [ABSTRACT FROM AUTHOR]

We study intersection theory for differential algebraic varieties. Particularly, we study families of differential hypersurface sections of arbitrary affine differential algebraic varieties over a differential field. We prove the differential analogue of Bertini's theorem, namely that for an arbitrary geometrically irreducible differential algebraic variety which is not an algebraic curve, generic hypersurface sections are geometrically irreducible and codimension one. Surprisingly, we prove a stronger result in the case that the order of the differential hypersurface is at least one; namely that the generic differential hypersurface sections of an irreducible differential algebraic variety are irreducible and codimension one. We also calculate the Kolchin polynomials of the intersections and prove several other results regarding intersections of differential algebraic varieties. [ABSTRACT FROM AUTHOR]

To study the dynamics of an anisotropic curvature flow with external driving force depending only on the normal vector, we focus on traveling waves composed of Jordan curves in R2. Here we call them compact traveling waves. The objective of this study is to investigate thoroughly the condition of the driving force for the existence of compact traveling waves to the anisotropic curvature flow. It is shown that all traveling waves are strictly convex and unstable, and that a compact traveling wave is unique, if they exist. To determine the existence of compact traveling waves, three cases are considered: if the driving force is positive, there exists a compact traveling wave; if it is negative, there is no traveling wave; if it is sign-changing, a positive answer is obtained under the assumption called '' admissible condition ''. We also obtain a necessary and sufficient condition for the existence of axisymmetric compact traveling waves. Lastly, we make reference to the inverse problem and non-convex compact traveling waves. [ABSTRACT FROM AUTHOR]

For parameters n,δ,B, and C, let X = (Xkl) be the random uniform contingency table whose first [nδ] rows and columns have margin [BCn] and the last n rows and columns have margin [Cn]. For every 0 < δ < 1, we establish a sharp phase transition of the limiting distribution of each entry of X at the critical value Bc = 1 + √1+1/C. In particular, for 1/2 < δ < 1, we show that the distribution of each entry converges to a geometric distribution in total variation distance whose mean depends sensitively on whether B < Bc or B > Bc. Our main result shows that E[X11] is uniformly bounded for B < Bc but has sharp asymptotic C(B-Bc) n1−δ for B > Bc. We also establish a strong law of large numbers for the row sums in top right and top left blocks. [ABSTRACT FROM AUTHOR]

Let R = C[x1, x2, . . ., xn]/(f1, . . ., fm) be a positively graded Artinian algebra. A long-standing conjecture in algebraic geometry, differential geometry, and rational homotopy theory is the non-existence of negative weight derivations on R. Alexsandrov conjectured that there are no negative weight derivations when R is a complete intersection algebra, and Yau conjectured there are no negative weight derivations on R when R is the moduli algebra of a weighted homogeneous hypersurface singularity. This problem is also important in rational homotopy theory and differential geometry. In this paper we prove the non-existence of negative weight derivations on R when the degrees of f1, . . .,fm are bounded below by a constant C depending only on the weights of x1, . . ., xn. Moreover this bound C is improved in several special cases. [ABSTRACT FROM AUTHOR]

In this paper, an intersection theory for generic differential polynomials is presented. The intersection of an irreducible differential variety of dimension d and order h with a generic differential hypersurface of order s is shown to be an irreducible variety of dimension d - 1 and order h + s. As a consequence, the dimension conjecture for generic differential polynomials is proved. Based on intersection theory, the Chow form for an irreducible differential variety is defined and most of the properties of the Chow form in the algebraic case are established for its differential counterpart. Furthermore, the generalized differential Chow form is defined and its properties are proved. As an application of the generalized differential Chow form, the differential resultant of n+1 generic differential polynomials in n variables is defined and properties similar to that of the Macaulay resultant for multivariate polynomials are proved. [ABSTRACT FROM AUTHOR]

Let M be a complete Riemannian manifold which is either compact or has a pole, and let φ be a positive smooth function on M. In the warped product M x φ ℝ, we study the flow by the mean curvature of a locally Lipschitz continuous graph on M and prove that the flow exists for all time and that the evolving hypersurface is C∞ for t > 0 and is a graph for all t. Moreover, under certain conditions, the flow has a well-defined limit. [ABSTRACT FROM AUTHOR]

The article explores the contributions, works, career, and life of Shiing-shen Chern, one of the giants in the history of geometry, who died in Tianjin, China, on December 3, 2004. One of Chern's major contribution in the field of mathematics is on the development of differential geometry and transcendental algebraic geometry. It also provides a brief description of Chern's six-page paper in the Annals of Mathematics on the intrinsic proof of the Gauss-Bonnet theorem.