1.
Nonconstant selfsimilar blow-up profile for the exponential reaction-diffusion equation - Fila, Marek; Pulkkinen, Aappo
We study the blow-up profile of radial solutions of a semilinear heat equation with an exponential source term. Our main aim is to show that solutions which can be continued beyond blow-up possess a nonconstant selfsimilar blow-up profile. For some particular solutions we determine this profile precisely.
2.
A note on relative duality for Voevodsky motives - Barbieri-Viale, Luca; Kahn, Bruno
Let $k$ be a perfect field which admits resolution of singularities in the sense of Friedlander and Voevodsky (for example, $k$ of characteristic $0$). Let $X$ be a smooth proper $k$-variety of pure dimension $n$ and $Y,Z$ two disjoint closed subsets of $X$. We prove an isomorphism \[ M(X-Z,Y)\simeq M(X-Y,Z)^*(n)[2n], \] where $M(X-Z,Y)$ and $M(X-Y,Z)$ are relative Voevodsky motives, defined in his triangulated category $\operatorname{DM}_{\rm gm}(k)$.
3.
Ricci curvature of affine connections - Kobayashi, Osamu
We show various examples of torsion-free affine connections which preserve volume elements and have definite Ricci curvature tensors.
4.
The main component of the toric Hilbert scheme - Chuvashova, Olga V.
Let $\boldsymbol{X}$ be an affine toric variety with big torus $\boldsymbol{T}\subset \boldsymbol{X}$ and let $T\subset\boldsymbol{T}$ be a subtorus. The general $T$-orbit closures in $\boldsymbol{X}$ and their flat limits are parametrized by the main component $H_0$ of the toric Hilbert scheme. Further, the quotient torus $\boldsymbol{T}/T$ acts on $H_0$ with a dense orbit. We describe the fan of this toric variety; this leads us to an integral analogue of the fiber polytope of Billera and Sturmfels. We also describe the relation of $H_0$ to the main component of the inverse limit of GIT quotients of $\boldsymbol{X}$ by $T$.
5.
Weak geometric structures on submanifolds of affine spaces - Opozda, Barbara
A few affine invariant structures depending only on the second fundamental form relative to arbitrary transversal bundles on submanifolds of the standard affine spaces are introduced. A notion of local strong convexity is proposed for arbitrary codimensional submanifolds. In the case of $n$-dimensional submanifolds of $2n$-dimensional real affine spaces, complex structures on the ambient spaces are used as a tool for studying real affine invariants.
6.
On the topology of minimal orbits in complex flag manifolds - Altomani, Andrea; Medori, Costantino; Nacinovich, Mauro
We compute the Euler-Poincaré characteristic of the homogeneous compact manifolds that can be described as minimal orbits for the action of a real form in a complex flag manifold.
7.
A note on toric Deligne-Mumford stacks - Perroni, Fabio
We give a new description of the data needed to specify a morphism from a scheme to a toric Deligne-Mumford stack. The description is given in terms of a collection of line bundles and sections which satisfy certain conditions. As applications, we characterize any toric Deligne-Mumford stack as a product of roots of line bundles over the rigidified stack, describe the torus action, describe morphisms between toric Deligne-Mumford stacks with complete coarse moduli spaces in terms of homogeneous polynomials, and compare two different definitions of toric stacks.
8.
On the defining equations of hypersurface purely elliptic singularities - Kanesaka, Naohiro
We investigate a class of isolated hypersurface singularities, the so-called purely elliptic singularities, of complex algebraic
varieties of dimension greater than or equal to two. We show that, for hypersurface purely elliptic singularities defined by nondegenerate polynomials, Calabi-Yau varieties arising among the irreducible components of the essential divisors are concretely associated with the defining equations of these singularities, and that the birational class of the Calabi-Yau varieties does not depend on the irreducible components.
9.
Numerically flat principal bundles - Biswas, Indranil; Subramanian, Swaminathan
Generalizing the notion of a numerically flat vector bundle over a Kähler manifold $M$, we define a numerically flat principal $G$-bundle over $M$, where $G$ is a semisimple complex algebraic group. It is proved that a principal $G$-bundle $E_G$ is numerically flat if and only if $\text{ad}(E_G)$ is numerically flat. Numerically flat bundles are also characterized using the notion of semistability.
10.
Compactifications of log morphisms - Grosse-Klönne, Elmar
We introduce the notion of a relative log scheme with boundary: a morphism of log schemes together with a (log schematically) dense open immersion of its source into a third log scheme. The sheaf of relative log differentials naturally extends to this compactification and there is a notion of smoothness for such data. We indicate how this weak sort of compactification may be used to develop useful de Rham and crystalline cohomology theories for semistable log schemes over the log point over a field which are not necessarily proper.
12.
Superposition operators on Dirichlet spaces - Fitzsimmons, Patrick J.
In the context of a strongly local Dirichlet space we show that if a function mapping the real line to itself (and fixing the origin) operates by composition on the left to map the Dirichlet space into itself, then the function is necessarily locally Lipschitz continuous. If, in addition, the Dirichlet space contains unbounded elements, then the function must be globally Lipschitz continuous. The proofs rely on a co-area formula for condenser potentials.
13.
Focusing of spherical nonlinear pulses in {$R\sp {1+3}$}, III. Sub and supercritical cases - Carles, Rémi; Rauch, Jeffrey
We study the validity of geometric optics in $L^\infty$ for nonlinear wave equations in three space dimensions whose solutions, pulse like, focus at a point. If the amplitude of the initial data is subcritical, then no nonlinear effect occurs at leading order. If the amplitude of the initial data is sufficiently big, then strong nonlinear effects occur; we study the cases where the equation is either dissipative or accretive. When the equation is dissipative, pulses are absorbed before reaching the focal point. When the equation is accretive, the family of pulses becomes unbounded.
14.
Twining characters and Kostant's homology formula - Naito, Satoshi
Let $\frak g$ be a symmetrizable generalized Kac-Moody algebra and $\frak n_-$ the sum of all its negative root spaces. We obtain a formula for the twining characters of the Lie algebra homology modules of $\frak n_-$ with coefficients in the irreducible highest weight $\frak g$-module $L(\Lambda)$ of symmetric, dominant integral highest weight $\Lambda$. This formula gives a new (and convincing) proof of the formula for the twining character of $L(\Lambda)$ above.
15.
A free boundary value problem of Euler system arising in supersonic flow past a curved cone - Chen, Shuxing
We study a free boundary value problem of the Euler system arising in the inviscid steady supersonic flow past a symmetric curved cone. The existence and stability of piesewise smooth weak entropy solutions was established, provided the cone is a small perturbation of its tangential cone with a vertex angle less than a given value determined by the parameters of the coming flow. Since the change of the entropy of the flow is also considered, the result in this paper gives a more precise description than previous ones on such problems.
16.
Mean curvature 1 surfaces in hyperbolic 3-space with low total curvature. II - Rossman, Wayne; Umehara, Masaaki; Yamada, Kotaro
In this work, complete constant mean curvature $1$ (\cmcone{}) surfaces in hyperbolic $3$-space with total absolute curvature at most $4\pi$ are classified. This classification suggests that the Cohn-Vossen inequality can be sharpened for surfaces with odd numbers of ends, and a proof of this is given.
17.
The behavior of the principal distributions on the graph of a homogeneous polynomial - Ando, Naoya
In this paper, we shall study the behavior of the principal distributions on the graph of a homogeneous polynomial in two variables such that the set of its umbilical points is finite. In particular, we shall present a method of describing the indices of the umbilical points and the point at infinity.
18.
Complex vector fields having orbits with bounded geometry - Scárdua, Bruno C. A.
Germs of holomorphic vector fields at the origin $0\in \co^{\kern1pt2}$ and polynomial vector fields on $\co^{\kern1pt2}$ are studied. Our aim is to classify these vector fields whose orbits have bounded geometry in a certain sense. Namely, we consider the following situations: (i) the volume of orbits is an integrable function, (ii) the orbits have sub-exponential growth, (iii) the total curvature of orbits is finite. In each case we classify these vector fields under some generic hypothesis on singularities. Applications to questions, concerning polynomial vector fields having closed orbits and complete polynomial vector fields, are given. We also give some applications...
19.
Birkhoff decompositions and Iwasawa decompositions for loop groups - Balan, Vladimir; Dorfmeister, Josef
Representations of arbitrary real or complex invertible matrices as products of matrices of special type have been used for many purposes. The matrix form of the Gram-Schmidt orthonormalization procedure and the Gauss elimination process are instances of such matrix factorizations. For arbitrary, finite-dimensional, semisimple Lie groups, the corresponding matrix factorizations are known as Iwasawa decomposition and Bruhat decomposition. The work of Matsuki and Rossmann has generalized the Iwasawa decomposition for the finite-dimensional, semisimple Lie groups. In infinite dimensions, for affine loop groups/Kac-Moody groups, the Bruhat decomposition has an, also classical, competitor, the Birkhoff decomposition. Both decompositions (in infinite dimensions), the...
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