Möbius characterization of hemispheres

In this paper we generalize the Möbius characterization of metric spheres as obtained in Foertsch and Schroeder [4] to a corresponding Möbius characterization of metric hemispheres.     

A refinement of Stone duality to skew Boolean algebras

We establish two theorems that refine the classical Stone duality between generalized Boolean algebras and locally compact Boolean spaces. In the first theorem, we prove that the category of left-handed skew Boolean algebras whose morphisms are proper skew Boolean algebra homomorphisms is equivalent to the category of étale spaces over locally compact Boolean spaces whose morphisms are étale space cohomomorphisms over continuous proper maps. In the second theorem, we prove that the category of left-handed skew Boolean -algebras whose morphisms are proper skew Boolean -algebra homomorphisms is equivalent to the category of étale spaces with compact clopen equalizers over locally compact Boolean spaces whose morphisms are injective étale space cohomomorphisms over continuous proper maps.     

The isogeny conjecture for <Emphasis Type="Italic">A</Emphasis>-motives

We prove the isogeny conjecture for A-motives over finitely generated fields K of transcendence degree ≤1. This conjecture says that for any semisimple A-motive M over K, there exist only finitely many isomorphism classes of A-motives M′ over K for which there exists a separable isogeny M′→M. The result is in precise analogy to known results for abelian varieties and for Drinfeld modules and will have strong consequences for the \mathfrak p{\mathfrak {p}}-adic and adelic Galois representations associated to M. The method makes essential use of the Harder–Narasimhan filtration for locally free coherent sheaves on an algebraic curve.     

Critical p-adic L-functions

We attach p-adic L-functions to critical modular forms and study them. We prove that those L-functions fit in a two-variables p-adic L-function defined locally everywhere on the eigencurve.     

The homomorphisms between the Dickson–Mùi algebras as modules over the Steenrod algebra

The Dickson–Mùi algebra consists of all invariants in the mod p cohomology of an elementary abelian p-group under the general linear group. It is a module over the Steenrod algebra, A{\mathcal {A}} . We determine explicitly all the A{\mathcal {A}} -module homomorphisms between the (reduced) Dickson–Mùi algebras and all the A{\mathcal {A}} -module automorphisms of the (reduced) Dickson–Mùi algebras. The algebra of all A{\mathcal {A}} -module endomorphisms of the (reduced) Dickson–Mùi algebra is claimed to be isomorphic to a quotient of the polynomial algebra on one indeterminate. We prove that the reduced Dickson–Mùi algebra is atomic in the meaning that if an A{\mathcal {A}} -module endomorphism of the algebra is non-zero on the least positive degree generator, then it is an automorphism. This particularly shows that the reduced Dickson–Mùi algebra is an indecomposable A{\mathcal {A}} -module. The similar results also hold for the odd characteristic Dickson algebras. In particular, the odd characteristic reduced Dickson algebra is atomic and therefore indecomposable as a module over the Steenrod algebra.     

Algebraicity of formal varieties and positivity of vector bundles

We propose a positivity condition for vector bundles on a projective variety and prove an algebraicity criterion for formal schemes. Then we apply the algebraicity criterion to the study of formal principle in algebraic geometry.     

Lattice triangulations of $\mathbb{E}^3 $ and of the 3-torus

This paper gives answers to a few questions concerning tilings of Euclidean spaces where the tiles are topological simplices with curvilinear edges. We investigate lattice triangulations of Euclidean 3-space in the sense that the vertices form a lattice of rank 3 and such that the triangulation is invariant under all translations of that lattice. This is the dual concept of a primitive lattice tiling where the tiles are not assumed to be Euclidean polyhedra but only topological polyhedra. In 3-space there is a unique standard lattice triangulation by Euclidean tetrahedra (and with straight edges) but there are infinitely many non-standard lattice triangulations where the tetrahedra necessarily have certain curvilinear edges. From the view-point of Discrete Differential Geometry this tells us that there are such triangulations of 3-space which do not carry any flat discrete metric which is equivariant under the lattice. Furthermore, we investigate lattice triangulations of the 3-dimensional torus as quotients by a sublattice. The standard triangulation admits such quotients with any number n ≥ 15 of vertices. The unique one with 15 vertices is neighborly, i.e., any two vertices are joined by an edge. It turns out that for any odd n ≥ 17 there is an n-vertex neighborly triangulation of the 3-torus as a quotient of a certain non-standard lattice triangulation. Combinatorially, one can obtain these neighborly 3-tori as slight modifications of the boundary complexes of the cyclic 4-polytopes. As a kind of combinatorial surgery, this is an interesting construction by itself.     

An Area Preserving Projection from the Regular Octahedron to the Sphere

In this paper, we propose an area preserving bijective map from the regular octahedron to the unit sphere ${\mathbb{S}^2}$ , both centered at the origin. The construction scheme consists of two steps. First, each face F _i of the octahedron is mapped to a curved planar triangle ${\mathcal{T}_i}$ of the same area. Afterwards, each ${\mathcal{T}_i}$ is mapped onto the sphere using the inverse Lambert azimuthal equal area projection with respect to a certain point of ${\mathbb{S}^2}$ . The proposed map is then used to construct uniform and refinable grids on a sphere, starting from any triangular uniform and refinable grid on the triangular faces of the octahedron.     

A review on probabilistic graphical models in evolutionary computation

Thanks to their inherent properties, probabilistic graphical models are one of the prime candidates for machine learning and decision making tasks especially in uncertain domains. Their capabilities, like representation, inference and learning, if used effectively, can greatly help to build intelligent systems that are able to act accordingly in different problem domains. Evolutionary algorithms is one such discipline that has employed probabilistic graphical models to improve the search for optimal solutions in complex problems. This paper shows how probabilistic graphical models have been used in evolutionary algorithms to improve their performance in solving complex problems. Specifically, we give a survey of probabilistic model building-based evolutionary algorithms, called estimation of distribution algorithms, and compare different methods for probabilistic modeling in these algorithms.     

Exponential Stability and Uniform Boundedness of Solutions for Nonautonomous Periodic Abstract Cauchy Problems. An Evolution Semigroup Approach

Let u _{μ, x, s} (., 0) be the solution of the following well-posed inhomogeneous Cauchy Problem on a complex Banach space X $$\left \{\begin{array}{ll}\dot{u}(t) = A(t)u(t) + e^{i\mu t}x, \quad t > s \\ u(s) = 0. \end{array} \right.$$ Here, x is a vector in X,?μ is a real number, q is a positive real number and A(·) is a q-periodic linear operator valued function. Under some natural assumptions on the evolution family ${\mathcal{U} = \{U(t, s): t \geq s\}}$ generated by the family {A(t)}, we prove that if for each?μ, each s?≥ 0 and every x the solution u _{μ, x, s} (·, 0) is bounded on R ₊ by a positive constant, depending only on x, then the family ${\mathcal{U}}$ is uniformly exponentially stable. The approach is based on the theory of evolution semigroups.