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Let $${\mathcal{R}}$$ be an o-minimal expansion of the real field. We introduce a class of Hausdorff limits, the T∞-limits over $${\mathcal{R}}$$ , that do not in general fall under the scope of Marker and Steinhorn’s definability-of-types theorem. We prove that if $${\mathcal{R}}$$ admits analytic cell decomposition, then every T∞-limit over $${\mathcal{R}}$$ is definable in...
In this paper, we will give a general but completely elementary description for hyperelliptic curves of genus three whose Jacobian varieties have endomorphisms by the real cyclotomic field $${{\mathbb{Q}} (\zeta_7 + \overline{\zeta}_7)}$$ . We study the algebraic correspondences on these curves which are lifts of algebraic correspondences on a conic in P2 associated with Poncelet 7-gons...
A lattice in a Euclidean space gives rise to facet-to-facet and space-filling convex polyhedral tilings called the Voronoi tiling and its dual Delaunay tiling of the Euclidean space. Given a subspace of the Euclidean space, we develop a systematic way of constructing facet-to-facet and space-filling convex polyhedral tilings of the subspace called the Namikawa tilings, which are generalization of...
Singularities of the plane sections of a general surface in three-space are well known, and counted; in particular, all plane sections are reduced. Fixing integers $$d,k$$ , we give formulas for the degree of the locus of surfaces of degree $$d$$ admitting a plane which is tangent along some curve of degree $$k$$ .
For a plane branch C with g Puiseux pairs, we determine the irreducible components of its jet schemes which correspond to the star (or rupture) and end divisors that appear on the dual graph of the minimal embedded desingularization of C. We exploit these informations to construct a Teissier type resolution of C embedded in $${\mathbb{C}^{g+1}}$$ , which is special in the sense that its restriction...
Generalized power series extend the notion of formal power series by considering exponents of each variable ranging in a well ordered set of positive real numbers. Generalized analytic functions are defined locally by the sum of convergent generalized power series with real coefficients. We prove a local monomialization result for these functions: they can be transformed into a monomial via a locally...
Using techniques from resolution of singularities, we show the existence of solutions for the Cauchy problem in the singular solution of a family of (p, q)-quasi-ordinary analytic PDEs.
In [1] we defined semi-monotone sets, as open bounded sets, definable in an o-minimal structure over the reals (e.g., real semialgebraic or subanalytic sets), and having connected intersections with all translated coordinate cones in $${\mathbb{R}^{n}}$$ . In this paper we develop this theory further by defining monotone functions and maps, and studying their fundamental geometric properties...
Let X denote a reduced algebraic variety and D a Weil divisor on X. The pair (X, D) is said to be semi-simple normal crossings (semi-snc) at $${a \in X}$$ if X is simple normal crossings at a (i.e., a simple normal crossings hypersurface, with respect to a local embedding in a smooth ambient variety), and D is induced by the restriction to X of a hypersurface that is simple normal crossings...
We state six axioms concerning any regularity property P in a given birational equivalence class of algebraic threefolds. Axiom 5 states the existence of a Local Uniformization in the sense of valuations for P. If Axioms 1 to 5 are satisfied by P, then the function field has a projective model which is everywhere regular with respect to P. Adding Axiom 6 ensures the existence of a P-Resolution of...
After a short review on foliations, we prove that a codimension 1 holomorphic foliation on $${\mathbb P^3_{\mathbb C}}$$ with simple singularities is given by a closed rational 1-form. The proof uses Hironaka-Matsumura prolongation theorem of formal objects.
We give a definition of Newton non-degeneracy independent of the system of generators defining the variety. This definition extends the notion of Newton non-degeneracy to varieties that are not necessarily complete intersection. As in the previous definition of non-degeneracy for complete intersection varieties, it is shown that the varieties satisfying our definition can be resolved with a toric...
In this paper, we apply fuzzy soft sets and fuzzy sets to algebraic hyperstructures to define soft hypermodules and fuzzy soft hypermodules. We will proceed to study their structural properties under operations such as union, intersection and “AND”. Also, Some of results about the level soft sets of fuzzy soft hypermodules are investigated. Finally, the theorems of homomorphic image and homomorphic...
Let $${\phi }$$ be an automorphism of prime order of a group G. If its fixed-point set $${\rm C}{_{\rm G}(\phi)}$$ is finite, then frequently G is nilpotent-by-finite. If $${\rm C}{_{\rm G}(\phi )}$$ is periodic, then sometimes the commutator subgroup [G, $${\phi }$$ ] of $${\phi }$$ is periodic-by-nilpotent, indeed occasionally G itself is periodic-by-nilpotent. Here, if...
We introduce the calculus of variations characterized by a Lagrangian containing both classical derivatives and hyperdifferential operators inspired from Tabasaki–Takebe–Toda lattice arguments. Necessary optimality conditions of the Euler–Lagrange type are obtained and proved. The Hamiltonian formalism is discussed and the isoperimetric problem is considered as well.
In this paper we point out that in spite of the possibility of defining weak curves “filling” the closed unit ball $$B_{X}$$ of any normed space $$X$$ is optimum, the existence of a continuous linear functional which does not attain its sup on the closed unit ball of a non-reflexive Banach space (James’s theorem) allows us to find weak neighborhoods producing an anomalous behavior of the...
In this article, we expose some results related with Nikodym’s boundedness theorem. In particular, it is shown that, if Ω is a compact k-dimensional interval in the euclidean space $${\mathbb{R}^k}$$ , $${\mathcal{A}}$$ is the algebra of the subsets of Ω which are Jordan measurable and $${(\mathcal{A}_n)}$$ is an increasing sequence of subsets of $${\mathcal{A}}$$ , whose union is...
As objects of study in functional analysis, Hilbert spaces stand out as special objects of study as do nuclear spaces (a’la Grothen-dieck) in view of a rich geometrical structure they possess as Banach and Frechet spaces, respectively. On the other hand, there is the class of Banach spaces including certain function spaces and sequence spaces which are distinguished by a poor geometrical structure...
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