The minimal resultant locus

Robert Rumely

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Displaying similar documents to “The minimal resultant locus”

On some properties of three-dimensional minimal sets in $ℝ^{4}$

Tien Duc Luu (2013)

Annales de la faculté des sciences de Toulouse Mathématiques

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We prove in this paper the Hölder regularity of Almgren minimal sets of dimension 3 in $ℝ^{4}$ around a $𝕐$ -point and the existence of a point of particular type of a Mumford-Shah minimal set in $ℝ^{4}$ , which is very close to a $𝕋$ . This will give a local description of minimal sets of dimension 3 in $ℝ^{4}$ around a singular point and a property of Mumford-Shah minimal sets in $ℝ^{4}$ .

Zero-set property of o-minimal indefinitely Peano differentiable functions

Andreas Fischer (2008)

Annales Polonici Mathematici

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Given an o-minimal expansion ℳ of a real closed field R which is not polynomially bounded. Let $^{\infty}$ denote the definable indefinitely Peano differentiable functions. If we further assume that ℳ admits $^{\infty}$ cell decomposition, each definable closed subset A of Rⁿ is the zero-set of a $^{\infty}$ function f:Rⁿ → R. This implies $^{\infty}$ approximation of definable continuous functions and gluing of $^{\infty}$ functions defined on closed definable sets.

O-minimal fields with standard part map

Jana Maříková (2010)

Fundamenta Mathematicae

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Let R be an o-minimal field and V a proper convex subring with residue field k and standard part (residue) map st: V → k. Let $k_{i n d}$ be the expansion of k by the standard parts of the definable relations in R. We investigate the definable sets in $k_{i n d}$ and conditions on (R,V) which imply o-minimality of $k_{i n d}$ . We also show that if R is ω-saturated and V is the convex hull of ℚ in R, then the sets definable in $k_{i n d}$ are exactly the standard parts of the sets definable in (R,V).

Definable stratification satisfying the Whitney property with exponent 1

Beata Kocel-Cynk (2007)

Annales Polonici Mathematici

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We prove that for a finite collection of sets $A ₁, . . ., A_{s} \subset ℝ^{k + n}$ definable in an o-minimal structure there exists a compatible definable stratification such that for any stratum the fibers of its projection onto $ℝ^{k}$ satisfy the Whitney property with exponent 1.

Product property for capacities in $ℂ^{N}$

Mirosław Baran, Leokadia Bialas-Ciez (2012)

Annales Polonici Mathematici

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The paper deals with logarithmic capacities, an important tool in pluripotential theory. We show that a class of capacities, which contains the L-capacity, has the following product property: $C_{ν} (E ₁ \times E ₂) = m i n (C_{ν ₁} (E ₁), C_{ν ₂} (E ₂))$ , where $E_{j}$ and $ν_{j}$ are respectively a compact set and a norm in $ℂ^{N_{j}}$ (j = 1,2), and ν is a norm in $ℂ^{N ₁ + N ₂}$ , ν = ν₁⊕ₚ ν₂ with some 1 ≤ p ≤ ∞. For a convex subset E of $ℂ^{N}$ , denote by C(E) the standard L-capacity and by $ω_{E}$ the minimal width of E, that is, the minimal Euclidean distance between two supporting hyperplanes...

Extending piecewise polynomial functions in two variables

Andreas Fischer, Murray Marshall (2013)

Annales de la faculté des sciences de Toulouse Mathématiques

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We study the extensibility of piecewise polynomial functions defined on closed subsets of $ℝ^{2}$ to all of $ℝ^{2}$ . The compact subsets of $ℝ^{2}$ on which every piecewise polynomial function is extensible to $ℝ^{2}$ can be characterized in terms of local quasi-convexity if they are definable in an o-minimal expansion of $ℝ$ . Even the noncompact closed definable subsets can be characterized if semialgebraic function germs at infinity are dense in the Hardy field of definable germs. We also present a piecewise...

Varieties of minimal rational tangents of codimension 1

Jun-Muk Hwang (2013)

Annales scientifiques de l'École Normale Supérieure

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Let $X$ be a uniruled projective manifold and let $x$ be a general point. The main result of [2] says that if the $(- K_{X})$ -degrees (i.e., the degrees with respect to the anti-canonical bundle of $X$ ) of all rational curves through $x$ are at least $dim X + 1$ , then $X$ is a projective space. In this paper, we study the structure of $X$ when the $(- K_{X})$ -degrees of all rational curves through $x$ are at least $dim X$ . Our study uses the projective variety $𝒞_{x} \subset ℙ T_{x} (X)$ , called the VMRT at $x$ , defined as the union of tangent directions to the...

On a magnetic characterization of spectral minimal partitions

Bernard Helffer, Thomas Hoffmann-Ostenhof (2013)

Journal of the European Mathematical Society

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Given a bounded open set $Ω$ in $ℝ^{n}$ (or in a Riemannian manifold) and a partition of $Ω$ by $k$ open sets $D_{j}$ , we consider the quantity ${𝚖𝚊𝚡}_{j} λ (D_{j})$ where $λ (D_{j})$ is the ground state energy of the Dirichlet realization of the Laplacian in $D_{j}$ . If we denote by $ℒ_{k} (Ω)$ the infimum over all the $k$ -partitions of ${𝚖𝚊𝚡}_{j} λ (D_{j})$ , a minimal $k$ -partition is then a partition which realizes the infimum. When $k = 2$ , we find the two nodal domains of a second eigenfunction, but the analysis of higher $k$ ’s is non trivial and quite interesting. In this...

On computation of minimal free resolutions over solvable polynomial algebras

Huishi Li (2015)

Commentationes Mathematicae Universitatis Carolinae

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Let $A = K [a_{1}, ..., a_{n}]$ be a (noncommutative) solvable polynomial algebra over a field $K$ in the sense of A. Kandri-Rody and V. Weispfenning [Non-commutative Gröbner bases in algebras of solvable type, J. Symbolic Comput. 9 (1990), 1–26]. This paper presents a comprehensive study on the computation of minimal free resolutions of modules over $A$ in the following two cases: (1) $A = ⨁_{p \in ℕ} A_{p}$ is an $ℕ$ -graded algebra with the degree-0 homogeneous part $A_{0} = K$ ; (2) $A$ is an $ℕ$ -filtered algebra with the filtration ${F_{p} A}_{p \in ℕ}$ determined by...

Minimality properties of Tsirelson type spaces

Denka Kutzarova, Denny H. Leung, Antonis Manoussakis, Wee-Kee Tang (2008)

Studia Mathematica

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We study minimality properties of partly modified mixed Tsirelson spaces. A Banach space with a normalized basis $(e_{k})$ is said to be subsequentially minimal if for every normalized block basis $(x_{k})$ of $(e_{k})$ , there is a further block basis $(y_{k})$ of $(x_{k})$ such that $(y_{k})$ is equivalent to a subsequence of $(e_{k})$ . Sufficient conditions are given for a partly modified mixed Tsirelson space to be subsequentially minimal, and connections with Bourgain’s ℓ¹-index are established. It is also shown that a large class of...

Renormings of $c_{0}$ and the minimal displacement problem

Łukasz Piasecki (2014)

Annales Universitatis Mariae Curie-Sklodowska, sectio A – Mathematica

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The aim of this paper is to show that for every Banach space $(X, ∥ \cdot ∥)$ containing asymptotically isometric copy of the space $c_{0}$ there is a bounded, closed and convex set $C \subset X$ with the Chebyshev radius $r (C) = 1$ such that for every $k \geq 1$ there exists a $k$ -contractive mapping $T : C \to C$ with $∥ x - T x ∥ > 1 - 1 / k$ for any $x \in C$ .

Structure theory for the group algebra of the symmetric group, with applications to polynomial identities for the octonions

Murray R. Bremner, Sara Madariaga, Luiz A. Peresi (2016)

Commentationes Mathematicae Universitatis Carolinae

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This is a survey paper on applications of the representation theory of the symmetric group to the theory of polynomial identities for associative and nonassociative algebras. In §1, we present a detailed review (with complete proofs) of the classical structure theory of the group algebra $𝔽 S_{n}$ of the symmetric group $S_{n}$ over a field $𝔽$ of characteristic 0 (or $p > n$ ). The goal is to obtain a constructive version of the isomorphism $ψ : ⨁_{λ} M_{d_{λ}} (𝔽) ⟶ 𝔽 S_{n}$ where $λ$ is a partition of $n$ and $d_{λ}$ counts the standard tableaux...

Computing the greatest $𝐗$ -eigenvector of a matrix in max-min algebra

Ján Plavka (2016)

Kybernetika

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A vector $x$ is said to be an eigenvector of a square max-min matrix $A$ if $A \otimes x = x$ . An eigenvector $x$ of $A$ is called the greatest $𝐗$ -eigenvector of $A$ if $x \in 𝐗 = {x; \underset{̲}{x} \leq x \leq \bar{x}}$ and $y \leq x$ for each eigenvector $y \in 𝐗$ . A max-min matrix $A$ is called strongly $𝐗$ -robust if the orbit $x, A \otimes x, A^{2} \otimes x, \dots$ reaches the greatest $𝐗$ -eigenvector with any starting vector of $𝐗$ . We suggest an $O (n^{3})$ algorithm for computing the greatest $𝐗$ -eigenvector of $A$ and study the strong $𝐗$ -robustness. The necessary and sufficient conditions for strong $𝐗$ -robustness are introduced...

On the real $X$ -ranks of points of $ℙ^{n} (ℝ)$ with respect to a real variety $X \subset ℙ^{n}$

Edoardo Ballico (2010)

Annales Universitatis Mariae Curie-Sklodowska, sectio A – Mathematica

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Let $X \subset ℙ^{n}$ be an integral and non-degenerate $m$ -dimensional variety defined over $ℝ$ . For any $P \in ℙ^{n} (ℝ)$ the real $X$ -rank $r_{X, ℝ} (P)$ is the minimal cardinality of $S \subset X (ℝ)$ such that $P \in 〈 S 〉$ . Here we extend to the real case an upper bound for the $X$ -rank due to Landsberg and Teitler.

Purity of level $m$ stratifications

Marc-Hubert Nicole, Adrian Vasiu, Torsten Wedhorn (2010)

Annales scientifiques de l'École Normale Supérieure

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Let $k$ be a field of characteristic $p > 0$ . Let $D_{m}$ be a ${BT}_{m}$ over $k$ (i.e., an $m$ -truncated Barsotti–Tate group over $k$ ). Let $S$ be a $k$ -scheme and let $X$ be a ${BT}_{m}$ over $S$ . Let $S_{D_{m}} (X)$ be the subscheme of $S$ which describes the locus where $X$ is locally for the fppf topology isomorphic to $D_{m}$ . If $p \geq 5$ , we show that $S_{D_{m}} (X)$ is pure in $S$ , i.e. the immersion $S_{D_{m}} (X) ↪ S$ is affine. For $p \in {2, 3}$ , we prove purity if $D_{m}$ satisfies a certain technical property depending only on its $p$ -torsion $D_{m} [p]$ . For $p \geq 5$ , we apply the developed techniques to show that...

Beyond two criteria for supersingularity: coefficients of division polynomials

Christophe Debry (2014)

Journal de Théorie des Nombres de Bordeaux

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Let $f (x)$ be a cubic, monic and separable polynomial over a field of characteristic $p \geq 3$ and let $E$ be the elliptic curve given by $y^{2} = f (x)$ . In this paper we prove that the coefficient at $x^{\frac{1}{2} p (p - 1)}$ in the $p$ –th division polynomial of $E$ equals the coefficient at $x^{p - 1}$ in $f {(x)}^{\frac{1}{2} (p - 1)}$ . For elliptic curves over a finite field of characteristic $p$ , the first coefficient is zero if and only if $E$ is supersingular, which by a classical criterion of Deuring (1941) is also equivalent to the vanishing of the second coefficient. So the...