A characterization of a minimal submanifold in $R^{n+p}$

Themis Koufogiorgos

Displaying similar documents to “A characterization of a minimal submanifold in $R^{n + p}$ ”

A note on minimal zero-sum sequences over ℤ

Papa A. Sissokho (2014)

Acta Arithmetica

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A zero-sum sequence over ℤ is a sequence with terms in ℤ that sum to 0. It is called minimal if it does not contain a proper zero-sum subsequence. Consider a minimal zero-sum sequence over ℤ with positive terms $a ₁, . . ., a_{h}$ and negative terms $b ₁, . . ., b_{k}$ . We prove that h ≤ ⌊σ⁺/k⌋ and k ≤ ⌊σ⁺/h⌋, where $σ ⁺ = \sum_{i = 1}^{h} a_{i} = - \sum_{j = 1}^{k} b_{j}$ . These bounds are tight and improve upon previous results. We also show a natural partial order structure on the collection of all minimal zero-sum sequences over the set i∈ ℤ : -n ≤ i ≤ n for any positive...

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 $ℝ^{4}$ .

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.

Weierstrass division theorem in definable $C^{\infty}$ germs in a polynomially bounded o-minimal structure

Abdelhafed Elkhadiri, Hassan Sfouli (2006)

Annales Polonici Mathematici

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We give some examples of polynomially bounded o-minimal expansions of the ordered field of real numbers where the Weierstrass division theorem does not hold in the ring of germs, at the origin of ℝⁿ, of definable $C^{\infty}$ functions.

A note on generalized projections in c₀

Beata Deręgowska, Barbara Lewandowska (2014)

Annales Polonici Mathematici

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Let V ⊂ Z be two subspaces of a Banach space X. We define the set of generalized projections by $_{V} (X, Z) : = {P \in (X, Z) : P |}_{V} = i d$ . Now let X = c₀ or $l_{\infty}^{m}$ , Z:= kerf for some f ∈ X* and $V : = Z \cap l ⁿ_{\infty}$ (n < m). The main goal of this paper is to discuss existence, uniqueness and strong uniqueness of a minimal generalized projection in this case. Also formulas for the relative generalized projection constant and the strong uniqueness constant will be given (cf. J. Blatter and E. W. Cheney [Ann. Mat. Pura Appl. 101 (1974), 215-227] and...

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.

On transitive submanifolds of $𝒞^{2}$ and $𝒞^{3}$

Alois Švec (1973)

Czechoslovak Mathematical Journal

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Global pinching theorems for minimal submanifolds in spheres

Kairen Cai (2003)

Colloquium Mathematicae

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Let M be a compact submanifold with parallel mean curvature vector embedded in the unit sphere $S^{n + p} (1)$ . By using the Sobolev inequalities of P. Li to get $L_{p}$ estimates for the norms of certain tensors related to the second fundamental form of M, we prove some rigidity theorems. Denote by H and ${| | σ | |}_{p}$ the mean curvature and the $L_{p}$ norm of the square length of the second fundamental form of M. We show that there is a constant C such that if ${| | σ | |}_{n / 2} < C$ , then M is a minimal submanifold in the sphere $S^{n + p - 1} (1 + H ²)$ with sectional...

The minimal resultant locus

Robert Rumely (2015)

Acta Arithmetica

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Let K be a complete, algebraically closed nonarchimedean valued field, and let φ(z) ∈ K(z) have degree d ≥ 2. We study how the resultant of φ varies under changes of coordinates. For γ ∈ GL₂(K), we show that the map $γ \mapsto o r d (R e s (φ^{γ}))$ factors through a function $o r d R e s_{φ} (\cdot)$ on the Berkovich projective line, which is piecewise affine and convex up. The minimal resultant is achieved either at a single point in $P ¹_{K}$ , or on a segment, and the minimal resultant locus is contained in the tree in $P ¹_{K}$ spanned by the fixed points...

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).

Almost invariant submanifolds for compact group actions

Alan Weinstein (2000)

Journal of the European Mathematical Society

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We define a $C^{1}$ distance between submanifolds of a riemannian manifold $M$ and show that, if a compact submanifold $N$ is not moved too much under the isometric action of a compact group $G$ , there is a $G$ -invariant submanifold $C^{1}$ -close to $N$ . The proof involves a procedure of averaging nearby submanifolds of riemannian manifolds in a symmetric way. The procedure combines averaging techniques of Cartan, Grove/Karcher, and de la Harpe/Karoubi with Whitney’s idea of realizing submanifolds as zeros...

Minimal models for $ℤ^{d}$ -actions

Bartosz Frej, Agata Kwaśnicka (2008)

Colloquium Mathematicae

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We prove that on a metrizable, compact, zero-dimensional space every $ℤ^{d}$ -action with no periodic points is measurably isomorphic to a minimal $ℤ^{d}$ -action with the same, i.e. affinely homeomorphic, simplex of measures.

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...

Minimal $𝒮$ -universality criteria may vary in size

Noam D. Elkies, Daniel M. Kane, Scott Duke Kominers (2013)

Journal de Théorie des Nombres de Bordeaux

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In this note, we give simple examples of sets $𝒮$ of quadratic forms that have minimal $𝒮$ -universality criteria of multiple cardinalities. This answers a question of Kim, Kim, and Oh [KKO05] in the negative.

The gap theorems for some extremal submanifolds in a unit sphere

Xi Guo and Lan Wu (2015)

Communications in Mathematics

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Let $M$ be an $n$ -dimensional submanifold in the unit sphere $S^{n + p}$ , we call $M$ a $k$ -extremal submanifold if it is a critical point of the functional $\int_{M} ρ^{2 k} d v$ . In this paper, we can study gap phenomenon for these submanifolds.

A discrepancy principle for Tikhonov regularization with approximately specified data

M. Thamban Nair, Eberhard Schock (1998)

Annales Polonici Mathematici

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Many discrepancy principles are known for choosing the parameter α in the regularized operator equation $(T * T + α I) x_{α}^{δ} = T * y^{δ}$ , $| y - y^{δ} | \leq δ$ , in order to approximate the minimal norm least-squares solution of the operator equation Tx = y. We consider a class of discrepancy principles for choosing the regularization parameter when T*T and $T * y^{δ}$ are approximated by Aₙ and $z ₙ^{δ}$ respectively with Aₙ not necessarily self-adjoint. This procedure generalizes the work of Engl and Neubauer (1985), and particular cases of the results...

Minimal systems and distributionally scrambled sets

Piotr Oprocha (2012)

Bulletin de la Société Mathématique de France

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In this paper we investigate numerous constructions of minimal systems from the point of view of $(ℱ_{1}, ℱ_{2})$ -chaos (but most of our results concern the particular cases of distributional chaos of type $1$ and $2$ ). We consider standard classes of systems, such as Toeplitz flows, Grillenberger $K$ -systems or Blanchard-Kwiatkowski extensions of the Chacón flow, proving that all of them are DC2. An example of DC1 minimal system with positive topological entropy is also introduced. The above mentioned results...

A uniform dichotomy for generic $SL (2, ℝ)$ cocycles over a minimal base

Artur Avila, Jairo Bochi (2007)

Bulletin de la Société Mathématique de France

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We consider continuous $SL (2, ℝ)$ -cocycles over a minimal homeomorphism of a compact set $K$ of finite dimension. We show that the generic cocycle either is uniformly hyperbolic or has uniform subexponential growth.

Bubbling on boundary submanifolds for the Lin–Ni–Takagi problem at higher critical exponents

Manuel del Pino, Fethi Mahmudi, Monica Musso (2014)

Journal of the European Mathematical Society

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Let $Ω$ be a bounded domain in $ℝ^{n}$ with smooth boundary $\partial Ω$ . We consider the equation $d^{2} Δ u - u + u^{\frac{n - k + 2}{n - k - 2}} = 0 in Ω$ , under zero Neumann boundary conditions, where $Ω$ is open, smooth and bounded and $d$ is a small positive parameter. We assume that there is a $k$ -dimensional closed, embedded minimal submanifold $K$ of $\partial Ω$ , which is non-degenerate, and certain weighted average of sectional curvatures of $\partial Ω$ is positive along $K$ . Then we prove the existence of a sequence $d = d_{j} \to 0$ and a positive solution $u_{d}$ such that $d^{2} {| \nabla u_{d} |}^{2} ⇀ S δ_{K} as d \to 0$ in the sense of measures,...

Ramification of the Gauss map of complete minimal surfaces in $ℝ^{m}$ on annular ends

Gerd Dethloff, Pham Hoang Ha, Pham Duc Thoan (2016)

Colloquium Mathematicae

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We study the ramification of the Gauss map of complete minimal surfaces in $ℝ^{m}$ on annular ends. This is a continuation of previous work of Dethloff-Ha (2014), which we extend here to targets of higher dimension.

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...

Centroaffine differential geometry and its relations to horizontal submanifolds

Luc Vrancken (2002)

Banach Center Publications

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We relate centroaffine immersions $f : M ⁿ \to ℝ^{n + 1}$ to horizontal immersions g of Mⁿ into $S_{n + 1}^{2 n + 1} (1)$ or $H_{n}^{2 n + 1} (- 1)$ . We also show that f is an equiaffine sphere, i.e. the centroaffine normal is a constant multiple of the Blaschke normal, if and only if g is minimal.

Best approximation in spaces of bounded linear operators

Grzegorz Lewicki

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CONTENTSChapter 0...............................................................................................................................................................................5 0.1. Introduction..................................................................................................................................................................5 0.2. Preliminary results.......................................................................................................................................................9Chapter...

Displaying similar documents to “A characterization of a minimal submanifold in R n + p ”

Displaying similar documents to “A characterization of a minimal submanifold in $R^{n + p}$ ”