Pieri-type intersection formulas and primary obstructions for decomposing 2-forms

Sinan Sertöz

Displaying similar documents to “Pieri-type intersection formulas and primary obstructions for decomposing 2-forms”

Liftings of 1-forms to $(J^{r} T ) $

Włodzimierz M. Mikulski (2002)

Colloquium Mathematicae

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Let $J^{r} T * M$ be the r-jet prolongation of the cotangent bundle of an n-dimensional manifold M and let $(J^{r} T * M) *$ be the dual vector bundle. For natural numbers r and n, a complete classification of all linear natural operators lifting 1-forms from M to 1-forms on $(J^{r} T * M) *$ is given.

Global classical solutions in a self-consistent chemotaxis(-Navier)-Stokes system

Yanjiang Li, Zhongqing Yu, Yumei Huang (2024)

Czechoslovak Mathematical Journal

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The self-consistent chemotaxis-fluid system $\{\begin{matrix} n_{t} + u \cdot \nabla n = Δ n - \nabla \cdot (n \nabla c) + \nabla \cdot (n \nabla φ), & x \in Ω, t > 0, \\ c_{t} + u \cdot \nabla c = Δ c - n c, & x \in Ω, t > 0, \\ u_{t} + κ (u \cdot \nabla) u + \nabla P = Δ u - n \nabla φ + n \nabla c, & x \in Ω, t > 0, \\ \nabla \cdot u = 0, & x \in Ω, t > 0, \end{matrix}$ is considered under no-flux boundary conditions for $n, c$ and the Dirichlet boundary condition for $u$ on a bounded smooth domain $Ω \subset ℝ^{N}$ $(N = 2, 3)$ , $κ \in {0, 1}$ . The existence of global bounded classical solutions is proved under a smallness assumption on $∥ c_{0} ∥_{L^{\infty} (Ω)}$ . Both the effect of gravity (potential force) on cells and the effect of the chemotactic force on fluid are considered here, and thus the coupling is stronger than the most studied chemotaxis-fluid...

On natural vector bundle morphisms $T_{A} \circ ⨂_{s}^{q} \to ⨂_{s}^{q} \circ T_{A}$ over $i d_{T_{A}}$

A. Ntyam, A. Mba (2009)

Annales Polonici Mathematici

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Some properties and applications of natural vector bundle morphisms $T_{A} \circ ⨂_{s}^{q} \to ⨂_{s}^{q} \circ T_{A}$ over $i d_{T_{A}}$ are presented.

Geometric stability of the cotangent bundle and the universal cover of a projective manifold

Frédéric Campana, Thomas Peternell (2011)

Bulletin de la Société Mathématique de France

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We first prove a strengthening of Miyaoka’s generic semi-positivity theorem: the quotients of the tensor powers of the cotangent bundle of a non-uniruled complex projective manifold $X$ have a pseudo-effective (instead of generically nef) determinant. A first consequence is that $X$ is of general type if its cotangent bundle contains a subsheaf with ‘big’ determinant. Among other applications, we deduce that if the universal cover of $X$ is not covered by compact positive-dimensional analytic...

On the connectivity of finite subset spaces

Jacob Mostovoy, Rustam Sadykov (2012)

Fundamenta Mathematicae

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We prove that the space $e x p_{k} ⋁ S^{m + 1}$ of nonempty subsets of cardinality at most k in a bouquet of m+1-dimensional spheres is (m+k-2)-connected. This, as shown by Tuffley, implies that the space $e x p_{k} X$ is (m+k-2)-connected for any m-connected cell complex X.

The generic dimension of the first derived system

Robert P. Buemi (1978)

Annales de l'institut Fourier

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Any $r$ -dimensional subbundle of the cotangent bundle on an $n$ -dimensional manifold $M$ partitions $M$ into subsets $M_{0}, ..., M_{m}$ ( $m$ being the minimum of $r$ and $C (n - r, 2)$ , the combinations of $n - r$ things taken 2 at a time). $M_{i}$ is the set on which the first derived systems of the subbundle has codimension $i$ . In this paper we prove the following: Theorem. Let $s \geq 2$ and let $Q$ be a generic $C^{s}$ $r$ -dimensional subbundle of the cotangent bundle of an $n$ -dimensional manifold $M$ . The codimension...

The natural operators lifting 1-forms to some vector bundle functors

J. Kurek, W. M. Mikulski (2002)

Colloquium Mathematicae

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Let F:ℳ f→ ℬ be a vector bundle functor. First we classify all natural operators $T_{| ℳ f ₙ} ⇝ T^{(0, 0)} (F_{| ℳ f ₙ}) *$ transforming vector fields to functions on the dual bundle functor $(F_{| ℳ f ₙ}) *$ . Next, we study the natural operators $T *_{| ℳ f ₙ} ⇝ T * (F_{| ℳ f ₙ}) *$ lifting 1-forms to $(F_{| ℳ f ₙ}) *$ . As an application we classify the natural operators $T *_{| ℳ f ₙ} ⇝ T * (F_{| ℳ f ₙ}) *$ for some well known vector bundle functors F.

Constructions on second order connections

J. Kurek, W. M. Mikulski (2007)

Annales Polonici Mathematici

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We classify all $ℱ ℳ_{m, n}$ -natural operators $: J ² ⟿ J ² V^{A}$ transforming second order connections Γ: Y → J²Y on a fibred manifold Y → M into second order connections $(Γ) : V^{A} Y \to J ² V^{A} Y$ on the vertical Weil bundle $V^{A} Y \to M$ corresponding to a Weil algebra A.

Natural operators lifting linear vector fields from a vector bundle to its r-jet prolongations

Włodzimierz M. Mikulski (2003)

Annales Polonici Mathematici

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All natural operators A transforming a linear vector field X on a vector bundle E into a vector field A(X) on the r-jet prolongation $J^{r} E$ of E are given. Similar results are deduced for the r-jet prolongations $J_{v}^{r} E$ and $J^{[r]} E$ in place of $J^{r} E$ .

Existence problem of the osculating planes of a curve in $R_{n}$

S. Topa (1964)

Annales Polonici Mathematici

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Lifting to the r-frame bundle by means of connections

J. Kurek, W. M. Mikulski (2010)

Annales Polonici Mathematici

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Let m and r be natural numbers and let $P^{r} : ℳ f_{m} \to ℱ ℳ$ be the rth order frame bundle functor. Let $F : ℳ f_{m} \to ℱ ℳ$ and $G : ℳ f_{k} \to ℱ ℳ$ be natural bundles, where $k = d i m (P^{r} ℝ^{m})$ . We describe all $ℳ f_{m}$ -natural operators A transforming sections σ of $F M \to M$ and classical linear connections ∇ on M into sections A(σ,∇) of $G (P^{r} M) \to P^{r} M$ . We apply this general classification result to many important natural bundles F and G and obtain many particular classifications.

The natural operators lifting horizontal 1-forms to some vector bundle functors on fibered manifolds

J. Kurek, W. M. Mikulski (2003)

Colloquium Mathematicae

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Let F:ℱ ℳ → ℬ be a vector bundle functor. First we classify all natural operators $T_{p r o j | ℱ ℳ_{m, n}} ⇝ T^{(0, 0)} (F_{| ℱ ℳ_{m, n}}) *$ transforming projectable vector fields on Y to functions on the dual bundle (FY)* for any $ℱ ℳ_{m, n}$ -object Y. Next, under some assumption on F we study natural operators $T *_{h o r | ℱ ℳ_{m, n}} ⇝ T * (F_{| ℱ ℳ_{m, n}}) *$ lifting horizontal 1-forms on Y to 1-forms on (FY)* for any Y as above. As an application we classify natural operators $T *_{h o r | ℱ ℳ_{m, n}} ⇝ T * (F_{| ℱ ℳ_{m, n}}) *$ for some vector bundle functors F on fibered manifolds.

The gradient flow of Higgs pairs

Jiayu Li, Xi Zhang (2011)

Journal of the European Mathematical Society

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We consider the gradient flow of the Yang–Mills–Higgs functional of Higgs pairs on a Hermitian vector bundle $(E, H_{0})$ over a Kähler surface $(M, ω)$ , and study the asymptotic behavior of the heat flow for Higgs pairs at infinity. The main result is that the gradient flow with initial condition $(A_{0}, φ_{0})$ converges, in an appropriate sense which takes into account bubbling phenomena, to a critical point $(A_{\infty}, φ_{\infty})$ of this functional. We also prove that the limiting Higgs pair $(A_{\infty}, φ_{\infty})$ can be extended smoothly to a vector bundle...

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.

General position properties in fiberwise geometric topology

Taras Banakh, Vesko Valov

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General position properties play a crucial role in geometric and infinite-dimensional topologies. Often such properties provide convenient tools for establishing various universality results. One of well-known general position properties is DDⁿ, the property of disjoint n-cells. Each Polish $L C^{n - 1}$ -space X possessing DDⁿ contains a topological copy of each n-dimensional compact metric space. This fact implies, in particular, the classical Lefschetz-Menger-Nöbeling-Pontryagin-Tolstova embedding...

Semistability of Frobenius direct images over curves

Vikram B. Mehta, Christian Pauly (2007)

Bulletin de la Société Mathématique de France

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Let $X$ be a smooth projective curve of genus $g \geq 2$ defined over an algebraically closed field $k$ of characteristic $p > 0$ . Given a semistable vector bundle $E$ over $X$ , we show that its direct image $F_{*} E$ under the Frobenius map $F$ of $X$ is again semistable. We deduce a numerical characterization of the stable rank- $p$ vector bundles $F_{*} L$ , where $L$ is a line bundle over $X$ .

Horizontal sections of connections on curves and transcendence

C. Gasbarri (2013)

Acta Arithmetica

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Let K be a number field, X be a smooth projective curve over it and D be a reduced divisor on X. Let (E,∇) be a vector bundle with connection having meromorphic singularities on D. Let $p_{1}, . . ., p_{s} \in X (K)$ and $X^{o} : = X ̅ ∖ D, p_{1}, . . ., p_{s}$ (the $p_{j}$ ’s may be in the support of D). Using tools from Nevanlinna theory and formal geometry, we give the definition of E-section of arithmetic type of the vector bundle E with respect to the points $p_{j}$ ; this is the natural generalization of the notion of E-function defined in Siegel-Shidlovskiĭ...

On the stratification of the orbit space for the action of automorphisms on connections

Witold Kondracki, Jan Rogulski

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CONTENTSIntroduction..................................................................................................................................................5§1. Basic notions and notation.....................................................................................................................7 1.1. Automorphisms of principal bundles....................................................................................................7 1.2. Connections and parallel...

The Group of Invertible Elements of the Algebra of Quaternions

Irina A. Kuzmina, Marie Chodorová (2016)

Acta Universitatis Palackianae Olomucensis. Facultas Rerum Naturalium. Mathematica

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We have, that all two-dimensional subspaces of the algebra of quaternions, containing a unit, are 2-dimensional subalgebras isomorphic to the algebra $ℂ$ of complex numbers. It was proved in the papers of N. E. Belova. In the present article we consider a 2-dimensional subalgebra $(i)$ of complex numbers with basis $1, i$ and we construct the principal locally trivial bundle which is isomorphic to the Hopf fibration.

On lifting of connections to Weil bundles

Jan Kurek, Włodzimierz M. Mikulski (2012)

Annales Polonici Mathematici

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We prove that the problem of finding all $ℳ f_{m}$ -natural operators $B : Q ⟿ Q T^{A}$ lifting classical linear connections ∇ on m-manifolds M to classical linear connections $B_{M} (\nabla)$ on the Weil bundle $T^{A} M$ corresponding to a p-dimensional (over ℝ) Weil algebra A is equivalent to the one of finding all $ℳ f_{m}$ -natural operators $C : Q ⟿ (T ¹_{p - 1}, T * \otimes T * \otimes T)$ transforming classical linear connections ∇ on m-manifolds M into base-preserving fibred maps $C_{M} (\nabla) : T ¹_{p - 1} M = ⨁_{M}^{p - 1} T M \to T * M \otimes T * M \otimes T M$ .

One-dimensional symmetry of periodic minimizers for a mean field equation

Chang-Shou Lin, Marcello Lucia (2007)

Annali della Scuola Normale Superiore di Pisa - Classe di Scienze

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We consider on a two-dimensional flat torus $T$ defined by a rectangular periodic cell the following equation $Δ u + ρ (\frac{e^{u}}{\int_{T} e^{u}} - \frac{1}{| T |}) = 0, \int_{T} u = 0 .$ It is well-known that the associated energy functional admits a minimizer for each $ρ \leq 8 π$ . The present paper shows that these minimizers depend actually only on one variable. As a consequence, setting $λ_{1} (T)$ to be the first eigenvalue of the Laplacian on the torus, the minimizers are identically zero whenever $ρ \leq min {8 π, λ_{1} (T) | T |}$ . Our results hold more generally for solutions that...

The 4-string braid group $B_{4}$ has property RD and exponential mesoscopic rank

Sylvain Barré, Mikaël Pichot (2011)

Bulletin de la Société Mathématique de France

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We prove that the braid group $B_{4}$ on 4 strings, its central quotient $B_{4} / 〈 z 〉$ , and the automorphism group $Aut (F_{2})$ of the free group $F_{2}$ on 2 generators, have the property RD of Haagerup–Jolissaint. We also prove that the braid group $B_{4}$ is a group of intermediate mesoscopic rank (of dimension 3). More precisely, we show that the above three groups have exponential mesoscopic rank, i.e., that they contain exponentially many large flat balls which are not included in flats.

A criterion for pure unrectifiability of sets (via universal vector bundle)

Silvano Delladio (2011)

Annales Polonici Mathematici

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Let m,n be positive integers such that m < n and let G(n,m) be the Grassmann manifold of all m-dimensional subspaces of ℝⁿ. For V ∈ G(n,m) let $π_{V}$ denote the orthogonal projection from ℝⁿ onto V. The following characterization of purely unrectifiable sets holds. Let A be an $^{m}$ -measurable subset of ℝⁿ with $^{m} (A) < \infty$ . Then A is purely m-unrectifiable if and only if there exists a null subset Z of the universal bundle $(V, v) | V \in G (n, m), v \in V$ such that, for all P ∈ A, one has $^{m (n - m)} (V \in G (n, m) | (V, π_{V} (P)) \in Z) > 0$ . One can replace “for all P ∈ A” by “for...

Liftings of forms to Weil bundles and the exterior derivative

Jacek Dębecki (2009)

Annales Polonici Mathematici

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In a previous paper we have given a complete description of linear liftings of p-forms on n-dimensional manifolds M to q-forms on $T^{A} M$ , where $T^{A}$ is a Weil functor, for all non-negative integers n, p and q, except the case p = n and q = 0. We now establish formulas connecting such liftings and the exterior derivative of forms. These formulas contain a boundary operator, which enables us to define a homology of the Weil algebra A. We next study the case p = n and q = 0 under the condition...

Pseudo-real principal Higgs bundles on compact Kähler manifolds

Indranil Biswas, Oscar García-Prada, Jacques Hurtubise (2014)

Annales de l’institut Fourier

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Let $X$ be a compact connected Kähler manifold equipped with an anti-holomorphic involution which is compatible with the Kähler structure. Let $G$ be a connected complex reductive affine algebraic group equipped with a real form $σ_{G}$ . We define pseudo-real principal $G$ -bundles on $X$ . These are generalizations of real algebraic principal $G$ -bundles over a real algebraic variety. Next we define stable, semistable and polystable pseudo-real principal $G$ -bundles. Their relationships with the usual...

Are All Smooth Solutions of Euclidean $S U (2)$ -Yang-Mills Equations Selfdual or Antiselfdual?

R. Flume (1979)

Recherche Coopérative sur Programme n°25

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Local cohomology of logarithmic forms

G. Denham, H. Schenck, M. Schulze, M. Wakefield, U. Walther (2013)

Annales de l’institut Fourier

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Let $Y$ be a divisor on a smooth algebraic variety $X$ . We investigate the geometry of the Jacobian scheme of $Y$ , homological invariants derived from logarithmic differential forms along $Y$ , and their relationship with the property that $Y$ be a free divisor. We consider arrangements of hyperplanes as a source of examples and counterexamples. In particular, we make a complete calculation of the local cohomology of logarithmic forms of generic hyperplane arrangements.

Simultaneous stabilization in $A_{ℝ} ()$

Raymond Mortini, Brett D. Wick (2009)

Studia Mathematica

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We study the problem of simultaneous stabilization for the algebra $A_{ℝ} ()$ . Invertible pairs $(f_{j}, g_{j})$ , j = 1,..., n, in a commutative unital algebra are called simultaneously stabilizable if there exists a pair (α,β) of elements such that $α f_{j} + β g_{j}$ is invertible in this algebra for j = 1,..., n. For n = 2, the simultaneous stabilization problem admits a positive solution for any data if and only if the Bass stable rank of the algebra is one. Since $A_{ℝ} ()$ has stable rank two, we are faced here with a different...

Expansions of o-minimal structures by sparse sets

Harvey Friedman, Chris Miller (2001)

Fundamenta Mathematicae

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Given an o-minimal expansion ℜ of the ordered additive group of real numbers and E ⊆ ℝ, we consider the extent to which basic metric and topological properties of subsets of ℝ definable in the expansion (ℜ,E) are inherited by the subsets of ℝ definable in certain expansions of (ℜ,E). In particular, suppose that $f (E^{m})$ has no interior for each m ∈ ℕ and $f : ℝ^{m} \to ℝ$ definable in ℜ, and that every subset of ℝ definable in (ℜ,E) has interior or is nowhere dense. Then every subset of ℝ definable in (ℜ,(S))...

The sizes of the classes of $H^{(N)}$ -sets

Václav Vlasák (2014)

Fundamenta Mathematicae

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The class of $H^{(N)}$ -sets forms an important subclass of the class of sets of uniqueness for trigonometric series. We investigate the size of this class which is reflected by the family of measures (called polar) annihilating all sets from the class. The main aim of this paper is to answer in the negative a question stated by Lyons, whether the polars of the classes of $H^{(N)}$ -sets are the same for all N ∈ ℕ. To prove our result we also present a new description of $H^{(N)}$ -sets.