714 research outputs found
Global Hypoellipticity for Strongly Invariant Operators
In this note, by analyzing the behavior at infinity of the matrix symbol of
an invariant operator with respect to a fixed elliptic operator, we obtain
a necessary and sufficient condition to guarantee that is globally
hypoelliptic. We also investigate relations between the global hypoellipticity
of and global subelliptic estimates.Comment: 20 page
Regularity of solutions to a Vekua-type equation on compact Lie groups
We present sufficient conditions to have global hypoellipticity for a class of Vekua-type operators defined on a compact Lie group. When the group has the property that every non-trivial representation is not self-dual we show that these sufficient conditions are also necessary. We also present results about the global solvability for this class of operators
Regularity of solutions to a Vekua-type equation on compact Lie groups
We present sufficient conditions to have global hypoellipticity for a class
of Vekua-type operators defined on a compact Lie group. When the group has the
property that every non-trivial representation is not self-dual we show that
these sufficient conditions are also necessary. We also present results about
the global solvability for this class of operators.Comment: 23 page
Hipoeliticidade global para operadores fortemente invariantes
Orientador : Prof. Dr. Alexandre KirilovDissertação (mestrado) - Universidade Federal do Paraná, Setor de Ciências Exatas, Programa de Pós-Graduação em Matemática. Defesa: Curitiba, 26/02/2016Inclui referências : f. 48-49Área de concentração: MatematicaResumo: A partir do conceito de operadores invariantes em relação a uma decomposição de um espaço de Hilbert em subespaços de dimensão finita, introduzimos o símbolo do operador em relação a essa decomposição. Esse símbolo é uma sequência de matrizes cujas propriedades permitem, por exemplo, afirmar se o operador está em alguma classe de Schatten-von Neumann e se é possível estende-lo a um operador limitado. Usamos esses resultados para decompor o espaço de Hilbert L2(M), sobre uma variedade suave compacta orientavel sem bordo M, como soma direta de autoespaços de um operador diferencial el?tico autoadjunto positivo e estudamos propriedades que os operadores invariantes possuem neste espaço. Por fim, obtemos resultados acerca da hipoeliticidade Global de operadores invariantes sobre M analisando seu símbolo.Abstract: From the idea of invariant operators relative to a fixed partition of a Hilbert space into a direct sum of finite dimensional subspaces, we introduce the operator's symbol relative to this decomposition. This symbol is a sequence of matrices whose properties allow us, for example, to state if the operator belong to some Schatten-von Neumann class and if it can be extended to a bounded operator. We apply this results to decompose the Hilbert space L2(M), where M is a orientable compact smooth manifold without boundary, as direct sum of eigenspaces of a positive self-adjoint elliptic differential operator and then we study some properties that the invariants operators have in this space. Finally, we obtain results about global hypoellipticity of invariant operators on M analyzing their symbol
Global properties for a class os operators on compact lie groups
Orientador: Dr. Alexandre KirilovCoorientador: Prof. Dr. Michael RuzhanskyTese (doutorado) - Universidade Federal do Paraná, Setor de Ciências Exatas, Programa de Pós-Graduação em Matemática. Defesa : Curitiba, 28/02/2020Inclui referências: p. 142-145Resumo: Esta tese apresenta condições necessárias e suficientes para a obtenção de hipoeliticidade global e resolubilidade global para uma classe de campos vetoriais definidos em um produto de grupos de Lie compactos. Tanto a hipoeliticidade global quanto a resolubilidade global sao estudadas no sentido usual das funcoes suaves, bem como em classes de Komatsu. Em vista da conjectura de Greenfield e Wallach sobre a nao existencia de campos vetoriais globalmente hipoelíticos senao definidos no toro, e estudada uma classe de exemplos que podem ser considerados como perturbacoes de ordem zero de campos vetoriais. Palavras-chave: grupos compactos, hipoeliticidade global, resolubilidade global, classes de Komatsu.Abstract: In this dissertation we present necessary and sufficient conditions to have global hypoellipticity and global solvability for a class of vector fields defined in a product of compact Lie groups. Both global hypoellipticity and solvability are studied in the usual smooth sense as in the sense of Komatsu. Considering the Greenfield's and Wallach's conjecture, about the non-existence of globally hypoelliptic vector fields out of tori, we also study classes of examples that can be considered as zeros-order perturbations of our vector fields. K eywords: compact groups, global hypoellipticity, global solvability, Komatsu classes
Global properties of vector fields on compact Lie groups in Komatsu classes. II. Normal forms
Let and be compact Lie groups, , and consider the operator \begin{equation*} L_{aq} = X_1 +
a(x_1)X_2 + q(x_1,x_2), \end{equation*} where and are
ultradifferentiable functions in the sense of Komatsu, and is real-valued.
We characterize completely the global hypoellipticity and the global
solvability of in the sense of Komatsu. For this, we present a
conjugation between and a constant-coefficient operator that preserves
these global properties in Komatsu classes. We also present examples of
globally hypoelliptic and globally solvable operators on and in the sense of Komatsu. In
particular, we give examples of differential operators which are not globally
-solvable, but are globally solvable in Gevrey spaces.Comment: 23 pages. arXiv admin note: text overlap with arXiv:1910.01922,
arXiv:1910.0005
Global Properties for first order differential operators on
In this paper, we study the global properties of a class of evolution-like
differential operator with a 0-order perturbation defined on the product of
tori and spheres , with
and non-negative integers. By varying the values of and , we show
that it is possible to recover results already known in the literature and
present new results. The main tool used in this study is Fourier analysis,
taken partially with respect to each copy of the torus and sphere. We obtain
necessary and sufficient conditions related to Diophantine inequalities, change
of sign and connectivity of level sets associated the operator's coefficients
Global analytic hypoellipticity for a class of evolution operators on
In this paper, we present necessary and sufficient conditions to have global
analytic hypoellipticity for a class of first-order operators defined on
. In the case of real-valued coefficients, we
prove that an operator in this class is conjugated to a constant-coefficient
operator satisfying a Diophantine condition, and that such conjugation
preserves the global analytic hypoellipticity. In the case where the imaginary
part of the coefficients is non-zero, we show that the operator is globally
analytic hypoelliptic if the Nirenberg-Treves condition () holds,
in addition to a Diophantine condition.Comment: 24 page
Pervasive gaps in Amazonian ecological research
Biodiversity loss is one of the main challenges of our time,1,2 and attempts to address it require a clear un derstanding of how ecological communities respond to environmental change across time and space.3,4
While the increasing availability of global databases on ecological communities has advanced our knowledge
of biodiversity sensitivity to environmental changes,5–7 vast areas of the tropics remain understudied.8–11 In
the American tropics, Amazonia stands out as the world’s most diverse rainforest and the primary source of
Neotropical biodiversity,12 but it remains among the least known forests in America and is often underrepre sented in biodiversity databases.13–15 To worsen this situation, human-induced modifications16,17 may elim inate pieces of the Amazon’s biodiversity puzzle before we can use them to understand how ecological com munities are responding. To increase generalization and applicability of biodiversity knowledge,18,19 it is thus
crucial to reduce biases in ecological research, particularly in regions projected to face the most pronounced
environmental changes. We integrate ecological community metadata of 7,694 sampling sites for multiple or ganism groups in a machine learning model framework to map the research probability across the Brazilian
Amazonia, while identifying the region’s vulnerability to environmental change. 15%–18% of the most ne glected areas in ecological research are expected to experience severe climate or land use changes by
2050. This means that unless we take immediate action, we will not be able to establish their current status,
much less monitor how it is changing and what is being lostinfo:eu-repo/semantics/publishedVersio
- …