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Shape index, Brouwer degree and Poincar\'e-Hopf theorem

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Héctor Barge, José M.R. Sanjurjo

Dynamical Systems · Submitted 11 Mar 2023

abstract compiled · 2026-09-23 · no language-model review

The authors' main claim We obtain new forms of the Poincar% \'{e}-Hopf theorem and of the Borsuk and Hirsch antipodal theorems.

how it is stated · asserted 1.00 read the record →

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arXiv ID2303.06472 (version 2)
Pith IDCURL633K, Pith's permanent record of this paper and its review
CitationH\'ector Barge, Jos\'e M.R. Sanjurjo. "Shape index, Brouwer degree and Poincar\'e-Hopf theorem." arXiv:2303.06472 (2023). https://pith.science/paper/CURL633K
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The compiled record

The paper's own sentences, arranged. Each label is drawn from a closed vocabulary; the number beside it is the probability the classifier gave that label. Numbers in the tables are the authors', checked by code. No sentence in this record was written by a model.

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Compiled by classifier-1.13.0 via TypeSafe API, question bank abstract-p1-20260923, 2026-09-23: 1 request, 3,534 input tokens.

What the abstract itself states, as probabilities: claims first 0.05 · states limitation 0.04 · human participants 0.01 · controlled comparison 0.01 · randomized assignment 0.01 · states quantitative result 0.08 · compares with prior results 0.13 · mentions code or data availability 0.01.

The abstract, sentence by sentence

arXiv listing abstract v2. Results and the main claim are underlined; .

In this paper we study the relationship of the Brouwer degree of a vector field with the dynamics of the induced flow. Analogous relations are studied for the index of a vector field. We obtain new forms of the Poincar% \'{e}-Hopf theorem and of the Borsuk and Hirsch antipodal theorems. As an application, we calculate the Brouwer degree of the vector field of the Lorenz equations in isolating blocks of the Lorenz strange set.

background contribution method result interpretation main claim

What the classifier reports for each sentence. The number is the probability it gave the label.
#sentencerolethe authors' own workstates the main conclusionstrength
s0 In this paper we study the relationship of the Brouwer degree of a vector field with the dynamics of the induced flow. contribution 0.93 0.85 0.07 asserted 0.68
s1 Analogous relations are studied for the index of a vector field. contribution 0.97 0.83 0.05 asserted 0.96
s2 We obtain new forms of the Poincar% \'{e}-Hopf theorem and of the Borsuk and Hirsch antipodal theorems. result 0.78 0.94 0.55 asserted 1.00
s3 As an application, we calculate the Brouwer degree of the vector field of the Lorenz equations in isolating blocks of the Lorenz strange set. result 0.74 0.95 0.13 asserted 1.00

What kind of paper the classifier reports this is: theoretical result 1.00 · evidence from proof or derivation 0.99.

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Every quoted sentence on this record is the authors'. Every label is from Pith's closed vocabulary, with the probability shown beside it. Every number in the measurements table is the authors', checked by code. Record as JSON

The paper
Original abstract

In this paper we study the relationship of the Brouwer degree of a vector field with the dynamics of the induced flow. Analogous relations are studied for the index of a vector field. We obtain new forms of the Poincar% \'{e}-Hopf theorem and of the Borsuk and Hirsch antipodal theorems. As an application, we calculate the Brouwer degree of the vector field of the Lorenz equations in isolating blocks of the Lorenz strange set.

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