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Circularly invariant uniformizable probability measures for linear transformations

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Chuang Xu

Dynamical Systems · Submitted 26 Sep 2016

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

The authors' main claim We show that there is a threshold constant $c$ depending only on the slope of the linear transformation such that there exists a CIUPM if and only if its support has a diameter at least as large as $c.$ Moreover, the CIUPM is unique up to translation if the diameter of the support equals $c.$

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arXiv ID1609.08204 (version 4)
Pith ID6R6MB4NT, Pith's permanent record of this paper and its review
CitationChuang Xu. "Circularly invariant uniformizable probability measures for linear transformations." arXiv:1609.08204 (2016). https://pith.science/paper/6R6MB4NT
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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, 2,405 input tokens.

What the abstract itself states, as probabilities: claims first 0.03 · states limitation 0.35 · human participants 0.02 · controlled comparison 0.02 · randomized assignment 0.01 · states quantitative result 0.63 · compares with prior results 0.04 · mentions code or data availability 0.01.

The abstract, sentence by sentence

arXiv listing abstract v4. Results and the main claim are underlined; .

In this paper, we prove a threshold result on the existence of a circularly invariant uniformizable probability measure (CIUPM) for linear transformations with non-zero slope on the line. We show that there is a threshold constant $c$ depending only on the slope of the linear transformation such that there exists a CIUPM if and only if its support has a diameter at least as large as $c.$ Moreover, the CIUPM is unique up to translation if the diameter of the support equals $c.$

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 prove a threshold result on the existence of a circularly invariant uniformizable probability measure (CIUPM) for linear transformations with non-zero slope on the line. contribution 0.88 0.95 0.65 asserted 1.00
s1 We show that there is a threshold constant $c$ depending only on the slope of the linear transformation such that there exists a CIUPM if and only if its support has a diameter at least as large as $c.$ Moreover, the CIUPM is unique up to translation if the diameter of the support equals $c.$ result 0.99 0.95 0.79 asserted 1.00

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

What was examined, and what was not

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  • references and citing sentences
  • relations to other papers
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  • P abstract
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  • S sections
  • T body sentences
  • C citations
  • R cross-paper relations

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 prove a threshold result on the existence of a circularly invariant uniformizable probability measure (CIUPM) for linear transformations with non-zero slope on the line. We show that there is a threshold constant $c$ depending only on the slope of the linear transformation such that there exists a CIUPM if and only if its support has a diameter at least as large as $c.$ Moreover, the CIUPM is unique up to translation if the diameter of the support equals $c.$

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