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Grid functions of nonstandard analysis in the theory of distributions and in partial differential equations

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Emanuele Bottazzi

Functional Analysis · Submitted 3 Apr 2017

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

The authors' main claim We will show that in the space of grid functions it is possible to formulate problems from many areas of functional analysis in a way that coherently generalizes the standard approaches.

how it is stated · hedged possibility 0.53 read the record →

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arXiv ID1704.00470 (version 2)
Pith IDC4NUOSMD, Pith's permanent record of this paper and its review
CitationEmanuele Bottazzi. "Grid functions of nonstandard analysis in the theory of distributions and in partial differential equations." arXiv:1704.00470 (2017). https://pith.science/paper/C4NUOSMD
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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, 3,578 input tokens.

What the abstract itself states, as probabilities: claims first 0.07 · states limitation 0.85 · human participants 0.02 · controlled comparison 0.01 · randomized assignment 0.01 · states quantitative result 0.02 · compares with prior results 0.35 · mentions code or data availability 0.02.

The abstract, sentence by sentence

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

We introduce the space of grid functions, a space of generalized functions of nonstandard analysis that provides a coherent generalization both of the space of distributions and of the space of Young measures. We will show that in the space of grid functions it is possible to formulate problems from many areas of functional analysis in a way that coherently generalizes the standard approaches. As an example, we discuss some applications of grid functions to the calculus of variations and to the nonlinear theory of distributions. Applications to nonlinear partial differential equations will be discussed in a subsequent paper.

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 We introduce the space of grid functions, a space of generalized functions of nonstandard analysis that provides a coherent generalization both of the space of distributions and of the space of Young measures. contribution 1.00 0.91 0.45 asserted 1.00
s1 We will show that in the space of grid functions it is possible to formulate problems from many areas of functional analysis in a way that coherently generalizes the standard approaches. contribution 0.73 0.67 0.41 hedged possibility 0.53
s2 As an example, we discuss some applications of grid functions to the calculus of variations and to the nonlinear theory of distributions. contribution 0.92 0.81 0.10 asserted 0.88
s3 Applications to nonlinear partial differential equations will be discussed in a subsequent paper. future work 1.00 0.17 0.04 asserted 0.89

What kind of paper the classifier reports this is: method or tool 0.94 · evidence from proof or derivation 0.85.

What was examined, and what was not

sources read
  • arXiv listing abstract v2
not examined
  • full text
  • figures and tables
  • references and citing sentences
  • relations to other papers
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  • P abstract
not run
  • 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

We introduce the space of grid functions, a space of generalized functions of nonstandard analysis that provides a coherent generalization both of the space of distributions and of the space of Young measures. We will show that in the space of grid functions it is possible to formulate problems from many areas of functional analysis in a way that coherently generalizes the standard approaches. As an example, we discuss some applications of grid functions to the calculus of variations and to the nonlinear theory of distributions. Applications to nonlinear partial differential equations will be discussed in a subsequent paper.

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1 arXiv paper cite this in 3 sentences

Each citing sentence is taken from the citing paper's LaTeX source and typed by Pith's classifier. A label appears at probability 0.8 or more (0.95 for support), a contest at 0.7 or more; the quoted words are the citing authors'. How each label was checked · JSON

Papers citing it · 1 sentence without a confident label

Cites it as background 1 paper

What it cites · 1 sentence without a confident label

Cites as background 1 paper
  • In [ref] and subsequent papers [ref] , the authors developed a theory of ultrafunctions, i.e. nonstandard vector spaces of a hyperfinite dimension that extend the space of distributions.

    Ultrafunctions and generalized solutions · Introduction · 0.94

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