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Record W2003907273 · doi:10.2478/s12175-014-0288-5

The L p primitive integral

2014· article· en· W2003907273 on OpenAlex

Why this work is in the frame

A frame that forgets how it found something cannot be audited. These are the routes that admitted this work.

affAt least one author lists a Canadian institution in the pinned OpenAlex snapshot.

Bibliographic record

VenueMathematica Slovaca · 2014
Typearticle
Languageen
FieldMathematics
TopicAdvanced Banach Space Theory
Canadian institutionsUniversity of the Fraser Valley
Fundersnot available
KeywordsMathematicsCombinatoricsLp spaceBanach spaceHardy spaceNorm (philosophy)Poisson kernelMathematical analysis

Abstract

fetched live from OpenAlex

Abstract For each 1 ≤ p < ∞, a space of integrable Schwartz distributions L′p, is defined by taking the distributional derivative of all functions in L p. Here, L p is with respect to Lebesgue measure on the real line. If f ∈ L′p such that f is the distributional derivative of F ∈ L p, then the integral is defined as $\int\limits_{ - \infty }^\infty {fG} = - \int\limits_{ - \infty }^\infty {F(x)g(x)dx} $, where g ∈ L q, $G(x) = \int\limits_0^x {g(t)dt} $ and 1/p + 1/q =1. A norm is ‖f‖p′ = ‖F‖p. The spaces L′p and L p are isometrically isomorphic. Distributions in L′p share many properties with functions in L p. Hence, L′p is reflexive, its dual space is identified with L q, there is a type of Hölder inequality, continuity in norm, convergence theorems, Gateaux derivative. It is a Banach lattice and abstract L-space. Convolutions and Fourier transforms are defined. Convolution with the Poisson kernel is well defined and provides a solution to the half plane Dirichlet problem, boundary values being taken on in the new norm. A product is defined that makes L′1 into a Banach algebra isometrically isomorphic to the convolution algebra on L 1. Spaces of higher order derivatives of L p functions are defined. These are also Banach spaces isometrically isomorphic to L p.

Fetched live from OpenAlex and de-inverted. Abstracts are not stored in this database: the inverted indexes are 8.6 GB of the frame’s 9.3 GB of text, and the host has 13 GB free.

Full frame distilled prediction

Teacher imitation

Not calibrated prevalence, not ground truth. Human validation pending. Learned from the 10,348 direct Codex labels and 10,348 direct Gemma labels. Candidate is the union of thresholded teacher heads; consensus is their intersection. These outputs are machine_predicted_unvalidated and are not human labels or direct frontier model labels.

metaresearch head score (Codex)0.001
metaresearch head score (Gemma)0.004
Version: codex-gemma-dda1882f352aValidation status: machine_predicted_unvalidated
Candidate categoriesnone
Consensus categoriesnone
DomainCandidate signal: none · Consensus signal: none
Study designCandidate signal: Theoretical or conceptual · Consensus signal: Theoretical or conceptual
GenreCandidate signal: Empirical · Consensus signal: none
Teacher disagreement score0.755
Threshold uncertainty score0.607

Codex and Gemma teacher scores by category

CategoryCodexGemma
Metaresearch0.0010.004
Meta-epidemiology (narrow)0.0000.000
Meta-epidemiology (broad)0.0000.000
Bibliometrics0.0000.000
Science and technology studies0.0000.000
Scholarly communication0.0000.000
Open science0.0010.000
Research integrity0.0000.000
Insufficient payload (model declined to judge)0.0000.000

Machine scores (provisional)

The two teacher heads of the student model, read on this work. A score orders the frame for review; it never asserts a category, and the validation status ships verbatim with every row.

Baseline scores from an immature model (maturity gate not passed, 7 training rounds). Scores rank; they never assert a category.

Opus teacher head0.024
GPT teacher head0.306
Teacher spread0.282 · how far apart the two teachers sit on this one work
Validation statusscore_only:v0-immature-baseline · verbatim from the scoring run: score_only means the number may rank works, and no category label ships from it