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Record W3213533742

The Sums of Integer Powers

2021· article· en· W3213533742 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

VenueStudent Research Proceedings · 2021
Typearticle
Languageen
FieldMathematics
TopicHistory and Theory of Mathematics
Canadian institutionsMacEwan University
Fundersnot available
KeywordsBernoulli's principleInteger (computer science)MathematicsSequence (biology)Bernoulli numberMatrix (chemical analysis)ComputationDiscrete mathematicsCombinatoricsAlgorithmComputer science
DOInot available

Abstract

fetched live from OpenAlex

An investigation of the origin of the formulas for the sums of integer powers was performed. A method for calculating the sums of the first n integers to the kth power, denoted Sk(n), was first derived by Jacques Bernoulli in the late 1600’s. Through the discovery of formulas for the computation of integer powers, a numeric sequence arose. This sequence has become known as the Bernoulli numbers. Bernoulli simultaneously derived a recursive algorithm that can generate Bernoulli numbers. This recursive relationship was the subject of the first computer program in 1843. Utilizing a relationship that Bernoulli observed, it is possible to then calculate all terms in a coefficient matrix that enables the calculation of formulas for Sk(n). This coefficient matrix was generated using computer software with two different methods. Numerical analysis techniques were applied to each generated matrix, and the relative error between the two matrices was compared. Department: Mathematical Sciences  Faculty Mentor: Dr. Christian Ivanescu

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)0.005
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: Empirical
Teacher disagreement score0.187
Threshold uncertainty score0.437

Codex and Gemma teacher scores by category

CategoryCodexGemma
Metaresearch0.0050.004
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 head0.198
GPT teacher head0.475
Teacher spread0.277 · 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