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A Three-Point Characterization of Central Symmetry

2004· article· en· W2028973787 on OpenAlex
G. D. Chakerian, M. S. Klamkin

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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

VenueAmerican Mathematical Monthly · 2004
Typearticle
Languageen
FieldDecision Sciences
TopicProbability and Risk Models
Canadian institutionsUniversity of Alberta
Fundersnot available
KeywordsSubsequenceMathematicsCombinatoricsSequence (biology)Permutation (music)Longest increasing subsequenceIndependent and identically distributed random variablesCentral limit theoremLimit (mathematics)Discrete mathematicsRandom permutationRandom variableMathematical analysisSymmetric groupStatistics

Abstract

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1. It is tempting to believe that any sequence (?n) that is C6saro-convergent in probability necessarily has a subsequence that is a.s. C6saro-convergent. This is not true however. As an example, take an independent identically distributed sequence that satisfies the weak but not the strong law of large numbers (e.g., any symmetric distribution without first moment but with tails slightly smaller than Cauchy). 2. If we drop the nonnegativity, Observation 1 becomes false. Consider, for example, the constants (1)n log n. But if every permutation of a sequence of arbitrarily signed random variables is a.s. C6saro-convergent to a finite limit, does the sequence satisfy condition (A)? We do not know. 3. S. D. Chatterji [4] has already given versions of the subsequence theorem for ?n in LP with p < 1, but with a factor n-'/P instead of n-1. A nice review paper is [5]. 4. More recently, E. P6ter [7] gave sufficient criteria describing general distributional limit laws for which a permutation invariant version of the subsequence principle holds, in the same way that Berkes's result improves Koml6s's theorem.

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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)0.002
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 score0.352
Threshold uncertainty score0.389

Codex and Gemma teacher scores by category

CategoryCodexGemma
Metaresearch0.0010.002
Meta-epidemiology (narrow)0.0000.000
Meta-epidemiology (broad)0.0010.000
Bibliometrics0.0000.001
Science and technology studies0.0000.001
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 head0.044
GPT teacher head0.317
Teacher spread0.272 · 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