On convexity analysis for discrete delta Riemann–Liouville fractional differences analytically and numerically
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Bibliographic record
Abstract
Abstract In this paper, we focus on the analytical and numerical convexity analysis of discrete delta Riemann–Liouville fractional differences. In the analytical part of this paper, we give a new formula for the discrete delta Riemann-Liouville fractional difference as an alternative definition. We establish a formula for the $\Delta ^{2}$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msup> <mml:mi>Δ</mml:mi> <mml:mn>2</mml:mn> </mml:msup> </mml:math> , which will be useful to obtain the convexity results. We examine the correlation between the positivity of $({}^{\mathrm{RL}}_{w_{0}}\Delta ^{\alpha} \mathrm{f} )( \mathrm{t})$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mo>(</mml:mo> <mml:msubsup> <mml:mrow /> <mml:msub> <mml:mi>w</mml:mi> <mml:mn>0</mml:mn> </mml:msub> <mml:mi>RL</mml:mi> </mml:msubsup> <mml:msup> <mml:mi>Δ</mml:mi> <mml:mi>α</mml:mi> </mml:msup> <mml:mi>f</mml:mi> <mml:mo>)</mml:mo> <mml:mo>(</mml:mo> <mml:mi>t</mml:mi> <mml:mo>)</mml:mo> </mml:math> and convexity of the function. In view of the basic lemmas, we define two decreasing subsets of $(2,3)$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mo>(</mml:mo> <mml:mn>2</mml:mn> <mml:mo>,</mml:mo> <mml:mn>3</mml:mn> <mml:mo>)</mml:mo> </mml:math> , $\mathscr{H}_{\mathrm{k},\epsilon}$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msub> <mml:mi>H</mml:mi> <mml:mrow> <mml:mi>k</mml:mi> <mml:mo>,</mml:mo> <mml:mi>ϵ</mml:mi> </mml:mrow> </mml:msub> </mml:math> and $\mathscr{M}_{\mathrm{k},\epsilon}$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msub> <mml:mi>M</mml:mi> <mml:mrow> <mml:mi>k</mml:mi> <mml:mo>,</mml:mo> <mml:mi>ϵ</mml:mi> </mml:mrow> </mml:msub> </mml:math> . The decrease of these sets allows us to obtain the relationship between the negative lower bound of $({}^{\mathrm{RL}}_{w_{0}}\Delta ^{\alpha} \mathrm{f} )( \mathrm{t})$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mo>(</mml:mo> <mml:msubsup> <mml:mrow /> <mml:msub> <mml:mi>w</mml:mi> <mml:mn>0</mml:mn> </mml:msub> <mml:mi>RL</mml:mi> </mml:msubsup> <mml:msup> <mml:mi>Δ</mml:mi> <mml:mi>α</mml:mi> </mml:msup> <mml:mi>f</mml:mi> <mml:mo>)</mml:mo> <mml:mo>(</mml:mo> <mml:mi>t</mml:mi> <mml:mo>)</mml:mo> </mml:math> and convexity of the function on a finite time set $\mathrm{N}_{w_{0}}^{\mathrm{P}}:=\{w_{0}, w_{0}+1, w_{0}+2,\dots , \mathrm{P}\}$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msubsup> <mml:mi>N</mml:mi> <mml:msub> <mml:mi>w</mml:mi> <mml:mn>0</mml:mn> </mml:msub> <mml:mi>P</mml:mi> </mml:msubsup> <mml:mo>:</mml:mo> <mml:mo>=</mml:mo> <mml:mo>{</mml:mo> <mml:msub> <mml:mi>w</mml:mi> <mml:mn>0</mml:mn> </mml:msub> <mml:mo>,</mml:mo> <mml:msub> <mml:mi>w</mml:mi> <mml:mn>0</mml:mn> </mml:msub> <mml:mo>+</mml:mo> <mml:mn>1</mml:mn> <mml:mo>,</mml:mo> <mml:msub> <mml:mi>w</mml:mi> <mml:mn>0</mml:mn> </mml:msub> <mml:mo>+</mml:mo> <mml:mn>2</mml:mn> <mml:mo>,</mml:mo> <mml:mo>…</mml:mo> <mml:mo>,</mml:mo> <mml:mi>P</mml:mi> <mml:mo>}</mml:mo> </mml:math> for some $\mathrm{P}\in \mathrm{N}_{w_{0}}:=\{w_{0}, w_{0}+1, w_{0}+2,\dots \}$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>P</mml:mi> <mml:mo>∈</mml:mo> <mml:msub> <mml:mi>N</mml:mi> <mml:msub> <mml:mi>w</mml:mi> <mml:mn>0</mml:mn> </mml:msub> </mml:msub> <mml:mo>:</mml:mo> <mml:mo>=</mml:mo> <mml:mo>{</mml:mo> <mml:msub> <mml:mi>w</mml:mi> <mml:mn>0</mml:mn> </mml:msub> <mml:mo>,</mml:mo> <mml:msub> <mml:mi>w</mml:mi> <mml:mn>0</mml:mn> </mml:msub> <mml:mo>+</mml:mo> <mml:mn>1</mml:mn> <mml:mo>,</mml:mo> <mml:msub> <mml:mi>w</mml:mi> <mml:mn>0</mml:mn> </mml:msub> <mml:mo>+</mml:mo> <mml:mn>2</mml:mn> <mml:mo>,</mml:mo> <mml:mo>…</mml:mo> <mml:mo>}</mml:mo> </mml:math> . The numerical part of the paper is dedicated to examinin the validity of the sets $\mathscr{H}_{\mathrm{k},\epsilon}$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msub> <mml:mi>H</mml:mi> <mml:mrow> <mml:mi>k</mml:mi> <mml:mo>,</mml:mo> <mml:mi>ϵ</mml:mi> </mml:mrow> </mml:msub> </mml:math> and $\mathscr{M}_{\mathrm{k},\epsilon}$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msub> <mml:mi>M</mml:mi> <mml:mrow> <mml:mi>k</mml:mi> <mml:mo>,</mml:mo> <mml:mi>ϵ</mml:mi> </mml:mrow> </mml:msub> </mml:math> for different values of k and ϵ . For this reason, we illustrate the domain of solutions via several figures explaining the validity of the main theorem.
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Full frame distilled prediction
Teacher imitationNot 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.
Codex and Gemma teacher scores by category
| Category | Codex | Gemma |
|---|---|---|
| Metaresearch | 0.000 | 0.000 |
| Meta-epidemiology (narrow) | 0.000 | 0.000 |
| Meta-epidemiology (broad) | 0.000 | 0.000 |
| Bibliometrics | 0.000 | 0.001 |
| Science and technology studies | 0.000 | 0.000 |
| Scholarly communication | 0.000 | 0.000 |
| Open science | 0.000 | 0.000 |
| Research integrity | 0.000 | 0.000 |
| Insufficient payload (model declined to judge) | 0.000 | 0.000 |
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.
score_only:v0-immature-baseline · verbatim from the scoring run: score_only means the number may rank works, and no category label ships from it