Geometric Analysis on Solutions of Some Differential Inequalities and within Restricted Classes of Holomorphic Functions
Bibliographic record
Abstract
Pars 1 and 2 are devoted to study of solutions of certain differential inequalities.\n\nNamely, in Part 1 we show that a germ of an analytic set (real or complex) admits \na Gagliardo-Nirenberg type inequality with a certain exponent s>=1. At a regular point \ns=1, and the inequality becomes classical. As our examples show, s can be strictly greater than 1 even for an isolated singularity. \n\nIn Part 2 we prove the property of unique continuation for solutions of differential inequality |\\Delta u|<=|Vu| for a large class of potentials V. This result can be applied to the problem of absence of positive eigenvalues for\nself-adjoint Schroedinger operator -\\Delta+V defined in the sense of the form sum. \nThe results of Part 2 are joint with Leonid Shartser.\n\n\nIn Parts 3 and 4 we derive the basic elements of complex function theory within \nsome subalgebras of holomorphic functions (including extension from submanifolds, corona type theorem, properties of divisors, approximation property). Our key instruments and results are the analogues of Cartan theorems A and B for the `coherent sheaves' on the maximal ideal spaces of these subalgebras, and of Oka-Cartan theorem on coherence of the sheaves of ideals of the corresponding complex analytic subsets. \n\nMore precisely, in Part 3 we consider the algebras of holomorphic functions on regular \ncoverings of complex manifolds whose restrictions to each fiber belong to a translation-invariant Banach subalgebra of bounded functions endowed with sup-norm.\nThe model examples of such subalgebras are Bohr's holomorphic almost periodic functions on tube domains, and all fibrewise bounded holomorphic functions on regular coverings of complex manifolds. \n\nIn Part 4 the primary object of study is the subalgebra of bounded holomorphic functions on the unit disk whose moduli can have only boundary discontinuities of the first kind.\nThe results of Parts 3 and 4 are joint with Alexander Brudnyi.
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How this classification was reachedexpand
Full frame machine prediction
Teacher imitationNot calibrated prevalence, not ground truth. Human validation pending. The Gemma side is a direct model label for every work in the frame, read from the title-only record. The Codex side is a classifier learned from the 10,348 direct Codex labels and calibrated to design-weighted sample rates; fields without enough sample support carry no Codex call. Candidate is the union of the two sides; consensus is their intersection. These outputs are machine_predicted_unvalidated and are not human labels.
Distilled classifier scores by category (both heads)
| Category | Codex | Gemma |
|---|---|---|
| Metaresearch | 0.001 | 0.002 |
| Meta-epidemiology (narrow) | 0.000 | 0.000 |
| Meta-epidemiology (broad) | 0.000 | 0.001 |
| Bibliometrics | 0.002 | 0.001 |
| Science and technology studies | 0.001 | 0.002 |
| Scholarly communication | 0.001 | 0.002 |
| Open science | 0.001 | 0.001 |
| Research integrity | 0.001 | 0.001 |
| Insufficient payload (model declined to judge) | 0.004 | 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 itClassification
machine, unvalidatedMachine predicted; a candidate call from one source (direct Gemma or distilled Codex), not a consensus.
How this classification was reached, model by model and score by score, is at the end of the page under "How this classification was reached".