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The paper compares two bodies of work on motion near the triangular Lagrange points. The first is the Páez–Eftymiopoulos normal-form program, which studies conservative Trojan motion in t…

The paper compares two bodies of work on motion near the triangular Lagrange points. The first is the Páez–Eftymiopoulos normal-form program, which studies conservative Trojan motion in the elliptic restricted three-body problem. It constructs a basic Hamiltonian, locates secondary resonances, maps chaotic transport, and measures escape statistics. The second is Doucette’s algebraic theory of Trojan dust clouds, which studies how a dissipative population of grains evolves under radiation pressure and Poynting–Robertson drag. The author argues that the two are complementary. The first supplies the conservative resonance skeleton: where stable motion is possible, where chaotic layers lie, and how orbits escape. The second supplies the dissipative and statistical layer: how slow drag moves grains through action space, what remains adiabatic between resonance crossings, and what steady cloud morphology results from continuous replenishment. The comparison supports the dust-cloud theory on several substantive points. The shared Poisson-algebra foundation is the working language of the normal-form computations. The hierarchy of timescales is real and usable. The finding of long-lived chaos without transport confirms the thin-thick distinction. Resonance-organized phase space is confirmed, with a simple linear count reproducing the secondary resonance families from one-to-five through one-to-twelve across the planetary mass range. Heavy-tailed escape statistics imply that a replenished cloud is dominated by long-lived grains. The Earth-Moon regime is supported from both conservative and dissipative directions. Persistence by replenishment is made quantitative. The comparison also requires three revisions to the conservative statements of the dust-cloud theory. A mass-ratio lifetime law should be replaced by a distance-based effective stability estimate. An eigenvalue gap should be replaced by a finite-order resonance margin. Exponential confinement claims should be restricted to resonance-free domains. The dissipative theorems are outside the scope of the conservative program and are neither confirmed nor refuted. The paper concludes that the two works fit into one program: the conservative structure of the Trojan region and the dissipative evolution of the dust that inhabits it can be computed together. It proposes joint calculations, including adding drag to the normal form, computing escape-time distributions with drag, and testing capture probabilities at gatekeeper resonances. Limitations include the absence of new numerical integrations, the qualitative nature of the linear resonance count, and the fact that the conservative program’s parameter range does not include the Earth-Moon system.  

Initial version of course from Oct 2025 with some minor updates, start of zenodo badge Oct 5, 2026

In contrast to the impressive dynamic current gain and cut-off frequency demonstrated by graphene field-effect transistors (GFETs), which compete with mature high-frequency (HF) technologies…

In contrast to the impressive dynamic current gain and cut-off frequency demonstrated by graphene field-effect transistors (GFETs), which compete with mature high-frequency (HF) technologies, GFETs still stay behind their HF counterparts in terms of power amplification and maximum oscillation frequency. This limitation is tipycally attributed to the absence of an energy bandgap, which leads to poor current saturation and hence high output conductance (gds), despite the exceptionally high carrier mobility attained in graphene and corresponding high transconductance (gm). Indeed, the intrinsic voltage gain of GFETs (Av, i= gm/gds) is not competitive with that of conventional high-frequency technologies. As a consequence of this back-of-the-envelope reasoning, the study of the most relevant metrics of amplifiers at higher frequencies, i.e. the power gain, has been often disregarded. In this work, a comprehensive analysis of the physics-based small-signal parameters of GFETs reveals that strong current saturation and the resulting low gds are not mandatory for achieving reasonable power gain levels in the GHz range, demonstrating that Av,i is not critical for HF performance evaluation. Specifically, we project a maximum stable gain exceeding 5 dB at 2.4 GHz in fabricated 1-µm-long GFETs with Av,i < 1. This work provides a new perspective on performance improvement in GFET-based power gain amplifiers, showing that high carrier mobility and selecting the bias point with a focus on transconductance, as the most relevant figure of merit for amplifier operation, are crucial, while microwave stability must also be properly addressed.

In the history of economic thinking, Adam Smith has enjoyed almost 250 centuries of reverence as the founder of “Perfect Competition”. According to conventional wisdom,

The root every row shares, to exist is to actuate, is proved in Appendix R unconditionally, at theorem grade, on no axiom and no posit: a theorem with no hypothesis on its constructed domain, and one …

The root every row shares, to exist is to actuate, is proved in Appendix R unconditionally, at theorem grade, on no axiom and no posit: a theorem with no hypothesis on its constructed domain, and one closed law for every root that grounds itself. Each row's closing theorem consumes one reading of that root and no other posit; on the Riemann row the functional equation is a field of the closing theorem's input type (ActualZeros.fold_closed), cited, not posited; on the Birch and Swinnerton-Dyer row the reflection law of the centred expansions enters the sign theorem's frame (SignedFrame, consumed by sign_fixes_curve_parity), cited, and neither input type of that row's closure, ActualCurves or ActualCurvesFull, carries a cited theorem as a field. The root alone decides no value, by theorem; the root read on each row is that row's value, by theorem; and the derivation of each value from the register alone is blocked, by theorem, each in the row's own kernel. The Offering Bit (Islam 2026a) stated a three-bit arc for every blocked problem row: a block, that an even register cannot decide an odd target; a hypothesis, that exactly one orientation bit is missing at a located seat; and a resolution, that the bit enters by deed. One Bit Across the Wall (Islam 2026b) typed twenty-three rows under that arc, and Nothing Escapes Least Erasure (Islam 2026j) graded them on two channels, kinetic and formal. This mini paper is the upgrade that reports the closure papers. By 5 October 2026 the author had published the first editions of seven closure papers, one per Millennium row; the editions of record, dated 6 October 2026, are the ones reported here, each with a Lean 4 kernel whose closing theorem, the one that prints the row's value from its act, rests on no axiom. We record them as the arc's results and grade them on the two channels. The block bit is proved on all seven rows, each paper formalizing its row's traditional barrier. The hypothesis bit is settled on all seven: on the Riemann, Hodge, Poincaré and Birch and Swinnerton-Dyer rows the seat is located in the kernel with its one missing orientation, and its parity where the row has one; on Navier–Stokes and Yang–Mills no seat is formalized, and the one bit, on Yang–Mills one per part, is carried through the record wall; on P versus NP no orientation sits at a seat: the complementation seat is proved vacant, and the hypothesis bit is the keyed separation bit, which no reading of the vanished seat's record returns. The resolution bit is supplied on all seven by the act, existence read on the row, proved equal to the row's value, and on each row its closure paper leaves nothing remaining on its closure. The kinetic channel is crossed by the act at premise grade, on the root proved beneath it, on every row but Poincaré, where that premise is discharged by Perelman's witness at the grade of its citation, a theorem of the literature carried as the witness, in the shape the kernel types (Witness, consumed by witness_gives_act), the kernel constructing no witness, so that the row stands at the grade of that citation (crossed; discharged, at the grade of its citation); the formal channel, the traditional route of derivation from the register's resources, is blocked by theorem on every open row, and Poincaré alone is crossed on it, by Perelman's offering, at the grade of its citation (Perelman 2002), full grade in the carried ledger of Islam (2026j). One cumulative ledger, the twenty-three rows of the earlier volume with the closure papers placed on top, each closure paper governing its row over the carried ledger whatever the edition dates, records every row's block, hypothesis and resolution bits, channels, remainder and physical witness; the target bit and the even register stand in Islam (2026b, Table 2), cited, not reprinted.

Archive creation (source, dichotomised, and review archives) now prefers pigz for gzip compression when it's on PATH -- parallel across CPU cores, versus the single-threaded stdlib gzip/zlib codec use…

Archive creation (source, dichotomised, and review archives) now prefers pigz for gzip compression when it's on PATH -- parallel across CPU cores, versus the single-threaded stdlib gzip/zlib codec used otherwise. The CLI reports "pigz found, using it." up front when it's available. Added CITATION.cff with the Zenodo DOI, so GitHub renders a "Cite this repository" button. See CHANGELOG.md for the full history.

The article focuses on issues related to the functioning of private universities in India. The key question discussed is: Can private universities operate ethically? Presents several recommendations f…

The article focuses on issues related to the functioning of private universities in India. The key question discussed is: Can private universities operate ethically? Presents several recommendations for Vice-Chancellors and Promoters of private universities.

Adam Smith’s An Inquiry into the Nature and Causes of the Wealth of Nations remains one of the foundational texts of political economy, not only because of its treatment of markets, labour, and …

Adam Smith’s An Inquiry into the Nature and Causes of the Wealth of Nations remains one of the foundational texts of political economy, not only because of its treatment of markets, labour, and commercial society, but also because of its systematic discussion of the fiscal functions of government.