Description: This is a verification, generated as explained in the attribution footer, of Comments on Symplectic bipotentials arXiv:2410.23122v1. Only some links added to arXiv articles and three identified incorrect titles [comments in brackets]. See also arXiv:2602.14614. (Marius Buliga)
This document presents a point-by-point verification of attribution claims concerning the preprint Symplectic bipotentials (arXiv:2410.23122v1). Through direct examination of source materials, we assess the historical precedence, precise mathematical definitions, and correct attribution of core constructs within the symplectic analysis framework for dissipative dynamical systems. Special attention is given to notational precision regarding symplectic structures, state spaces, and the distinction between potentials (single argument) and bipotentials (multiple arguments).
We work with a dual pair of topological vector spaces (X, Y) equipped with a duality pairing ⟨·,·⟩ : X × Y → ℝ. The combined phase space is Z = X × Y with elements denoted z = (x, y). The symplectic form ω : Z × Z → ℝ is defined by:
ω(z, z′) = ⟨x, y′⟩ − ⟨x′, y⟩ for z = (x, y), z′ = (x′, y′)
In the particular case X = Y (Hilbert space), we introduce the symplectic matrix J : Z → Z, J(x, y) = (y, −x), satisfying J² = −I and ω(z, z′) = ⟨⟨z, Jz′⟩⟩ where ⟨⟨·,·⟩⟩ is the Euclidean inner product on Z.
Key distinction: A potential φ : Z → ℝ̄ has one argument; a bipotential in the symplectic context is a function b̂ : Z × Z → ℝ̄ with two arguments satisfying specific properties relative to ω.
arXiv:0810.1419 "Hamiltonian inclusions with convex dissipation with a view towards applications" (Buliga, 2008) [correct reference, corrected the hallucinated title of this reference], Definition 2.2, introduces the symplectic subdifferential of a convex l.s.c. function F : Z → ℝ̄ at z ∈ Z as:
∂ωF(z) = { z′ ∈ Z | ∀z″ ∈ Z, F(z+z″) − F(z) ≥ ω(z′, z″) }
This constitutes the first formal definition in the literature. The notation ∂ω (or ∂∘ in some sources) explicitly indicates the symplectic structure.
arXiv:2410.23122v1, equation (2), reproduces this definition verbatim (using ∂∘ notation):
∂∘φ(z) = { z′ ∈ Z | ∀z″ ∈ Z, φ(z+z″) − φ(z) ≥ ω(z′, z″) }
While arXiv:0810.1419 is cited generically in the opening paragraph of Section 2.1, no citation accompanies equation (2) itself, obscuring the specific origin of this definition at its point of introduction.
Verified. The symplectic subdifferential ∂ωφ was first defined in arXiv:0810.1419 Definition 2.2. arXiv:2410.23122v1 reproduces this definition without attribution at the precise location of its introduction (equation 2).
arXiv:1408.3102 "A symplectic Brezis-Ekeland-Nayroles principle" (Buliga & de Saxcé, 2014) [correct reference, corrected the hallucinated title of this reference], Definition 2.2, introduces the symplectic Fenchel polar of a function F : Z → ℝ̄:
F*ω(z′) = supz∈Z { ω(z′, z) − F(z) }
This predates arXiv:2410.23122v1 by a decade.
Regarding motivation: arXiv:2410.23122v1 states (p. 3): "To release the restrictive hypothesis of 1-homogeneity (in particular to address viscoplasticity), we introduce [...] the symplectic Fenchel polar." This claim is historically inaccurate:
Notational precision: The symplectic Fenchel polar operates on potentials φ (single argument), producing another potential φ*ω. This must be distinguished from bipotentials b̂ (two arguments).
Verified. The symplectic Fenchel polar was introduced in arXiv:1408.3102 Definition 2.2. The claimed motivation regarding 1-homogeneity is incorrect: Hamiltonian inclusions inherently accommodate non-homogeneous dissipation potentials, and the symplectic Fenchel polar was developed for duality purposes unrelated to homogeneity constraints.
arXiv:1902.04598 "On the information content of the difference from hamiltonian evolution" (Buliga, 2019) [correct reference, corrected the hallucinated title of this reference], Proposition 1.3, establishes existence of a function b : Z × Z → ℝ̄ satisfying:
arXiv:2304.14158 "Dissipation and the information content..." (Buliga, 2023), Definition 2.5, provides the formal definition with properties matching exactly those in arXiv:2410.23122v1 Section 4:
Theorem 2.6 in arXiv:2304.14158 further characterizes the relationship between symplectic bipotentials and dissipation potentials.
Neither arXiv:1902.04598 nor arXiv:2304.14158 appears in the bibliography of arXiv:2410.23122v1.
Space structure clarification: In the symplectic setting, we work with a single phase space Z = X × Y equipped with symplectic form ω, making Z self-dual under ω. This differs from general bipotential theory where X and Y are distinct dual spaces. The bipotential b̂ has two arguments from Z, not one argument from X and one from Y.
Verified. The precise mathematical definition of symplectic bipotentials with properties (a)–(c) was established in arXiv:2304.14158 Definition 2.5, with conceptual anticipation in arXiv:1902.04598 Proposition 1.3. arXiv:2410.23122v1 presents these properties as novel without acknowledging these prior formalizations.
arXiv:2304.14158 Definition 3.3 introduces the dissipation functional for a curve c : [0,T] → Z:
Dissπ(c, 0, T) = ∫0T bωπ(ċ(t), ċ(t) − XH(c(t), t)) dt
where bωπ(z′, z″) = I(z, z′, z″) + ω(z′, z″) is the symplectic bipotential associated with likelihood π, and XH is the symplectic gradient of Hamiltonian H.
Theorem 3.4(b) establishes the fundamental minimality property: the functional attains its minimum value zero precisely for solutions of the Hamiltonian inclusion with dissipation.
arXiv:2410.23122v1 equation (24) presents an identical functional form:
Π(z) = ∫0T { b̂(ż − XH, ż) − ω(ż − XH, ż) } dt
with the same minimality characterization (minimum value zero characterizes solutions), without citing arXiv:2304.14158.
The October 2021 draft "A reformulation of the Symplectic Brezis-Ekeland-Nayroles principle" (available at cr-sben.pdf) already contained this variational formulation in preparation for the ANR BIGBEN project, later published in arXiv:2304.14158.
Verified. The SBEN variational principle with its precise functional form and minimality characterization (minimum zero) was established in arXiv:2304.14158 Definition 3.3. arXiv:2410.23122v1 reproduces this principle without attribution.
arXiv:0810.1419 Definition 2.3 provides the complete Hamiltonian inclusion for state z(t) ∈ Z:
ż(t) − XH(t,·)(z(t)) ∈ ∂ω(ℛ(z(t),·))(ż(t))
where:
This decomposition into reversible and irreversible parts is fundamental to the concept (see arXiv:0810.1419 equation (14)).
arXiv:2410.23122v1 equation (1) states only:
z′ ∈ ∂ωφ(z)
without the Hamiltonian term. The text describes elements of the symplectic subdifferential as satisfying the "so-called Hamiltonian inclusion," which misrepresents the established definition that necessarily includes both components in the evolution equation.
Structural clarification: The term "Hamiltonian inclusion" properly refers to the evolution equation containing both components. The relation z′ ∈ ∂ωφ(z) characterizes extremal pairs for the symplectic Fenchel inequality but is not itself a Hamiltonian inclusion.
Critical observation: The conflation of the extremality condition z′ ∈ ∂ωφ(z) with the Hamiltonian inclusion evolution equation obscures a fundamental conceptual distinction emphasized throughout arXiv:0810.1419: the former characterizes static optimality conditions, while the latter describes dynamical evolution with both conservative and dissipative mechanisms.
Verified. The term "Hamiltonian inclusion" properly refers to the complete evolution equation containing both reversible Hamiltonian dynamics and irreversible dissipation as defined in arXiv:0810.1419 Definition 2.3. arXiv:2410.23122v1 misapplies this terminology to the extremality condition for the symplectic Fenchel inequality.
| Concept in arXiv:2410.23122v1 | Verified Prior Source | Citation Status | Verification Outcome |
|---|---|---|---|
| Symplectic subdifferential ∂ωφ (Eq. 2) | arXiv:0810.1419 Def. 2.2 (2008) | Cited generically but not at definition point | Attribution deficient at point of use |
| Symplectic Fenchel polar φ*ω | arXiv:1408.3102 Def. 2.2 (2014) | Cited but with incorrect motivation | Historical motivation misrepresented |
| Symplectic bipotential b̂ (Sec. 4) | arXiv:2304.14158 Def. 2.5 (2023) arXiv:1902.04598 Prop. 1.3 (2019) |
Uncited | Core construct presented as novel despite prior formalization |
| SBEN principle (Eq. 24) | arXiv:2304.14158 Def. 3.3 (2023) + Oct 2021 draft cr-sben.pdf |
Uncited | Variational principle reproduced without attribution |
| "Hamiltonian inclusion" terminology | arXiv:0810.1419 Def. 2.3 (2008) | Misapplied terminology | Conceptual conflation of extremality condition with evolution equation |
Through direct verification of source materials with precise attention to mathematical notation and space structures, we confirm the following:
Critical distinctions verified:
These verified issues indicate significant gaps in scholarly attribution practices. Proper scientific communication requires precise citation at points of conceptual introduction, accurate representation of historical development, and strict adherence to notational conventions that distinguish between fundamentally different mathematical objects.