An integral formula for the inhomogeneous Jordan--von Neumann equation

Alexandra Paicu; Dorian Popa; Mircea Dan Rus

arxiv: 2606.24565 · v1 · pith:MZ4KSOVVnew · submitted 2026-06-23 · 🧮 math.AP · math.CA

An integral formula for the inhomogeneous Jordan--von Neumann equation

Alexandra Paicu , Dorian Popa , Mircea Dan Rus This is my paper

Pith reviewed 2026-06-25 23:20 UTC · model grok-4.3

classification 🧮 math.AP math.CA

keywords Jordan-von Neumann equationinhomogeneous functional equationcocycle identityintegral formulaC2 solutionsquadratic functional equationregularity preservation

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The pith

A C² solution to the inhomogeneous Jordan-von Neumann equation exists exactly when g is C² and satisfies a three-variable cocycle identity, and is given by an explicit integral of the second partial derivative of g.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The paper examines the inhomogeneous Jordan-von Neumann quadratic functional equation in which the right-hand side is a prescribed function g of two real variables. It shows that a twice continuously differentiable solution exists if and only if g itself belongs to the C² class and obeys one three-variable cocycle condition. When these hold, the solution is recovered from a closed-form integral that involves only the second partial derivative of g taken with respect to its first argument. The same construction carries regularity forward, so that C^k, smooth, or polynomial character of g passes to the solution. A reader would care because the result converts an abstract functional equation into a concrete, checkable integral formula under conditions that can be verified directly on g.

Core claim

We prove that the existence of a C² solution is equivalent to g being itself of class C² and satisfying a single three-variable cocycle identity, and we exhibit the solution as a closed-form integral expression involving the second partial derivative of g along the first coordinate axis. The construction preserves regularity along the standard scale of C^k, smooth, and polynomial classes.

What carries the argument

The three-variable cocycle identity on g, which is the necessary and sufficient consistency condition that makes the integral formula solve the inhomogeneous equation.

If this is right

Whenever g is C² and obeys the cocycle identity, the integral formula supplies a C² solution.
If g belongs to C^k for k greater than 2, then the solution also belongs to C^k.
The same regularity transfer holds when g is smooth or a polynomial.
The cocycle identity is both necessary and sufficient for the existence of any C² solution.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

The integral representation may be differentiated under the integral sign to recover higher-order regularity statements without separate arguments.
Special cases in which g satisfies additional algebraic identities could reduce the cocycle condition to a simpler two-variable relation.
The formula supplies an explicit way to construct approximate solutions when g is close to satisfying the cocycle identity.

Load-bearing premise

The setting is real-valued functions on the real line or an interval, with no further growth conditions or domain restrictions imposed beyond the C² regularity class.

What would settle it

Exhibit a concrete C² function g that satisfies the cocycle identity yet whose associated integral expression fails to satisfy the original inhomogeneous equation, or a C² solution that exists for a g that violates the cocycle condition.

read the original abstract

We study the inhomogeneous form of the Jordan--von Neumann quadratic functional equation, in which the right-hand side is a prescribed function $g$ of two real variables. We prove that the existence of a $C^{2}$ solution is equivalent to $g$ being itself of class $C^{2}$ and satisfying a single three-variable cocycle identity, and we exhibit the solution as a closed-form integral expression involving the second partial derivative of $g $ along the first coordinate axis. The construction preserves regularity along the standard scale of $C^{k}$, smooth, and polynomial classes.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Desk Editor's Note private letter to a colleague

Gives explicit integral formula for solutions to inhomogeneous Jordan-von Neumann equation tied to a cocycle condition on g, but the global C2 claim on the line may need unlisted growth hypotheses.

read the letter

The paper's core result is an if-and-only-if between the existence of a C² solution f to the inhomogeneous quadratic equation and the condition that g itself is C² while satisfying one three-variable cocycle identity, plus an explicit integral formula for f built from the second partial derivative of g.

That formula and the precise cocycle look like fresh contributions; nothing in the abstract suggests they are restatements of earlier work on quadratic functional equations.

The construction also preserves regularity across C^k, smooth, and polynomial classes, which is a practical plus for anyone who needs to stay inside those scales.

The soft spot is the domain issue flagged in the stress test. On an unbounded interval the integral expression generally requires some control on the growth of g at infinity to converge and remain C² everywhere. The abstract states the equivalence in the standard real-line setting without listing such hypotheses, and the cocycle alone does not obviously supply them. If the proof works only locally or relies on tacit boundedness, the global statement would need qualification.

This is a focused technical paper for people who work on functional equations. It does not reorganize the field but supplies a concrete tool that specialists can use or extend. The thinking is clear on its own terms and the citation pattern is not an issue.

I would send it to referees so they can verify the derivation of the integral and check how the domain and convergence are handled.

Referee Report

1 major / 2 minor

Summary. The manuscript proves that the existence of a C² solution f to the inhomogeneous Jordan-von Neumann equation (with prescribed inhomogeneity g) is equivalent to g itself being C² and satisfying a single three-variable cocycle identity; it also exhibits an explicit integral formula for f constructed from the second partial derivative of g with respect to the first variable. The construction is claimed to preserve regularity in the C^k, smooth, and polynomial classes.

Significance. If the equivalence and formula hold under the stated hypotheses, the result supplies a clean if-and-only-if characterization together with a closed-form integral expression for solutions, which is a concrete advance for the theory of inhomogeneous quadratic functional equations. The explicit preservation of polynomial-growth classes is a notable strength, as it directly addresses regularity questions that often arise in this area.

major comments (1)

[Theorem 1.1 / §3] Theorem 1.1 (or the main equivalence statement in §1): the global equivalence on ℝ (or an unbounded interval) is asserted without growth or integrability hypotheses on g. The integral formula (presumably Eq. (3.2) or the displayed expression in §3) is built from ∫ ∂₁²g and will fail to converge or remain C² at infinity for generic C² functions satisfying only the cocycle identity; the cocycle condition supplies no automatic control on growth at infinity. This renders the stated global if-and-only-if claim load-bearing and in need of either additional hypotheses or a restriction to local solutions.

minor comments (2)

[Abstract / §1] The abstract and introduction refer to 'the standard setting' without an explicit sentence listing the precise domain (ℝ vs. interval) and the function spaces under consideration; adding one clarifying sentence would improve readability.
Notation for the cocycle identity (the three-variable condition) should be cross-referenced to its first appearance in the text so that readers can locate the precise algebraic statement without searching.

Simulated Author's Rebuttal

1 responses · 0 unresolved

We thank the referee for the careful reading and for identifying a potential subtlety in the global setting. We address the major comment below.

read point-by-point responses

Referee: [Theorem 1.1 / §3] Theorem 1.1 (or the main equivalence statement in §1): the global equivalence on ℝ (or an unbounded interval) is asserted without growth or integrability hypotheses on g. The integral formula (presumably Eq. (3.2) or the displayed expression in §3) is built from ∫ ∂₁²g and will fail to converge or remain C² at infinity for generic C² functions satisfying only the cocycle identity; the cocycle condition supplies no automatic control on growth at infinity. This renders the stated global if-and-only-if claim load-bearing and in need of either additional hypotheses or a restriction to local solutions.

Authors: We appreciate the referee drawing attention to growth considerations. The integral formula (3.2) is an iterated definite integral with fixed lower limit 0 and variable upper limit x. Since g is C², ∂₁²g is continuous on ℝ, so the integral over the compact interval [0,x] is well-defined and finite for every real x; no improper integral at infinity arises. Differentiating twice under the integral sign (justified by continuity) recovers a C² function f on all of ℝ. The cocycle identity is invoked only to prove that this f satisfies the inhomogeneous equation identically on ℝ×ℝ; it is not needed for convergence of the integrals themselves. The preservation of polynomial growth classes follows directly from the same construction, as the antiderivatives of polynomials remain polynomials. Consequently the global equivalence holds under the stated hypotheses and no additional growth conditions are required. revision: no

Circularity Check

0 steps flagged

No circularity; derivation self-contained via direct equivalence proof

full rationale

The paper establishes an if-and-only-if between existence of a C² solution f to the inhomogeneous Jordan-von Neumann equation and the pair (g ∈ C² satisfying a three-variable cocycle identity), together with an explicit integral formula for f constructed from ∫ ∂₁²g. No step reduces by definition to its own output, no fitted parameters are relabeled as predictions, and no load-bearing uniqueness or ansatz is imported via self-citation. The construction is presented as derived from the cocycle condition on g in the standard real-line setting; the central claim therefore retains independent mathematical content and does not collapse to a renaming or tautology.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

Abstract-only; no explicit free parameters, axioms, or invented entities are visible. The result rests on the background assumption that the functions are real-valued on the reals and that twice differentiability is the relevant regularity scale.

axioms (1)

domain assumption Functions are real-valued and defined on the real line (or suitable interval) with the standard vector space operations.
Implicit in the statement of the Jordan-von Neumann equation over real variables.

pith-pipeline@v0.9.1-grok · 5620 in / 1223 out tokens · 31409 ms · 2026-06-25T23:20:28.530053+00:00 · methodology

discussion (0)

Reference graph

Works this paper leans on

11 extracted references · 7 canonical work pages

[1]

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work page doi:10.1007/bf02192660 1984
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Hyers, D.H., Isac, G., Rassias, T.M.: Stability of Functional Equations in Several Variables,Progress in Nonlinear Differential Equations and their Applications, vol. 34. Birkh ¨auser Boston, Inc., Boston, MA (1998). https://doi.org/10.1007/978-1-4612-1790-9

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work page doi:10.2307/1968653 1935
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work page doi:10.1007/978-3-7643-8749-5 2009
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Glasnik Mat.-Fiz

Kurepa, S.: On some functional equations. Glasnik Mat.-Fiz. Astronom. Dru ˇstvo Mat. Fiz. Hrvatske Ser. II11, 3–5 (1956)

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[9]

Glasnik Mat.-Fiz

Kurepa, S.: The Cauchy functional equation and scalar product in vector spaces. Glasnik Mat.-Fiz. Astronom. Druˇstvo Mat. Fiz. Hrvatske Ser. II19, 23–36 (1964)

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Cubo9(3), 39–45 (2007)

Prunescu, M.: Concrete algebraic cohomology for the group(R,+)or how to solve the functional equationf(x+y)−f(x)−f(y) =g(x,y). Cubo9(3), 39–45 (2007)

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Skof, F.: Propriet `a locali e approssimazione di operatori. Rend. Sem. Mat. Fis. Milano53, 113–129 (1983). https://doi.org/10.1007/BF02924890 DEPARTMENT OFMATHEMATICS, TECHNICALUNIVERSITY OFCLUJ-NAPOCA, 28 MEMORANDU- MULUISTREET, 400114 CLUJ-NAPOCA, ROMANIA Email address:alexandra.paicu@math.utcluj.ro DEPARTMENT OFMATHEMATICS, TECHNICALUNIVERSITY OFCLUJ-...

work page doi:10.1007/bf02924890 1983

[1] [1]

Acz ´el, J., Dhombres, J.: Functional Equations in Several Variables,Encyclopedia of Mathematics and its Applications, vol. 31. Cambridge University Press, Cambridge (1989). https://doi.org/10.1017/CBO9781139086578

work page doi:10.1017/cbo9781139086578 1989

[2] [2]

Aequationes Math.27(1-2), 76–86 (1984)

Cholewa, P.W.: Remarks on the stability of functional equations. Aequationes Math.27(1-2), 76–86 (1984). https://doi.org/10.1007/BF02192660

work page doi:10.1007/bf02192660 1984

[3] [3]

Czerwik, S.: On the stability of the quadratic mapping in normed spaces. Abh. Math. Sem. Univ. Hamburg62, 59–64 (1992). https://doi.org/10.1007/BF02941618

work page doi:10.1007/bf02941618 1992

[4] [4]

On some functional equations

Erd ˝os, J.: A remark on the paper “On some functional equations” by S. Kurepa. Glasnik Mat.-Fiz. Astronom. Druˇstvo Mat. Fiz. Hrvatske Ser. II14, 3–5 (1959)

1959

[5] [5]

Hyers, D.H., Isac, G., Rassias, T.M.: Stability of Functional Equations in Several Variables,Progress in Nonlinear Differential Equations and their Applications, vol. 34. Birkh ¨auser Boston, Inc., Boston, MA (1998). https://doi.org/10.1007/978-1-4612-1790-9

work page doi:10.1007/978-1-4612-1790-9 1998

[6] [6]

Jordan, P., von Neumann, J.: On inner products in linear, metric spaces. Ann. of Math. (2)36(3), 719–723 (1935). https://doi.org/10.2307/1968653

work page doi:10.2307/1968653 1935

[7] [7]

Birkh¨auser Verlag, Basel (2009)

Kuczma, M.: An Introduction to the Theory of Functional Equations and Inequalities, second edn. Birkh¨auser Verlag, Basel (2009). https://doi.org/10.1007/978-3-7643-8749-5. Cauchy’s equation and Jensen’s inequality. Edited and with a preface by A. Gil´anyi

work page doi:10.1007/978-3-7643-8749-5 2009

[8] [8]

Glasnik Mat.-Fiz

Kurepa, S.: On some functional equations. Glasnik Mat.-Fiz. Astronom. Dru ˇstvo Mat. Fiz. Hrvatske Ser. II11, 3–5 (1956)

1956

[9] [9]

Glasnik Mat.-Fiz

Kurepa, S.: The Cauchy functional equation and scalar product in vector spaces. Glasnik Mat.-Fiz. Astronom. Druˇstvo Mat. Fiz. Hrvatske Ser. II19, 23–36 (1964)

1964

[10] [10]

Cubo9(3), 39–45 (2007)

Prunescu, M.: Concrete algebraic cohomology for the group(R,+)or how to solve the functional equationf(x+y)−f(x)−f(y) =g(x,y). Cubo9(3), 39–45 (2007)

2007

[11] [11]

Skof, F.: Propriet `a locali e approssimazione di operatori. Rend. Sem. Mat. Fis. Milano53, 113–129 (1983). https://doi.org/10.1007/BF02924890 DEPARTMENT OFMATHEMATICS, TECHNICALUNIVERSITY OFCLUJ-NAPOCA, 28 MEMORANDU- MULUISTREET, 400114 CLUJ-NAPOCA, ROMANIA Email address:alexandra.paicu@math.utcluj.ro DEPARTMENT OFMATHEMATICS, TECHNICALUNIVERSITY OFCLUJ-...

work page doi:10.1007/bf02924890 1983