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arxiv: 1907.09941 · v1 · pith:EBPY7N3Inew · submitted 2019-07-20 · ⚛️ physics.class-ph

The number of independent invariants for m unit vectors and n symmetric second order tensors is 2m+ 6n-3

M.H.B.M. Shariff This is my paper

Pith reviewed 2026-05-24 18:26 UTC · model grok-4.3

classification ⚛️ physics.class-ph
keywords invariantsunit vectorssymmetric tensorsintegrity basisorthogonal groupcontinuum mechanicsanisotropic materials
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0 comments X

The pith

The number of independent invariants for m unit vectors and n symmetric tensors is at most 2m+6n-3

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

The paper proves that the number of independent invariants formed from m unit vectors and n symmetric second-order tensors cannot exceed 2m + 6n - 3. This count matters in continuum mechanics because constitutive equations for anisotropic materials are built from these invariants, and the total determines how many separate experiments must hold all but one invariant fixed to map the full dependence. The work also gives relations that connect the classical invariants appearing in the minimal integrity basis. A sympathetic reader would apply the bound to keep constitutive models from containing more independent variables than necessary.

Core claim

The number of independent invariants for a set of n symmetric tensors and m unit vectors is at most 2m+6n-3.

What carries the argument

The minimal integrity basis generated by the action of the orthogonal group on the m vectors and n tensors

If this is right

  • Constitutive equations can be written with no more than this number of independent scalar arguments.
  • Experimental identification of material response requires at most this many controlled invariants.
  • Redundancies among classical invariants can be removed using the supplied relations.

Where Pith is reading between the lines

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

  • The same orbit-dimension counting may apply to other finite or Lie groups acting on tensors of higher order.
  • Existing lists of invariants in the literature can be checked for completeness against this numerical ceiling.

Load-bearing premise

The invariants in question are those unchanged under the orthogonal group acting on three-dimensional Euclidean space containing the given vectors and tensors.

What would settle it

A concrete choice of m and n together with more than 2m+6n-3 functionally independent scalar functions that remain unchanged under all orthogonal transformations would disprove the bound.

read the original abstract

Anisotropic invariants play an important role in continuum mechanics. Knowing the number of independent invariants is crucial in modelling and in a rigorous construction of a constitutive equation for a particular material, where it is determined by doing tests that hold all, except one, of the independent invariants constant so that the dependence in the one invariant can be identified. Hence, the aim of this paper is to prove that the number of independent invariants for a set of $n$ symmetric tensors and $m$ unit vectors is at most 2m+ 6n-3. We also give relations between classical invariants in the corresponding minimal integrity basis.

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.

Referee Report

2 major / 1 minor

Summary. The manuscript claims to prove that the number of independent invariants for a set of m unit vectors and n symmetric second-order tensors under the orthogonal group O(3) is at most 2m + 6n - 3. It also provides relations between classical invariants in the corresponding minimal integrity basis. The work is motivated by the role of anisotropic invariants in continuum mechanics for constitutive equation construction via targeted experiments.

Significance. If the result holds with appropriate qualification for all non-negative integers m and n, it would be useful in continuum mechanics by bounding the number of independent parameters that must be identified experimentally when building invariant-based constitutive models. The explicit relations among invariants in the integrity basis would further support practical implementations.

major comments (2)
  1. [Abstract] Abstract and title: The claimed bound 2m+6n-3 is negative for small m,n (e.g., m=0 n=0 yields -3; m=1 n=0 yields -1), yet the number of independent invariants is always a non-negative integer and equals 0 whenever the configuration-space dimension 2m+6n is less than dim(O(3))=3. The statement that this number 'is at most 2m+6n-3' (abstract) or 'is 2m+6n-3' (title) is therefore false in these regimes, since a non-negative integer cannot be ≤ a negative number. Any derivation that subtracts 3 from the configuration dimension without a max(0,·) or case distinction cannot support the stated general claim.
  2. [Abstract] The central result is presented as a general bound without domain restriction or the necessary max(0,2m+6n-3) correction required by the orbit-dimension formula for the quotient space. This must be corrected for the claim to be accurate.
minor comments (1)
  1. [Abstract] The abstract states that relations between classical invariants are given but does not indicate the section or theorem in which they appear.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the detailed and constructive report. The comments correctly identify that our stated bound requires qualification for small m and n. We address each point below and will revise the manuscript to correct the presentation.

read point-by-point responses

  1. Referee: [Abstract] Abstract and title: The claimed bound 2m+6n-3 is negative for small m,n (e.g., m=0 n=0 yields -3; m=1 n=0 yields -1), yet the number of independent invariants is always a non-negative integer and equals 0 whenever the configuration-space dimension 2m+6n is less than dim(O(3))=3. The statement that this number 'is at most 2m+6n-3' (abstract) or 'is 2m+6n-3' (title) is therefore false in these regimes, since a non-negative integer cannot be ≤ a negative number. Any derivation that subtracts 3 from the configuration dimension without a max(0,·) or case distinction cannot support the stated general claim.

    Authors: We agree with this observation. The expression 2m + 6n - 3 arises as the difference between the dimension of the configuration space (2m + 6n) and the dimension of O(3) (equal to 3). This difference gives the dimension of the orbit space only when 2m + 6n ≥ 3 and the group action is free on a dense open set; otherwise the number of independent invariants is zero. The manuscript presents the bound without explicit domain restriction, which is inaccurate for the indicated small values. We will revise the title to read 'at most max(0, 2m + 6n - 3)' and update the abstract and introduction to include the necessary case distinction or max(0, ·) formulation, together with a brief explanation of the underlying orbit-dimension reasoning. revision: yes

  2. Referee: [Abstract] The central result is presented as a general bound without domain restriction or the necessary max(0,2m+6n-3) correction required by the orbit-dimension formula for the quotient space. This must be corrected for the claim to be accurate.

    Authors: We accept the referee's point. The central claim is an upper bound derived from the generic dimension of the quotient, but the derivation implicitly assumes sufficient configuration-space dimension. We will add an explicit statement in the abstract, title, and main text that the number of independent invariants is at most max(0, 2m + 6n - 3) for non-negative integers m, n, and we will include a short paragraph explaining when the bound is attained and when it is replaced by zero. These changes will be incorporated in the revised version. revision: yes

Circularity Check

0 steps flagged

No significant circularity; derivation presented as independent proof

full rationale

The paper frames its central claim as a mathematical proof that the number of independent invariants is at most 2m+6n-3, derived from the orthogonal group action on the configuration space of m unit vectors and n symmetric tensors. No load-bearing steps reduce by definition, by renaming a fit as a prediction, or via self-citation chains that substitute for external verification. The abstract and context indicate a self-contained argument in invariant theory, with the bound offered as an upper estimate from dimension considerations. Any issues with the bound becoming negative for small m,n pertain to correctness or completeness of the statement rather than circularity in the derivation chain.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

Only the abstract is available, so the ledger records the minimal background assumptions implied by the claim.

axioms (1)
  • domain assumption Invariants are defined with respect to the standard orthogonal group action on vectors and symmetric tensors in three-dimensional space.
    Implicit in any continuum-mechanics treatment of anisotropic invariants.

pith-pipeline@v0.9.0 · 5637 in / 1201 out tokens · 22748 ms · 2026-05-24T18:26:26.751948+00:00 · methodology

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Reference graph

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