Uniqueness in multivariate curve resolution, re-tuned

Rabeea Razaq; Robert Rajko

REVIEW 4 major objections 6 minor 14 references

The paper argues that the classic resolution theorems for multivariate curve resolution are necessary but not sufficient for unique resolution, and that a data-based uniqueness rule is the correct criterion.

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2026-07-31 23:49 UTC pith:5QPG6ET5

load-bearing objection A clearly written re-statement of the authors' prior results on uniqueness in MCR; the core claim is likely right, but the paper's main new illustration is unverifiable as submitted, and the tone undercuts its value. the 4 major comments →

arxiv 2607.23280 v1 pith:5QPG6ET5 submitted 2026-07-25 stat.CO

Uniqueness in multivariate curve resolution, re-tuned

Robert Rajko , Rabeea Razaq This is my paper

classification stat.CO

keywords multivariate curve resolutionuniquenessManne's theoremsdata-based uniquenessgeneral rule for uniquenessrotational ambiguitydualitynonnegative matrix factorization

verification ladder T0 review T1 audit T2 compute T3 formal T4 reserved

The pith

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

This paper re-examines the conditions under which multivariate curve resolution (MCR) yields a unique decomposition of a mixed spectroscopic signal. It argues that Manne's three resolution theorems, which rely on selective and zero-component windows, are necessary but not sufficient: a new counterexample satisfies both Theorem 1 and Theorem 2 yet still admits multiple feasible solutions. The paper presents the data-based uniqueness (DBU) theorem as the correct unification of Manne's conditions, and introduces the general rule for uniqueness (GRU) based on duality between concentration and spectral spaces. A practical consequence is that MCR software relying solely on Manne's conditions may report non-unique solutions as resolved.

Core claim

The central discovery is that Manne's theorems are necessary but not sufficient: a profile can satisfy Theorems 1 and 2 and still be non-unique because the spectral mode has no zero or selective subwindows. The data-based uniqueness (DBU) theorem is presented as the correct unification, requiring a selective window in one mode and a zero-contribution sub-window in the other. The paper also affirms the general rule for uniqueness (GRU): fixing the subspace of all complementary components in one space yields a unique solution for the analyte in the other.

What carries the argument

The central objects are Manne's resolution theorems, the data-based uniqueness (DBU) theorem, and the general rule for uniqueness (GRU). Manne's theorems use local rank information: Theorem 1 requires a zero-component subwindow for each interferent, Theorem 2 requires a selective subwindow for the analyte, and Theorem 3 claims these conditions are also necessary. DBU reformulates uniqueness in data terms, requiring a selective window in one mode and a zero-contribution sub-window in the other. GRU expresses uniqueness through duality: fixing the subspace of all complementary components in one space fixes the analyte in the dual space. The key mechanism is the counterexample in Figure 1, whic

Load-bearing premise

The load-bearing premise is that the counterexample in Figure 1 is a valid decomposition with no zero or selective subwindow in the spectral mode, which the paper has not yet independently validated.

What would settle it

In the spectral profiles of Figure 1, look for any wavelength where all but one component's contribution is zero; if such a selective subwindow exists, Manne's Theorem 2 would apply in the spectral direction and the counterexample would collapse. Alternatively, once the PredUnix validation is published, running it on the Figure 1 data must return a uniqueness value of exactly zero for each profile if the decomposition is indeed non-unique.

Watch this falsifier — get emailed when new claim-graph text bears on it.

If this is right

Any MCR analysis that uses only Manne's conditions as a uniqueness check can report a non-unique solution as resolved; DBU should be used instead.
The data-based uniqueness theorem unifies Manne's Theorems 1 and 2, so those theorems should be treated as necessary, not sufficient, conditions.
The general rule for uniqueness extends the duality argument to constraints like trilinearity, equality, and correspondence, predicting uniqueness in systems that satisfy the subspace-fixing condition.
The complementarity and coupling theorems are equivalent to the minimal constrained duality when nonnegativity is applied, meaning they are not a separate approach.
The particular solution based strategy enables detection of all uniquely resolved profiles from a single MCR-ALS run under non-negativity, without extra computation.

Where Pith is reading between the lines

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

If the counterexample is validated, it would be the clearest published demonstration against Manne's sufficiency; if validation reveals an overlooked subwindow, the main argument loses its demonstrative force.
A direct testable extension: apply the DBU check to MCR results from published studies that relied on Manne's conditions and see whether non-unique solutions were missed.
The paper's dismissal of MVSA assumes its 'at least p-1 zero-abundance pixels' condition is rarely met; this could be checked systematically on real tablet data.
The equivalence between polar cones and subwindow conditions suggests DBU could be checked via linear feasibility rather than searching windows.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit.

Desk Editor's Note

A clearly written re-statement of the authors' prior results on uniqueness in MCR; the core claim is likely right, but the paper's main new illustration is unverifiable as submitted, and the tone undercuts its value.

read the letter

Colleague—the short version is: this is a review-ish re-tuning of the Rajkó school's work on uniqueness in MCR, and the central point (Manne's Theorems 1 and 2 are necessary but not sufficient for profile uniqueness) is almost certainly correct—it was already made in Rajkó et al. (2015) and formalized in Gillis & Rajkó (2023). What this paper adds is a pedagogical repackaging, some new figures, and a direct answer to a confusion that apparently surfaced at CAC2026. If you work in chemometrics, it is a useful summary of the state of the art from one authoritative group.

What it does well: the paper is honest about what is new and what is not—it calls itself a re-tuned summary. It gives a workable explanation of DBU and GRU and connects them to prior work. The illustrative figures (Figs 2 and 3) of cone duality are helpful, and the authors make an effort to translate between chemometrics and convex geometry. The section on complementarity/coupling theorems is a fair warning that those theorems are not distinct from duality when nonnegativity is enforced.

The biggest soft spot is exactly the one your stress-test flagged: Figure 1 is presented as a new counterexample, but the validation of PredUnix is deferred to a future paper. The caption and text do not give enough detail for a reader to reproduce or check the claim. That is a missing support for a load-bearing illustration. Because the paper already points to Rajkó et al. (2015, Fig. 8) as an existing counterexample, the overall claim does not collapse if Figure 1 fails, but the paper's own new contribution is then just an unverifiable picture.

Second, the polemical passages, especially the extended critique of Abdollahi & Tauler (2011) and the MVSA remarks, are not fully derived. Some claims there may be correct, but as written they read as scores being settled, and Fig. 4 (a screenshot of a reviewer's report) has no scientific place in a methods paper. That will distract referees.

Also, the text says "The details on the mentioned PredUnix and its validation algorithms will be published elsewhere"—so the paper should either include that material or explicitly say it is a teaser. As is, the central claim leans on a citation to the authors' own 2015 work, which is fine, but the paper doesn't add much beyond the published record.

Who is this for? People in MCR who need a compact, opinionated guide to uniqueness conditions, and who want to know why local-rank heuristics are not enough. It is not for a general statistical audience. It deserves a serious referee—but that referee should send it back with a request to either (a) make Figure 1 self-contained by adding the PredUnix validation or deleting it and relying on the 2015 counterexample, and (b) remove or considerably tone down the extraneous polemics. If those changes are made, it could be a useful tutorial to publish. If not, I'd leave it on arXiv.

Referee Report

4 major / 6 minor

Summary. The paper argues that Manne's 1995 resolution theorems for hyphenated chromatography are necessary but not sufficient for uniqueness in multivariate curve resolution (MCR). It presents a 're-tuned' summary of prior work by the authors and collaborators, including data-based uniqueness (DBU), the general rule for uniqueness (GRU), minimal constrained duality, and a particular-solution based approach. The central assertion is that a profile satisfying Manne's Theorems 1 and 2 may still not be uniquely resolved, and that DBU provides the correct unification. The paper introduces a new counterexample (Figure 1) and critiques earlier work by Abdollahi & Tauler and Sawall et al., while also including an unusual figure showing a reviewer's comment.

Significance. If the central claim holds—that Manne's conditions are insufficient for uniqueness—then any MCR analysis that relies solely on those conditions may report non-unique solutions as unique. This is an important correction in chemometrics. The paper's strength is that the claim is already supported by a published peer-reviewed counterexample (Rajkó et al. 2015, Fig. 8), and the manuscript synthesizes a coherent body of work including a rigorous mathematical treatment in Gillis & Rajkó (2023). However, the new illustrative counterexample is not self-contained, and the paper's polemical sections undermine its scientific weight.

major comments (4)

[Section 2, Figure 1] The new counterexample is not verifiable in this manuscript. The text explicitly states, 'The details on the mentioned PredUnix and its validation algorithms will be published elsewhere.' Without those details, readers cannot confirm that the displayed elution profiles satisfy Manne's Theorems 1 and 2, that no zero-component or selective subwindows exist for spectra, or that the solution is truly non-unique. Since Figure 1 is presented as a novel illustration of the paper's central claim, this is a load-bearing gap. The manuscript should either include the full validation as an appendix or remove the figure and rely on the published counterexample (Rajkó et al. 2015, Fig. 8).
[Section 4] The claimed equivalence between complementarity-coupling theorems and minimal constrained duality is asserted without a proof or a precise statement. The text says, 'If nonnegativity constraint for both ways of a data matrix is applied, the complementarity-coupling approach will be totally equivalent to the faces of the polyhedral cone and its dual in convex geometry world,' but no derivation is given. Similarly, the assertion that MVSA 'will fail' on the data of Abdollahi and Tauler (2011) is not substantiated with data or analysis. These claims are central to the paper's argument that the convex-geometric duality framework subsumes other approaches, so they must be either proven or accompanied by a clear reference to where the proof appears.
[Section 4, Figure 4] The inclusion of a reviewer's comment as Figure 4 is not a scientific contribution and is not appropriate for a journal article. It appears to be a private communication and does not support any technical claim. This should be removed, as it distracts from the paper's scientific content and could be seen as an attempt to discredit a reviewer rather than to advance the argument.
[Section 4, paragraphs on Abdollahi & Tauler] The distinction between 'ambiguity of the model' and 'ambiguity of the method' is argued via Eqs. (3)-(5), but the argument is incomplete. The equations show that for any invertible transformation matrix T, a new feasible decomposition exists, which is a property of the bilinear model. However, the text does not close the logical gap: if an MCR algorithm returns one of these many solutions, the non-uniqueness of the output is still an algorithmic issue. The paper should explicitly connect this distinction to the definitions of uniqueness in Section 2 and explain why the method is not the source of ambiguity. As written, the claim that 'the rotational ambiguities belong to the bilinear model and not to the methods' is not fully established.

minor comments (6)

[Figure 1 caption] The sentence 'Note, though the green component has zero values, but there is no zero-component subwindow regarding to Th1' is grammatically garbled and confusing. Please rewrite for clarity.
[Equation (1)] The typesetting of Eq. (1) is corrupted with long arrow lines, making it unreadable. Please fix the formatting so that the duality between points and hyperplanes is clearly presented.
[Section 4] The remark 'Rotation is for if the object can rotate, rotational is for if somebody makes the object rotate' is a linguistic digression that does not affect the technical content. If kept, it should be tied to the scientific discussion; otherwise, consider removing it.
[Section 5] There is a typo: 'mathematical proves' should be 'mathematical proofs'.
[Section 6] The 'principle of explosion' analogy in the conclusion is undeveloped and does not follow from the preceding sections. Either connect it explicitly to the paper's argument or delete it.
[General] Several long block quotes (e.g., from Karimvand et al. and Akbari Lakeh et al.) interrupt the narrative. Consider paraphrasing them and retaining only the key sentences to improve readability.

Circularity Check

2 steps flagged

The central claim that Manne's theorems are insufficient is loaded from Rajkó et al. (2015) and from Figure 1, whose PredUnix validation is explicitly deferred; the argument reduces to self-citation and an unverifiable illustration.

specific steps

self citation load bearing [Section 2 (Data-based uniqueness), paragraph quoting Rajkó et al. (2015)]
"Rajkó et al. (2015) stated the following “Differences between the data-based uniqueness and profile based uniqueness were described, and we have shown that Manne's theorems are not sufficient in general.” … Figure 8 of their paper demonstrated a counterexample against Manne's theorems … The details on the mentioned PredUnix and its validation algorithms will be published elsewhere."

The paper's headline assertion—'Manne's theorems are not sufficient in general'—is the central result, but the only support provided is a quote from the first author's 2015 paper plus a new figure whose validation algorithm is explicitly deferred to a future publication. The reader is asked to accept the insufficiency claim on the authority of the authors' own earlier counterexample, which is not re-derived or independently checkable here. The argument thus reduces to a self-citation chain rather than a self-contained derivation.
other [Figure 1 caption and following text in Section 2]
"Showing how to generate a counterexample against Manne's theorems: the elution profiles of blue and red components fulfill Theorems 1 and 2 (Th1 and Th2, resp.), i.e., there are zero-component (Th1) and selective (Th2) subwindows, but there is no any zero-component or selective subwindow for spectra. … Much slower two validation methods (for a unique profile, the corresponding value should be zero in the inserted tables) show that the PredUnix algorithm worked properly and fast. … The details on the mentioned PredUnix and its validation algorithms will be published elsewhere."

The counterexample is load-bearing for the paper's central claim, yet its verification is not contained in the manuscript: the algorithm (PredUnix) and its validation are said to be published elsewhere. The caption also asserts an apparent contradiction ('green component has zero values, but there is no zero-component subwindow') that is not resolved in the text. Without the deferred validation, the figure cannot be checked, so the key demonstration of non-uniqueness rests on an unverifiable assertion rather than a proof.

full rationale

The manuscript is explicitly a 're-tuned summary' rather than a fresh derivation, and most technical content is quoted from the authors' prior publications. The central assertion—that satisfying Manne's Theorems 1 and 2 does not guarantee uniqueness—is taken verbatim from Rajkó et al. (2015) and supported by Figure 8 of that earlier paper. The only new evidence, Figure 1, is presented as a counterexample, but the caption's claims that the profiles satisfy the theorems and that PredUnix validation gives zero are not checkable because 'the details ... will be published elsewhere.' Thus the chain that would establish the headline result is not self-contained: the conclusion is imported from a self-citation and from a deferred validation. This is not a case of a fitted parameter renamed as a prediction, nor a definitional equivalence, but it is load-bearing self-citation and an omitted proof. The other sections (GRU, duality, particular-solution uniqueness) are likewise summaries of the same authors' works, but they do not add a separate circular step beyond the central insufficiency claim. Score 6 reflects partial circularity: some independent content exists in the cited peer-reviewed counterexample, but the present paper does not supply it.

Axiom & Free-Parameter Ledger

0 free parameters · 5 axioms · 0 invented entities

The paper introduces no new free parameters or entities. It relies on standard convex geometry results and on a set of prior characterizations (DBU, GRU) from the authors' own publications, plus an unverified illustrative counterexample (Figure 1, validation deferred).

axioms (5)

domain assumption Bilinear model R = U D V^T with non-negativity constraints for both modes
Standard model for MCR; invoked throughout, e.g., Eq. (3)-(5).
standard math Polar cone duality: the dual of a facet is an extremal ray, generalized to face–face correspondence
Used in Section 4; cited to Stoer & Witzgall (2012) and Tam (1976).
domain assumption Data-based uniqueness (DBU) and general rule for uniqueness (GRU) as correct characterizations
These are the paper's central criteria, taken from the authors' prior papers Rajkó et al. (2015), Akbari Lakeh et al. (2022), Karimvand et al. (2020), and assumed as valid.
domain assumption Manne's Theorem 3 states the conditions are necessary; the paper accepts its necessity claim.
The paper disputes sufficiency but accepts necessity, using it in the definition of DBU.
ad hoc to paper Figure 1 correctly depicts a non-unique system satisfying Manne's conditions
The counterexample is load-bearing for the claim that Manne's theorems are insufficient, but validation is deferred to a future publication ('details ... published elsewhere').

pith-pipeline@v1.3.0-alltime-deepseek · 7834 in / 9751 out tokens · 79467 ms · 2026-07-31T23:49:55.425999+00:00 · methodology

0 comments

read the original abstract

There are many misunderstandings about the term and interpretation of uniqueness in multivariate curve resolution tasks. In CAC2026 Tarragona, it turned out that even mathematicians do not properly construe Manne's theorems. So we decided to provide this re-tuned summary, using several original quotes, their explanations, and newly created illustrative figures to get the points across.

Figures

Figures reproduced from arXiv: 2607.23280 by Rabeea Razaq, Robert Rajko.

**Figure 1.** Figure 1: Showing how to generate a counterexample against Manne’s theorems: the elution profiles of blue and red components fulfill Theorems 1 and 2 (Th1 and Th2, resp.), i.e., there are zero-component (Th1) and selective (Th2) subwindows, but there is no any zero-component or selective subwindow for spectra. Note, though the green component has zero values, but there is no zero-component subwindow regarding to Th1… view at source ↗

**Figure 2.** Figure 2: Visual representation of the minimal constrained duality for a three-component system (the original rotatable Matlab fig file can be downloaded) [PITH_FULL_IMAGE:figures/full_fig_p005_2.png] view at source ↗

**Figure 3.** Figure 3: Visual representation of the minimal constrained duality for a four-component system (the original rotatable Matlab fig file can be downloaded) ”The method based on the duality concept [37,38] was finally preferred in this work for its simplicity and speed, allowing the direct analytical calculation of the limits or boundaries of the band of feasible solutions.” – There is nothing about the exact equations… view at source ↗

**Figure 4.** Figure 4: Part of the unpredictable review comments for the manuscript of the paper of Gillis and Rajkó (2023) a) When none of the resolved profiles contain any zeroregions, there is no unique solution. b) When none of the resolved profiles of a specific mode (e.g., concentration mode) contain any zero-regions, there is no unique solution. c) When none of the profiles of a species contain any zero-regions, there is… view at source ↗

discussion (0)

Reference graph

Works this paper leans on

14 extracted references · 10 canonical work pages

[1]

1995 , issn =

On the resolution problem in hyphenated chromatography , journal =. 1995 , issn =. doi:10.1016/0169-7439(95)80009-X , author =

work page doi:10.1016/0169-7439(95)80009-x 1995
[2]

2015 , issn =

Definition and detection of data-based uniqueness in evaluating bilinear (two-way) chemical measurements , journal =. 2015 , issn =. doi:10.1016/j.aca.2014.12.017 , author =

work page doi:10.1016/j.aca.2014.12.017 2015
[3]

A general rule for uniqueness in self-modeling curve resolution methods , journal =

Karimvand, Somaiyeh Khodadadi and Lakeh, Mahsa Akbari and Tavakkoli, Elnaz and Ghaffari, Mahdiyeh and Omidikia, Nematollah and Abad, Saeed Khalili Ali and Rajk. A general rule for uniqueness in self-modeling curve resolution methods , journal =. doi:10.1002/cem.3268 , abstract =

work page doi:10.1002/cem.3268
[4]

2005 , issn =

Duality in multivariate receptor models , journal =. 2005 , issn =. doi:10.1016/j.chemolab.2005.04.001 , author =

work page doi:10.1016/j.chemolab.2005.04.001 2005
[5]

Natural duality in minimal constrained self modeling curve resolution , journal =

Rajk. Natural duality in minimal constrained self modeling curve resolution , journal =. doi:10.1002/cem.999 , abstract =

work page doi:10.1002/cem.999
[6]

2012 , publisher=

Convexity and Optimization in Finite Dimensions I , author=. 2012 , publisher=

2012
[7]

2011 , issn =

Uniqueness and rotation ambiguities in Multivariate Curve Resolution methods , journal =. 2011 , issn =. doi:10.1016/j.chemolab.2011.05.009 , author =

work page doi:10.1016/j.chemolab.2011.05.009 2011
[8]

Studies on the adaptability of different

Rajk. Studies on the adaptability of different. Journal of Chemometrics , volume =. doi:10.1002/cem.1221 , abstract =

work page doi:10.1002/cem.1221
[9]

Journal of Chemometrics , volume =

Sawall, Mathias and Fischer, Christian and Heller, Detlef and Neymeyr, Klaus , title =. Journal of Chemometrics , volume =. doi:https://doi.org/10.1002/cem.2458 , abstract =

work page doi:10.1002/cem.2458
[10]

Journal of the Australian Mathematical Society , author=

A note on polyhedral cones , volume=. Journal of the Australian Mathematical Society , author=. 1976 , pages=. doi:10.1017/S1446788700016311 , number=

doi:10.1017/s1446788700016311 1976
[11]

2022 , issn =

Predicting the uniqueness of single non-negative profiles estimated by multivariate curve resolution methods , journal =. 2022 , issn =. doi:https://doi.org/10.1016/j.aca.2022.339575 , author =

arXiv 2022
[12]

Partial Identifiability for Nonnegative Matrix Factorization , journal =

Gillis, Nicolas and Rajk\'. Partial Identifiability for Nonnegative Matrix Factorization , journal =. 2023 , doi =

2023
[13]

Finding the basic feasible region , journal =

Extension of self-modeling curve resolution to mixtures of more than three components: Part 1. Finding the basic feasible region , journal =. 1990 , issn =. doi:https://doi.org/10.1016/0169-7439(90)80136-T , author =

work page doi:10.1016/0169-7439(90)80136-t 1990
[14]

Finding the complete solution , journal =

Extension of self-modeling curve resolution to mixtures of more than three components: Part 2. Finding the complete solution , journal =. 1999 , issn =. doi:https://doi.org/10.1016/S0169-7439(99)00029-5 , author =

work page doi:10.1016/s0169-7439(99)00029-5 1999

[1] [1]

1995 , issn =

On the resolution problem in hyphenated chromatography , journal =. 1995 , issn =. doi:10.1016/0169-7439(95)80009-X , author =

work page doi:10.1016/0169-7439(95)80009-x 1995

[2] [2]

2015 , issn =

Definition and detection of data-based uniqueness in evaluating bilinear (two-way) chemical measurements , journal =. 2015 , issn =. doi:10.1016/j.aca.2014.12.017 , author =

work page doi:10.1016/j.aca.2014.12.017 2015

[3] [3]

A general rule for uniqueness in self-modeling curve resolution methods , journal =

Karimvand, Somaiyeh Khodadadi and Lakeh, Mahsa Akbari and Tavakkoli, Elnaz and Ghaffari, Mahdiyeh and Omidikia, Nematollah and Abad, Saeed Khalili Ali and Rajk. A general rule for uniqueness in self-modeling curve resolution methods , journal =. doi:10.1002/cem.3268 , abstract =

work page doi:10.1002/cem.3268

[4] [4]

2005 , issn =

Duality in multivariate receptor models , journal =. 2005 , issn =. doi:10.1016/j.chemolab.2005.04.001 , author =

work page doi:10.1016/j.chemolab.2005.04.001 2005

[5] [5]

Natural duality in minimal constrained self modeling curve resolution , journal =

Rajk. Natural duality in minimal constrained self modeling curve resolution , journal =. doi:10.1002/cem.999 , abstract =

work page doi:10.1002/cem.999

[6] [6]

2012 , publisher=

Convexity and Optimization in Finite Dimensions I , author=. 2012 , publisher=

2012

[7] [7]

2011 , issn =

Uniqueness and rotation ambiguities in Multivariate Curve Resolution methods , journal =. 2011 , issn =. doi:10.1016/j.chemolab.2011.05.009 , author =

work page doi:10.1016/j.chemolab.2011.05.009 2011

[8] [8]

Studies on the adaptability of different

Rajk. Studies on the adaptability of different. Journal of Chemometrics , volume =. doi:10.1002/cem.1221 , abstract =

work page doi:10.1002/cem.1221

[9] [9]

Journal of Chemometrics , volume =

Sawall, Mathias and Fischer, Christian and Heller, Detlef and Neymeyr, Klaus , title =. Journal of Chemometrics , volume =. doi:https://doi.org/10.1002/cem.2458 , abstract =

work page doi:10.1002/cem.2458

[10] [10]

Journal of the Australian Mathematical Society , author=

A note on polyhedral cones , volume=. Journal of the Australian Mathematical Society , author=. 1976 , pages=. doi:10.1017/S1446788700016311 , number=

doi:10.1017/s1446788700016311 1976

[11] [11]

2022 , issn =

Predicting the uniqueness of single non-negative profiles estimated by multivariate curve resolution methods , journal =. 2022 , issn =. doi:https://doi.org/10.1016/j.aca.2022.339575 , author =

arXiv 2022

[12] [12]

Partial Identifiability for Nonnegative Matrix Factorization , journal =

Gillis, Nicolas and Rajk\'. Partial Identifiability for Nonnegative Matrix Factorization , journal =. 2023 , doi =

2023

[13] [13]

Finding the basic feasible region , journal =

Extension of self-modeling curve resolution to mixtures of more than three components: Part 1. Finding the basic feasible region , journal =. 1990 , issn =. doi:https://doi.org/10.1016/0169-7439(90)80136-T , author =

work page doi:10.1016/0169-7439(90)80136-t 1990

[14] [14]

Finding the complete solution , journal =

Extension of self-modeling curve resolution to mixtures of more than three components: Part 2. Finding the complete solution , journal =. 1999 , issn =. doi:https://doi.org/10.1016/S0169-7439(99)00029-5 , author =

work page doi:10.1016/s0169-7439(99)00029-5 1999