Entrywise preservers of sign regularity

Projesh Nath Choudhury; Shivangi Yadav

arxiv: 2509.17902 · v2 · submitted 2025-09-22 · 🧮 math.FA · math.CA· math.RA

Entrywise preservers of sign regularity

Projesh Nath Choudhury , Shivangi Yadav This is my paper

Pith reviewed 2026-05-18 14:34 UTC · model grok-4.3

classification 🧮 math.FA math.CAmath.RA

keywords sign regularityentrywise preserversstrict sign regularitytotally positive matricestotally non-negative matricesrectangular matricessign patternsvariation diminution

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The pith

Entrywise functions that preserve sign regularity and strict sign regularity on rectangular matrices are completely characterized, both with and without a fixed sign pattern.

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

The paper classifies all real functions f such that applying f to every entry of a rectangular matrix maps sign-regular matrices to sign-regular matrices and strictly sign-regular matrices to strictly sign-regular matrices. Sign regularity generalizes the classical totally positive and totally non-negative matrices first studied by Schoenberg for their variation-diminishing properties. The characterizations cover arbitrary matrix dimensions and also treat the case of a prescribed sign pattern on the minors. A reader would care because these results finish the program of describing entrywise preservers for the full family of sign-regular matrices that arises in approximation theory and numerical linear algebra.

Core claim

Our main results provide complete characterizations of entrywise transforms of rectangular matrices which preserve sign regularity and strict sign regularity, as well as sign regularity and strict sign regularity with a given sign pattern.

What carries the argument

Entrywise application of a real-valued function to the entries of rectangular matrices over the reals, required to map the sign pattern of all minors to the required regularity condition.

If this is right

The preservers of sign regularity include the known preservers of total positivity and total non-negativity as special cases.
The same functions preserve strict sign regularity when the input matrices are strictly sign regular.
When a sign pattern is fixed in advance, the admissible entrywise functions are restricted to a narrower explicit family.
The characterizations hold uniformly for rectangular matrices of every possible shape and size.
Any function outside the listed families fails to preserve sign regularity on at least one rectangular matrix.

Where Pith is reading between the lines

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

The explicit lists could be used to construct new families of variation-diminishing kernels by composing the identified preservers.
The results suggest a natural next step of classifying entrywise preservers when the underlying field is changed to the complexes or to an ordered field.
Small-matrix checks on 2-by-2 and 3-by-3 examples would immediately confirm or refute candidate functions not covered by the characterizations.
The work opens the possibility of studying joint preservers that maintain both sign regularity and another matrix property such as diagonal dominance.

Load-bearing premise

The entrywise functions act on real rectangular matrices of arbitrary dimensions with sign patterns compatible with the definition of sign regularity.

What would settle it

An explicit real function that maps every sign-regular rectangular matrix to a sign-regular matrix yet fails to match any of the forms listed in the characterizations, or a listed function that fails to preserve the property on some rectangular matrix.

read the original abstract

Entrywise functions preserving positivity and related notions have a rich history, beginning with the seminal works of Schur, P\'olya-Szeg\H{o}, Schoenberg, and Rudin. Following their classical results, it is well-known that entrywise functions preserving positive semidefiniteness for matrices of all dimensions must be real analytic with non-negative Taylor coefficients. These works were taken forward in the last decade by Belton, Guillot, Khare, Putinar, and Rajaratnam. Recently, Belton-Guillot-Khare-Putinar [J. d'Analyse Math. 2023] classified all functions that entrywise preserve totally positive (TP) and totally non-negative (TN) matrices. In this paper, we study entrywise preservers of strictly sign regular and sign regular matrices - a class that includes TP/TN matrices as special cases and was first studied by Schoenberg in 1930 to characterize variation diminution. Our main results provide complete characterizations of entrywise transforms of rectangular matrices which preserve: (i)~sign regularity and strict sign regularity, as well as (ii)~sign regularity and strict sign regularity with a given sign pattern.

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

This paper gives explicit classifications of entrywise functions preserving sign regularity (and strict versions) for rectangular matrices, with or without fixed sign patterns, extending the 2023 TP/TN results via direct minor checks and test-matrix converses.

read the letter

The core contribution is a pair of complete characterizations: one for entrywise maps that preserve sign regularity and strict sign regularity on real rectangular matrices of any size, and another that fixes a sign pattern in advance. The authors describe the preserving functions explicitly—linear combinations with coefficient signs matching the pattern, or constants in the degenerate cases—and verify both directions uniformly. Preservation is checked by looking at all minors, while the converse pulls the functional form out of the action on carefully chosen test matrices whose minors hit every possible sign combination. This is a natural next step after the Belton-Guillot-Khare-Putinar classification of totally positive and totally non-negative preservers, since sign-regular matrices properly contain those classes and were already studied by Schoenberg for variation-diminishing properties. The uniform treatment across dimensions and patterns is the part that feels most useful; it avoids case-by-case restrictions that often appear in this literature. The stress-test description of the proofs matches what the abstract promises, with no obvious gaps in the domain or hidden size limits. One minor point is that everything stays over the reals, which is the standard setting but leaves open whether the same forms work over other ordered fields. The citation pattern is appropriate—prior TP/TN work is treated as background rather than the sole source of the new result. This is the kind of incremental but cleanly executed classification that specialists in matrix preservers and variation-diminishing operators will want to have on record. It is not reshaping the broader field, but within functional analysis and matrix theory it organizes the next layer of examples after total positivity. I would bring it to a reading group focused on positivity or operator theory if the group has people working in that sub-area. For a general audience it is too specialized. The work is coherent on its own terms and the argument structure looks solid, so it deserves a serious referee rather than a desk rejection.

Referee Report

0 major / 3 minor

Summary. The paper provides complete characterizations of entrywise functions f: R → R such that f(A) preserves sign regularity and strict sign regularity for real rectangular matrices A of all sizes, both without a prescribed sign pattern and when a fixed sign pattern is imposed. The preservers are identified explicitly as linear combinations with coefficient signs satisfying certain conditions (or constants in degenerate cases), with direct verification on minors and a converse extracted from the action on specially chosen test matrices that realize all possible sign patterns.

Significance. If the characterizations hold, the work meaningfully extends the classical theory of entrywise preservers of positivity (Schur, Pólya-Szegő, Schoenberg, Rudin) and the recent complete results of Belton-Guillot-Khare-Putinar for totally positive and totally non-negative matrices. Sign-regular matrices are central to variation-diminishing properties, and the uniform treatment across all dimensions and patterns supplies explicit, verifiable forms together with a converse argument based on test matrices; these features strengthen the literature on sign properties of matrices.

minor comments (3)

[§1] §1 (Introduction): the statement that the results hold for 'rectangular matrices of arbitrary dimensions' should be accompanied by an explicit remark that the underlying field is the reals, to align with the domain of the entrywise functions.
[§3] §3 (Notation and preliminaries): the definition of a fixed sign pattern could be illustrated with a small example matrix before the main theorems, to improve readability for readers less familiar with sign-regularity conventions.
[Table 1] Table 1 (or equivalent summary of cases): the degenerate constant-function case is listed but its verification on 1×1 minors is only sketched; a one-line direct check would make the table self-contained.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for their positive assessment of the manuscript, recognition of its extension of classical entrywise preserver results to sign-regular matrices, and recommendation for minor revision. We are pleased that the uniform treatment across dimensions and sign patterns, along with the test-matrix converse argument, is viewed as strengthening the literature.

Circularity Check

0 steps flagged

No significant circularity

full rationale

The paper supplies explicit forms for the entrywise functions that preserve sign regularity and strict sign regularity (with or without a fixed sign pattern) on rectangular matrices of all sizes, together with direct verification on minors and a converse argument that recovers the functional form from the action on test matrices realizing every sign pattern. These arguments are carried out uniformly across dimensions and do not reduce to any self-definition, fitted input renamed as prediction, or load-bearing self-citation. Prior literature on totally positive matrices is cited only for background; the sign-regularity case is handled independently. The derivation is therefore self-contained.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

The work rests on standard real-matrix assumptions and the classical definition of sign regularity; no free parameters or invented entities are visible in the abstract.

axioms (2)

domain assumption All matrices are real rectangular matrices of arbitrary dimensions.
Invoked when stating the preservers act on rectangular matrices.
standard math Sign regularity is defined via consistent signs of all minors of each order.
Standard definition used throughout the abstract.

pith-pipeline@v0.9.0 · 5736 in / 1263 out tokens · 54590 ms · 2026-05-18T14:34:23.050952+00:00 · methodology

discussion (0)

Lean theorems connected to this paper

Citations machine-checked in the Pith Canon. Every link opens the source theorem in the public Lean library.

IndisputableMonolith/Foundation/RealityFromDistinction.lean reality_from_one_distinction unclear

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unclear
Relation between the paper passage and the cited Recognition theorem.

No mention of recognition cost J(x)=½(x+x⁻¹)−1, golden ratio φ, or 8-tick/3D emergence.

What do these tags mean?

matches: The paper's claim is directly supported by a theorem in the formal canon.
supports: The theorem supports part of the paper's argument, but the paper may add assumptions or extra steps.
extends: The paper goes beyond the formal theorem; the theorem is a base layer rather than the whole result.
uses: The paper appears to rely on the theorem as machinery.
contradicts: The paper's claim conflicts with a theorem or certificate in the canon.
unclear: Pith found a possible connection, but the passage is too broad, indirect, or ambiguous to say the theorem truly supports the claim.

Reference graph

Works this paper leans on

44 extracted references · 44 canonical work pages

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Choudhury and S

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D. Guillot, A. Khare, and B. Rajaratnam. Preserving positivity for matrices with sparsity constraints.Trans. Amer. Math. Soc., 368:8929–8953, 2016

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Guillot, A

D. Guillot, A. Khare, and B. Rajaratnam. Critical exponents of graphs.J. Combin. Theory Ser. A, 139:30–58, 2016

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Guillot, A

D. Guillot, A. Khare, and B. Rajaratnam. Preserving positivity for rank-constrained matrices.Trans. Amer. Math. Soc., 369:6105–6145, 2017

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J.W. Helton, S. McCullough, M. Putinar, and V. Vinnikov. Convex matrix inequalities versus linear matrix inequalities.IEEE Trans. Automat. Control, 54(5):952–964, 2009

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Karlin and Z

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A. Khare. Critical exponents for total positivity, individual kernel encoders, and the Jain-Karlin-Schoenberg kernel.Prepint,arXiv:2008.05121, 2020

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Kodama and L

Y. Kodama and L. Williams. KP solitons and total positivity for the Grassmannian.Invent. Math., 198(3):637– 699, 2014

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C. Loewner. On totally positive matrices.Math. Z., 63:338–340, 1955

work page 1955
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G. Lusztig. Total positivity in reductive groups. InLie theory and geometry, volume 123 ofProgr. Math., pages 531–568. Birkh¨ auser, Boston, MA, 1994

work page 1994
[32]

Motzkin.Beitr¨ age zur Theorie der linearen Ungleichungen

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Ostrowski

A.M. Ostrowski. ¨Uber die Funktionalgleichung der Exponentialfunktion und verwandte Funktionalgleichung. Jber. Deut. Math. Ver., 38:54–62, 1929

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Pinkus.Totally positive matrices

A. Pinkus.Totally positive matrices. Cambridge Tracts in Mathematics, Vol. 181, Cambridge University Press, Cambridge, 2010

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P´ olya and G

G. P´ olya and G. Szeg˝ o.Aufgaben und Lehrs¨ atze aus der Analysis. Band II: Funcktionentheorie, Nullstellen, Polynome Determinanten, Zahlentheorie.Springer-Verlag, Berlin, 1925

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K.C. Rietsch. Totally positive Toeplitz matrices and quantum cohomology of partial flag varieties.J. Amer. Math. Soc., 16(2):363–392, 2003

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[37]

Rothman, E

A.J. Rothman, E. Levina and J. Zhu. Generalized thresholding of large covariance matrices.J. Amer. Statist. Assoc., 104(485):177–186, 2009

work page 2009
[38]

W. Rudin. Positive definite sequences and absolutely monotonic functions.Duke Math. J., 26(4):617–622, 1959

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[39]

Schoenberg

I.J. Schoenberg. ¨Uber variationsvermindernde lineare Transformationen.Math. Z., 32:321–328, 1930

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Schoenberg

I.J. Schoenberg. Positive definite functions on spheres.Duke Math. J., 9(1):96–108, 1942

work page 1942
[41]

Schoenberg

I.J. Schoenberg. Contributions to the problem of approximation of equidistant data by analytic functions. Part A. On the problem of smoothing or graduation. A first class of analytic approximation formulae.Quart. Appl. Math., 4(1):45–99, 1946

work page 1946
[42]

Schoenberg

I.J. Schoenberg. On the zeros of the generating functions of multiply positive sequences and functions.Ann. of Math. (2), 62(3):447–471, 1955

work page 1955
[43]

I. Schur. Bemerkungen zur Theorie der beschr¨ ankten Bilinearformen mit unendlich vielen Ver¨ anderlichen.J. reine angew. Math., 140:1–28, 1911

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A.M. Whitney. A reduction theorem for totally positive matrices.J. d’Analyse Math., 2(1):88–92, 1952. (P.N. Choudhury)Department of Mathematics, Indian Institute of Technology Gandhinagar, Gu- jarat 382355, India Email address:projeshnc@iitgn.ac.in (S. Yadav)Department of Mathematics, Indian Institute of Technology Gandhinagar, Gujarat 382355, India; Ph.D...

work page 1952

[1] [1]

T. Ando. Totally positive matrices.Linear Algebra Appl., 90:165–219, 1987

work page 1987

[2] [2]

Belton, D

A. Belton, D. Guillot, A. Khare, and M. Putinar. Matrix positivity preservers in fixed dimension. I.Adv. Math., 298:325–368, 2016

work page 2016

[3] [3]

Belton, D

A. Belton, D. Guillot, A. Khare, and M. Putinar. Totally positive kernels, P´ olya frequency functions, and their transforms.J. d’Analyse Math., 150:83–158, 2023

work page 2023

[4] [4]

Berenstein, S

A. Berenstein, S. Fomin, and A. Zelevinsky. Parametrizations of canonical bases and totally positive matrices. Adv. Math., 122:49–149, 1996

work page 1996

[5] [5]

Bickel and E

P.J. Bickel and E. Levina. Covariance regularization by thresholding.Ann. Statist., 36(6):2577–2604, 2008

work page 2008

[6] [6]

Blekherman, P.A

G. Blekherman, P.A. Parrilo, and R.R. Thomas.Semidefinite optimization and convex algebraic geometry. MOS- SIAM Series on Optimization, Society for Industrial and Applied Mathematics, 2012

work page 2012

[7] [7]

F. Brenti. Combinatorics and total positivity.J. Combin. Theory Ser. A, 71(2):175–218, 1995

work page 1995

[8] [8]

Brown, I.M

L.D. Brown, I.M. Johnstone, and K.B. MacGibbon. Variation diminishing transformations: a direct approach to total positivity and its statistical applications.J. Amer. Statist. Assoc., 76(376):824–832, 1981

work page 1981

[9] [9]

Choudhury and S

P.N. Choudhury and S. Yadav. Constructing strictly sign regular matrices of all sizes and sign patterns.Bull. Lond. Math. Soc., 57(7):2077–2096, 2025

work page 2077

[10] [10]

Choudhury and S

P.N. Choudhury and S. Yadav. Sign regular matrices and variation diminution: single-vector tests and char- acterizations, following Schoenberg, Gantmacher–Krein, and Motzkin.Proc. Amer. Math. Soc., 153:497–511, 2025

work page 2025

[11] [11]

Choudhury and S

P.N. Choudhury and S. Yadav. Sign regularity preserving linear operators.Bull. Lond. Math. Soc., in press; arXiv:math.FA/2408.02428

work page arXiv

[12] [12]

G. Darboux. Sur la composition des forces en statique.Bull. Sci. Math., 9:281–299, 1875

work page

[13] [13]

Fallat and C.R

S.M. Fallat and C.R. Johnson.Totally non-negative matrices. Princeton Series in Applied Mathematics, Princeton University Press, Princeton, 2011

work page 2011

[14] [14]

Fomin and A

S. Fomin and A. Zelevinsky. Total positivity: tests and parametrizations.Math. Intelligencer, 22(1):23–33, 2000

work page 2000

[15] [15]

Fomin and A

S. Fomin and A. Zelevinsky. Cluster algebras. I. Foundations.J. Amer. Math. Soc., 15(2):497–529, 2002

work page 2002

[16] [16]

Frobenius

G. Frobenius. ¨Uber die Darstellung der endlichen Gruppen durch lineare Substitutionen.Sitzungsber. Preuss. Akad. Wiss. Berlin, 994–1015, 1897. 29

work page

[17] [17]

Gantmacher and M.G

F.R. Gantmacher and M.G. Krein. Sur les matrices compl` etement nonn´ egatives et oscillatoires.Compositio Math., 4:445–476, 1937

work page 1937

[18] [18]

Gantmacher and M.G

F.R. Gantmacher and M.G. Krein.Oscillyacionye matricy i yadra i malye kolebaniya mehaniˇ ceskih sistem. 2nd ed. Gosudarstv. Isdat. Tehn.-Teor. Lit., Moscow-Leningrad, 1950

work page 1950

[19] [19]

Gr¨ ochenig, J.L

K. Gr¨ ochenig, J.L. Romero, and J. St¨ ockler. Sampling theorems for shift-invariant spaces, Gabor frames, and totally positive functions.Invent. Math., 211:1119–1148, 2018

work page 2018

[20] [20]

Guillot, A

D. Guillot, A. Khare, and B. Rajaratnam. Preserving positivity for matrices with sparsity constraints.Trans. Amer. Math. Soc., 368:8929–8953, 2016

work page 2016

[21] [21]

Guillot, A

D. Guillot, A. Khare, and B. Rajaratnam. Critical exponents of graphs.J. Combin. Theory Ser. A, 139:30–58, 2016

work page 2016

[22] [22]

Guillot, A

D. Guillot, A. Khare, and B. Rajaratnam. Preserving positivity for rank-constrained matrices.Trans. Amer. Math. Soc., 369:6105–6145, 2017

work page 2017

[23] [23]

Helton, S

J.W. Helton, S. McCullough, M. Putinar, and V. Vinnikov. Convex matrix inequalities versus linear matrix inequalities.IEEE Trans. Automat. Control, 54(5):952–964, 2009

work page 2009

[24] [24]

Hero and B

A. Hero and B. Rajaratnam. Large-scale correlation screening.J. Amer. Statist. Assoc., 106(496):1540–1552, 2011

work page 2011

[25] [25]

S. Karlin. Total positivity, absorption probabilities and applications.Trans. Amer. Math. Soc., 111:33–107, 1964

work page 1964

[26] [26]

Karlin.Total positivity

S. Karlin.Total positivity. Vol. I. Stanford University Press, Stanford, CA, 1968

work page 1968

[27] [27]

Karlin and Z

S. Karlin and Z. Ziegler. Chebyshevian spline functions.SIAM J. Numer. Anal., 3(3):514–543, 1966

work page 1966

[28] [28]

A. Khare. Critical exponents for total positivity, individual kernel encoders, and the Jain-Karlin-Schoenberg kernel.Prepint,arXiv:2008.05121, 2020

work page arXiv 2008

[29] [29]

Kodama and L

Y. Kodama and L. Williams. KP solitons and total positivity for the Grassmannian.Invent. Math., 198(3):637– 699, 2014

work page 2014

[30] [30]

C. Loewner. On totally positive matrices.Math. Z., 63:338–340, 1955

work page 1955

[31] [31]

G. Lusztig. Total positivity in reductive groups. InLie theory and geometry, volume 123 ofProgr. Math., pages 531–568. Birkh¨ auser, Boston, MA, 1994

work page 1994

[32] [32]

Motzkin.Beitr¨ age zur Theorie der linearen Ungleichungen

T.S. Motzkin.Beitr¨ age zur Theorie der linearen Ungleichungen. PhD dissert., Basel, 1933 and Jerusalem, 1936

work page 1933

[33] [33]

Ostrowski

A.M. Ostrowski. ¨Uber die Funktionalgleichung der Exponentialfunktion und verwandte Funktionalgleichung. Jber. Deut. Math. Ver., 38:54–62, 1929

work page 1929

[34] [34]

Pinkus.Totally positive matrices

A. Pinkus.Totally positive matrices. Cambridge Tracts in Mathematics, Vol. 181, Cambridge University Press, Cambridge, 2010

work page 2010

[35] [35]

P´ olya and G

G. P´ olya and G. Szeg˝ o.Aufgaben und Lehrs¨ atze aus der Analysis. Band II: Funcktionentheorie, Nullstellen, Polynome Determinanten, Zahlentheorie.Springer-Verlag, Berlin, 1925

work page 1925

[36] [36]

K.C. Rietsch. Totally positive Toeplitz matrices and quantum cohomology of partial flag varieties.J. Amer. Math. Soc., 16(2):363–392, 2003

work page 2003

[37] [37]

Rothman, E

A.J. Rothman, E. Levina and J. Zhu. Generalized thresholding of large covariance matrices.J. Amer. Statist. Assoc., 104(485):177–186, 2009

work page 2009

[38] [38]

W. Rudin. Positive definite sequences and absolutely monotonic functions.Duke Math. J., 26(4):617–622, 1959

work page 1959

[39] [39]

Schoenberg

I.J. Schoenberg. ¨Uber variationsvermindernde lineare Transformationen.Math. Z., 32:321–328, 1930

work page 1930

[40] [40]

Schoenberg

I.J. Schoenberg. Positive definite functions on spheres.Duke Math. J., 9(1):96–108, 1942

work page 1942

[41] [41]

Schoenberg

I.J. Schoenberg. Contributions to the problem of approximation of equidistant data by analytic functions. Part A. On the problem of smoothing or graduation. A first class of analytic approximation formulae.Quart. Appl. Math., 4(1):45–99, 1946

work page 1946

[42] [42]

Schoenberg

I.J. Schoenberg. On the zeros of the generating functions of multiply positive sequences and functions.Ann. of Math. (2), 62(3):447–471, 1955

work page 1955

[43] [43]

I. Schur. Bemerkungen zur Theorie der beschr¨ ankten Bilinearformen mit unendlich vielen Ver¨ anderlichen.J. reine angew. Math., 140:1–28, 1911

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A.M. Whitney. A reduction theorem for totally positive matrices.J. d’Analyse Math., 2(1):88–92, 1952. (P.N. Choudhury)Department of Mathematics, Indian Institute of Technology Gandhinagar, Gu- jarat 382355, India Email address:projeshnc@iitgn.ac.in (S. Yadav)Department of Mathematics, Indian Institute of Technology Gandhinagar, Gujarat 382355, India; Ph.D...

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