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An integer analogue of Carathéodory's theorem

1986
*
Journal of combinatorial theory. Series B (Print)
*

The result is used to give

doi:10.1016/0095-8956(86)90064-x
fatcat:xsbfyf7sgfaoze2ohpteqc76oi
*an*upper bound on the number*of*nonzero variables needed in optimal solutions to*integer*programs associated with totally dual integral systems. ... We prove a*theorem*on Hilbert bases analogous to Carathtodory's*theorem*for convex cones. ... If x E C, then x can be expressed as a nonnegative linear combination*of*dim C vectors in (a,,..., ak). We study*an**integer**analogue**of*this result. ...##
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A counterexample to an integer analogue of Carathéodory's theorem

1999
*
Journal für die Reine und Angewandte Mathematik
*

"

doi:10.1515/crll.1999.045
fatcat:s6ijhdnyovacnfegab2fwo6oxu
*of*the German Science Foundation (DFG). combination (and thus having a nice counterpart to*Carathéodory's**theorem*) has already been raised by Cook, Fonlupt&Schrijver [CFS86] . ... Hence (ICP) (and the stronger properties) do not even hold in the class*of*cones generated by 0/1-vectors. can be achieved by*an**integer*vector y ∈ Z m for each*integer*vector c ∈ Z n for which the optima ...##
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Carathéodory bounds for integer cones

2006
*
Operations Research Letters
*

*Integer*

*Analogues*

*of*

*Carathéodory's*

*Theorem*X ⊆ d

*an*integral Hilbert basis if cone(X) ∩ d = int_cone(X) X Hilbert basis and cone(X) pointed: (Bruns, Gubeladze, Henk, Martin, Weismantel 1999) • What ... • | X| 2 d − 1 (Cook, Fonlupt & Schrijver 1986) • | X| 2 d − 2 (Sebő 1990) • | X| d disproved

*Integer*

*Analogues*

*of*

*Carathéodory's*

*Theorem*X ⊆ d

*an*integral Hilbert basis if cone(X) ∩ d = int_cone(X) ...

*Integer*Programming

*Theorem*. Given IP min{c T y | A y = b, y 0, y

*integer*}, where A ∈ d×n and c ∈ n with with optimal value γ. ...

##
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Book announcements

1989
*
Discrete Applied Mathematics
*

Behaviour

doi:10.1016/0166-218x(89)90020-6
fatcat:kqa33ru6szamnduuuwcrqavuhu
*of*total dual integrality under operations.*An**integer**analogue**of**Caratheodory's**theorem*. Another characterization*of*total dual integrality. ... Regular matroids and signing*of*{0, I}-matrices. Chain groups.*An*upper bound*of*Heller. Unimodular matrices more generally. Balanced matrices. ...##
###
Page 1128 of Mathematical Reviews Vol. 29, Issue 6
[page]

1965
*
Mathematical Reviews
*

representation is a complete

*analogue**of*the Paley- Wiener*theorem*. ... However, the proof*of*the mapping*theorem*is now more direct and perspicuous than in*Carathéodory’s*version. ...##
###
Page 288 of American Mathematical Society. Bulletin of the American Mathematical Society Vol. 55, Issue 3
[page]

1949
*
American Mathematical Society. Bulletin of the American Mathematical Society
*

Every such a yields

*an**analogue**of*the particular case. A complete genesis*of*unconditionally additive set properties is obtained in terms*of*ascending set-point properties. ...*An*immediate generalization*of*a well known*theorem*, namely, that a linear set is the disjoint sum*of*a dense-in-itself and a scattered set, leads to the problem*of*the genesis*of**an*unconditionally additive ...##
###
Page 6742 of Mathematical Reviews Vol. , Issue 92m
[page]

1992
*
Mathematical Reviews
*

For differential forms having such functions as coefficients, we give

*analogues**of*exterior differentiation and integration over chains and prove*an**analogue**of*the abstract Stokes*theorem*.” 28 MEASURE ... Let § be the power set*of*the positive*integers*N and let ba be the space*of*all finitely additive set functions yw: — R which have finite variation. ...##
###
Unimodular functions and uniform boundedness

1989
*
Publicacions matemàtiques
*

SHAPIRO In this paper we study the role that unimodular functions play in deciding the uniform boundedness

doi:10.5565/publmat_33189_11
fatcat:mq2mjjid7vbwdcga77sjfju7cu
*of*sets*of*continuous linear functionals on various function spaacs . ...*An*interesting question is whether*Theorem*4 has*an**analogue*when M is norm closed and <~C L'(X, Es)* . ... Using a technique similar to that used in the proof*of**Theorem*1, we prove the following generalization*of**Carathéodory's**Theorem*.*Theorem*3. ...##
###
Page 516 of Mathematical Reviews Vol. 34, Issue 3
[page]

1967
*
Mathematical Reviews
*

Math. 145 (1915), 177-223] begins with accessible boundary elements and proceeds to full generality by

*an**analogue**of*Dedekind sections. ... Let {n,} be*an*increasing sequence*of*non-negative*integers*with n)=0 and write N°(t)=>,, <, 1. The following result is obtained. Suppose that f(z)= >}. ...##
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The theorems of Carathéodory and Gluskin for $0<p<1$

1995
*
Proceedings of the American Mathematical Society
*

In this note we prove the p-convex

doi:10.1090/s0002-9939-1995-1242074-2
fatcat:lez2fxrymrd3ljzb4kfvbxhv54
*analogue**of*both*Caratheodory's*convexity*theorem*and Gluskin's*theorem*concerning the diameter*of*Minkowski compactum. ... Acknowledgments The authors are indebted to the referee for showing them the simpler proof*of**Theorem*1 which notably simplifies a previous one. ... Proof*of**Theorem*1. Let x e p-conv(A), x ^ 0. Let N be the smallest*integer*so that x in the p-convex hull*of*a subset {Px, ... , P^}*of*A. ...##
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Non-negative hereditary polynomials in a free *−algebra

2005
*
Mathematische Zeitschrift
*

We prove a non-negative-stellensatz and a null-stellensatz for a class

doi:10.1007/s00209-004-0758-5
fatcat:hibxapr465c3xe3crl7uvt75hm
*of*polynomials called hereditary polynomials in a free * -algebra. ... Proof: This is*an*application*of**Carathéodory's**theorem*, see for instance [9] for a similar derivation in the commutative case. ... The proof can be reduced to the self-adjoint case treated by the*analogue**of**Theorem*1.1 via the decomposition 2q = (q + q * ) + i[(q − q * )/i]. ...##
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Unitary equivalence of multiplication operators on the Bergman spaces of polygons

2021
*
Canadian mathematical bulletin
*

In this paper, we will show that the unitary equivalence

doi:10.4153/s0008439521000151
fatcat:dupcfbt3njdehm7mkqgumqvnom
*of*two multiplication operators on the Bergman spaces on polygons depends on the geometry*of*the polygon. ... A remark*of**Theorem*1.1 is in order. Firstly, there is no*analogue**of**Theorem*1.1 in the case*of*L 2 a (D). Secondly, the condition that both g and h are holomorphic on Σ can not be removed. ... Unitary equivalence*of*multiplication operators on the Bergman spaces 5 To prove*Theorem*1.1, we will use*Carathéodory's**theorem*and*Theorem*2.3 to establish the following lemma. ...##
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Integral decomposition of polyhedra and some applications in mixed integer programming

2003
*
Mathematical programming
*

It is shown that there is a finite subset

doi:10.1007/s10107-002-0315-0
fatcat:plmqcvqt3vhyxihr6wajj2r5ce
*of*this family that generates the entire family. Moreover,*an**integer**analogue**of**Carathéodory's**theorem*carries over to this general setting. ... This paper addresses the question*of*decomposing*an*infinite family*of*rational polyhedra in*an**integer*fashion. ... In the same vein it is shown in this paper that*an**integer**analogue**of**Carathéodory's**theorem*that is known to hold for the integral points in a rational polyhedral cone carries over to the more general ...##
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A weak Krasnosel\cprime skiĭ\ theorem in ${\bf R}\sp d$

1988
*
Proceedings of the American Mathematical Society
*

Let S be a compact, locally starshaped set in Rd, and let k be a fixed

doi:10.1090/s0002-9939-1988-0962828-3
fatcat:ssds5zy3f5anvfpmkkn76eakra
*integer*, 0 < k < d. ... If k = 0 or k = d -1, then each point*of*S sees via S some point*of*F. Moreover, if k = 1, then F can be chosen so that F n S is convex. ... Since S is locally starshaped, there is*an**integer*/ such that [í,í¿] Ç S for i > 1. ...##
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Page 2200 of Mathematical Reviews Vol. , Issue 87d
[page]

1987
*
Mathematical Reviews
*

(NL-TILB-E)

*An**integer**analogue**of**Carathéodory’s**theorem*. J. Combin. Theory Ser. B 40 (1986), no. 1, 63-70. ... A proof*of*this is easily given as*an*applica- tion*of*Kakutani’s fixed point*theorem*and a*theorem**of*Wey! on polyhedral sets. ...
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