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An integer analogue of Carathéodory's theorem

W Cook, J Fonlupt, A Schrijver
1986 Journal of combinatorial theory. Series B (Print)
The result is used to give 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.  ...

A counterexample to an integer analogue of Carathéodory's theorem

Winfried Bruns, Joseph Gubeladze, Martin Henk, Alexander Martin, Robert Weismantel
1999 Journal für die Reine und Angewandte Mathematik
" 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  ...

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.  ...

Book announcements

1989 Discrete Applied Mathematics
Behaviour 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

J. Fernández, S. Hui H. Shapiro, H. Shapiro
1989 Publicacions matemàtiques
SHAPIRO In this paper we study the role that unimodular functions play in deciding the uniform boundedness 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)= >}.  ...

The theorems of Carathéodory and Gluskin for $0<p<1$

Jes{ús Bastero, Julio Bernu{és, Ana Pe{ña
1995 Proceedings of the American Mathematical Society
In this note we prove the p-convex 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.  ...

Non-negative hereditary polynomials in a free *−algebra

J. William Helton, Scott A. McCullough, Mihai Putinar
2005 Mathematische Zeitschrift
We prove a non-negative-stellensatz and a null-stellensatz for a class 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].  ...

Unitary equivalence of multiplication operators on the Bergman spaces of polygons

Hansong Huang, Dechao Zheng
In this paper, we will show that the unitary equivalence 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.  ...

Integral decomposition of polyhedra and some applications in mixed integer programming

Martin Henk, Matthias Köppe, Robert Weismantel
2003 Mathematical programming
It is shown that there is a finite subset 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  ...

A weak Krasnosel\cprime skiĭ\ theorem in ${\bf R}\sp d$

Marilyn Breen
1988 Proceedings of the American Mathematical Society
Let S be a compact, locally starshaped set in Rd, and let k be a fixed 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.  ...

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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