Maximal Unimodular Systems of Vectors

Данилов, В. И.; Grishukhin, Viatcheslav

doi:10.1006/eujc.1999.0298

Cited by 21 publications

(25 citation statements)

References 7 publications

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“… Seymour () characterized unimodular sets of vectors; Danilov and Grishukhin's () extension fully describes, up to basis change, all maximal such sets (there are finitely many for any n ). …”

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confidence: 99%

Understanding Preferences: “Demand Types”, and the Existence of Equilibrium With Indivisibilities

Baldwin

Klemperer

2019

ECTA

105

124

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An Equivalence Theorem between geometric structures and utility functions allows new methods for understanding preferences. Our classification of valuations into “Demand Types” incorporates existing definitions (substitutes, complements, “strong substitutes,” etc.) and permits new ones. Our Unimodularity Theorem generalizes previous results about when competitive equilibrium exists for any set of agents whose valuations are all of a “demand type.” Contrary to popular belief, equilibrium is guaranteed for more classes of purely‐complements than of purely‐substitutes, preferences. Our Intersection Count Theorem checks equilibrium existence for combinations of agents with specific valuations by counting the intersection points of geometric objects. Applications include matching and coalition‐formation, and the “Product‐Mix Auction” introduced by the Bank of England in response to the financial crisis.

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confidence: 99%

Understanding Preferences: “Demand Types”, and the Existence of Equilibrium With Indivisibilities

Baldwin

Klemperer

2019

ECTA

105

124

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“…The cographic system (K 3,3 ∨ e) * is a subsystem of the maximal cographic system W 5 of dimension 5. For details see [1].…”

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confidence: 98%

There are Exactly 222 L-types of Primitive Five-dimensional Lattices

Engel

Grishukhin

2002

European Journal of Combinatorics

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Enumerations of combinatorial types (i.e., L-types) of five-dimensional primitive lattices was performed in [6] and [3]. But both results differ in one L-type. We show explicitly the L-type missed in [6]. In 1976, Ryshkov and Baranovskii published the paper [6], where a list of 221 L-types of primitive five-dimensional (5D) lattices was given. Recently, the first author of this paper using a computer repeated the enumeration of 5D primitive parallelohedra. The result is given in [3]. But he obtained 222 combinatorial types. In this paper, we compare the results of both these papers and give explicitly the L-type missed in [6].The lists of L-types are contained in Table III of [6] and Table 1 of [3]. Unfortunately, the direct comparison of these tables is not possible, since the parameters used for describing L-types in the tables are distinct.The paper [6] describes an L-type by a symbol (corresponding to a C-type) and by a set of repartition pairs of indices of the set I 5 = {1, 2, 3, 4, 5} (indicating an L-type domain inside the C-type domain).For the Voronoi polytope P V of each combinatorial type, the paper [3] gives numbers N i of its i-faces, 0 ≤ i ≤ 4, and subordination indices (indicating how much (k − 1)-faces are contained in a k-face). In addition, Table 1 Every L-type symbol γ of [6] describes completely the corresponding L-type domain D(γ ). Theoretically, using the symbol γ , one can write out explicitly the system of inequalities describing the domain D(γ ) and find a representative form f ∈ D(γ ) and its Gram matrix. Then, applying the algorithm of [3] to the Gram matrix, one can determine the corresponding combinatorial type from Table 1 of [3]. But this is a huge amount of work. Hence we proceed in another way.It turns out that it is not difficult to find the parameters p and q of a given L-type using the symbol of the L-type from [6]. In particular, the parameter q of a lattice is equal also to the number of its laminar hyperplanes. This is so, since there is a one-to-one correspondence between closed zones of the Voronoi polytope P V and laminar hyperplanes of the corresponding lattice (see, for example, [2]).All Voronoi polytopes of an n-dimensional lattice L form the Voronoi partition of R n . The dual to the Voronoi partition is the Delaunay partition consisting of Delaunay polytopes. Let the origin (that we suppose to be a point of L) be a vertex of a Delaunay polytope P D of the Delaunay partition. Let F be a facet of P D and H be the hyperplane supporting F. The intersection of H with the Delaunay partition related to L is the Delaunay partition of H related to the lattice L ∩ H . Each Delaunay polytope of L ∩ H is the intersection of H with a Delaunay polytope of L. The hyperplane H is called lamina or laminar hyperplane if each Delaunay polytope of the sublattice L ∩ H is a facet of a Delaunay polytope of L.Proposition 14.2 of [6] describes how to find all q laminar hyperplanes of a lattice of a given L-type using the symbol of the L-type. Inspection of all symbols shows that q ≥ 10 f...

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“…50 Seymour (1980) developed a characterisation of unimodular sets of vectors; Danilov and Grishukhin (1999) extended this to give a full characterisation of all maximal such sets.…”

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confidence: 99%

Understanding Preferences: 'Demand Types', and The Existence of Equilibrium with Indivisibilities

Baldwin

Klemperer

2015

SSRN Journal

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We propose new techniques for understanding agents' valuations. Our classification into "demand types", incorporates existing definitions (substitutes, complements, "strong substitutes", etc.) and permits new ones. Our Unimodularity Theorem generalises previous results about when competitive equilibrium exists for any set of agents whose valuations are all of a "demand type" for indivisible goods. Contrary to popular belief, equilibrium is guaranteed for more classes of purely-complements, than of purely-substitutes, preferences. Our Intersection Count Theorem checks equilibrium existence for combinations of agents with specific valuations by counting the intersection points of geometric objects. Applications include matching and coalition-formation; and the Product-Mix Auction, introduced by the Bank of England in response to the financial crisis.

show abstract

Maximal Unimodular Systems of Vectors

Cited by 21 publications

References 7 publications

Understanding Preferences: “Demand Types”, and the Existence of Equilibrium With Indivisibilities

Understanding Preferences: “Demand Types”, and the Existence of Equilibrium With Indivisibilities

There are Exactly 222 L-types of Primitive Five-dimensional Lattices

Understanding Preferences: 'Demand Types', and The Existence of Equilibrium with Indivisibilities

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