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Small circulant complex Hadamard matrices of Butson type ; Schlenker, Jean-Marc in European Journal of Combinatorics (2016), 51 We study the circulant complex Hadamard matrices of order nn whose entries are llth roots of unity. For n=ln=l prime we prove that the only such matrix, up to equivalence, is the Fourier matrix, while for ... [more ▼] We study the circulant complex Hadamard matrices of order nn whose entries are llth roots of unity. For n=ln=l prime we prove that the only such matrix, up to equivalence, is the Fourier matrix, while for n=p+q,l=pqn=p+q,l=pq with p,qp,q distinct primes there is no such matrix. We then provide a list of equivalence classes of such matrices, for small values of n,ln,l. [less ▲] Detailed reference viewed: 99 (10 UL)On the infinitesimal rigidity of weakly convex polyhedra ; Schlenker, Jean-Marc in European Journal of Combinatorics (2010), 31(4), 1080--1090 The main motivation here is a question: whether any polyhedron which can be subdivided into convex pieces without adding a vertex, and which has the same vertices as a convex polyhedron, is ... [more ▼] The main motivation here is a question: whether any polyhedron which can be subdivided into convex pieces without adding a vertex, and which has the same vertices as a convex polyhedron, is infinitesimally rigid. We prove that it is indeed the case for two classes of polyhedra: those obtained from a convex polyhedron by ``denting'' at most two edges at a common vertex, and suspensions with a natural subdivision. [less ▲] Detailed reference viewed: 69 (11 UL)Minors of Boolean functions with respect to clique functions and hypergraph homomorphisms Lehtonen, Erkko ; in European Journal of Combinatorics (2010), 31(8), 1981-1995 Each clone C on a fixed base set A induces a quasi-order on the set of all operations on A by the following rule: f is a C-minor of g if f can be obtained by substituting operations from C for the ... [more ▼] Each clone C on a fixed base set A induces a quasi-order on the set of all operations on A by the following rule: f is a C-minor of g if f can be obtained by substituting operations from C for the variables of g. By making use of a representation of Boolean functions by hypergraphs and hypergraph homomorphisms, it is shown that a clone C on {0,1} has the property that the corresponding C-minor partial order is universal if and only if C is one of the countably many clones of clique functions or the clone of self-dual monotone functions. Furthermore, the C-minor partial orders are dense when C is a clone of clique functions. [less ▲] Detailed reference viewed: 51 (0 UL)Labeled posets are universal Lehtonen, Erkko in European Journal of Combinatorics (2008), 29(2), 493-506 Partially ordered sets labeled with k labels (k-posets) and their homomorphisms are examined. The homomorphicity order of finite k-posets is shown to be a distributive lattice. Homomorphicity orders of ... [more ▼] Partially ordered sets labeled with k labels (k-posets) and their homomorphisms are examined. The homomorphicity order of finite k-posets is shown to be a distributive lattice. Homomorphicity orders of finite k-posets and k-lattices are shown to be universal in the sense that every countable poset can be embedded into them. Labeled posets are represented by directed graphs, and a categorical isomorphism between k-posets and their digraph representations is established. [less ▲] Detailed reference viewed: 35 (0 UL) |
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