On the Asymptotic Behaviour of General Partition Functions
... the right hand side in (1.5) can be as large as we wish for m fixed, and r large enough. We remark that the lim sup in (1.1) cannot be replaced by lim inf; to show this we shall have to consider sets A that are very irregularly distributed, similar to the counterexample given in [3]. We hope to retu ...
... the right hand side in (1.5) can be as large as we wish for m fixed, and r large enough. We remark that the lim sup in (1.1) cannot be replaced by lim inf; to show this we shall have to consider sets A that are very irregularly distributed, similar to the counterexample given in [3]. We hope to retu ...
19 Feb 2010
... to you, then to shuffle them thoroughly. Arrange the cards on a table face up, in rows of three. Ask your friend what column the card is in; call that number α. Now collect the cards, making sure they remain in the same order as they were when you dealt them. Arrange them on a a table face up again, ...
... to you, then to shuffle them thoroughly. Arrange the cards on a table face up, in rows of three. Ask your friend what column the card is in; call that number α. Now collect the cards, making sure they remain in the same order as they were when you dealt them. Arrange them on a a table face up again, ...
A Course on Convex Geometry
... vert P . If the vertices are affinely independent, the polytope P is called a simplex. More precisely, P is called an r-simplex, if it has precisely r + 1 affinely independent vertices. Remarks. (1) For a polytope P , we have P = conv vert P . This can be seen directly from an inductive argument (se ...
... vert P . If the vertices are affinely independent, the polytope P is called a simplex. More precisely, P is called an r-simplex, if it has precisely r + 1 affinely independent vertices. Remarks. (1) For a polytope P , we have P = conv vert P . This can be seen directly from an inductive argument (se ...
Schauder Hats for the 2-variable Fragment of BL
... and let S be a k-simplex of U . Then the starring of U at S, in symbols U ∗ S, is the set of simplices obtained as follows. 1) Put in U ∗ S all simplices of U not containing S. 2) Display v1 , . . . , vk the vertices of S. Then, for each d ∈ {1, . . . , k − 1} and each d-dimensional face T of S, dis ...
... and let S be a k-simplex of U . Then the starring of U at S, in symbols U ∗ S, is the set of simplices obtained as follows. 1) Put in U ∗ S all simplices of U not containing S. 2) Display v1 , . . . , vk the vertices of S. Then, for each d ∈ {1, . . . , k − 1} and each d-dimensional face T of S, dis ...
The bounded derived category of an algebra with radical squared zero
... and a cycle if w is non-trivial, reduced and closed. The degree ∂(w) of a walk w is defined as follows. We first define ∂(w) = 0, 1, or −1 in case w is a trivial path, an arrow, or the inverse of an arrow respectively, and then extend this definition to all walks in Q by ∂(uv) = ∂(u) + ∂(v) whenever ...
... and a cycle if w is non-trivial, reduced and closed. The degree ∂(w) of a walk w is defined as follows. We first define ∂(w) = 0, 1, or −1 in case w is a trivial path, an arrow, or the inverse of an arrow respectively, and then extend this definition to all walks in Q by ∂(uv) = ∂(u) + ∂(v) whenever ...
Hankel Matrices: From Words to Graphs
... Used by Freedman, Lovász and Schrijver, 2007, for characterizing multiplicative graph parameters over the real numbers • k-unions (connections, connection matrices) Used by Freedman, Lovász, Schrijver and Szegedy, 2007ff, for characterizing various forms and partition functions. • Joins, cartesian ...
... Used by Freedman, Lovász and Schrijver, 2007, for characterizing multiplicative graph parameters over the real numbers • k-unions (connections, connection matrices) Used by Freedman, Lovász, Schrijver and Szegedy, 2007ff, for characterizing various forms and partition functions. • Joins, cartesian ...
Automorphism groups of cyclic codes Rolf Bienert · Benjamin Klopsch
... that every finite group arises as the automorphism group of a suitable binary linear code; cf. [9]. The question which finite permutation groups, i.e. finite groups with a fixed faithful permutation representation, arise as automorphism groups of binary linear codes is more subtle; a possible approa ...
... that every finite group arises as the automorphism group of a suitable binary linear code; cf. [9]. The question which finite permutation groups, i.e. finite groups with a fixed faithful permutation representation, arise as automorphism groups of binary linear codes is more subtle; a possible approa ...
Exponential sums with multiplicative coefficients
... (ii) If q < N ~ and (a,q)= 1, then there is an f e o ~ for which [S(a/q)[>>Nq -~. (iii) If N(log N)- 3 < Q < N, then there are a, q, f s u c h that Q - 3 N Q- 1 < q < Q, (a, q)= 1,fe ~- and [S(a/q)l >>(Nq) ~. In fact, in each of the above the f we construct is totally multiplicative and satisfies If ...
... (ii) If q < N ~ and (a,q)= 1, then there is an f e o ~ for which [S(a/q)[>>Nq -~. (iii) If N(log N)- 3 < Q < N, then there are a, q, f s u c h that Q - 3 N Q- 1 < q < Q, (a, q)= 1,fe ~- and [S(a/q)l >>(Nq) ~. In fact, in each of the above the f we construct is totally multiplicative and satisfies If ...
Birkhoff's representation theorem
This is about lattice theory. For other similarly named results, see Birkhoff's theorem (disambiguation).In mathematics, Birkhoff's representation theorem for distributive lattices states that the elements of any finite distributive lattice can be represented as finite sets, in such a way that the lattice operations correspond to unions and intersections of sets. The theorem can be interpreted as providing a one-to-one correspondence between distributive lattices and partial orders, between quasi-ordinal knowledge spaces and preorders, or between finite topological spaces and preorders. It is named after Garrett Birkhoff, who published a proof of it in 1937.The name “Birkhoff's representation theorem” has also been applied to two other results of Birkhoff, one from 1935 on the representation of Boolean algebras as families of sets closed under union, intersection, and complement (so-called fields of sets, closely related to the rings of sets used by Birkhoff to represent distributive lattices), and Birkhoff's HSP theorem representing algebras as products of irreducible algebras. Birkhoff's representation theorem has also been called the fundamental theorem for finite distributive lattices.