Báo cáo toán học: "n-Color partition theoretic interpretations of some mock theta functions" doc

... definitions of mock theta functions and their order the reader is referred to [6]. The object of this paper is to provide new number theoretic interpretations of the following mock theta functions: Ψ(q)= ∞  m=1 q m 2 (q; ... aq n+i ) , for any constant a. We remark that Ψ(q) is of order 3 while the remaining three are of order 5. Number theoretic interpretations of...

Ngày tải lên: 07/08/2014, 08:20

... structural properties and that the natural way to describe the structure of these graphs is in terms of graph homomorphisms. One infinite sequence of cores of dense maximal triangle-free graphs was known. All graphs ... 4-colourable, in contrast to a conjecture of Jin [8], saying that there are graphs of arbitrarily large chromatic number with this property. Another infinite sequenc...

Ngày tải lên: 07/08/2014, 06:22

... 0-element) of the ζ-matrix of S byacolumnofall1’s. By the argument of the preceding section, the permanent of R(S)isthenumberof -chains in S that contain the 0 element. The permutations of [ |S|] ... element. Then the permanent of R is the number of chains of S that contain the 0 element, and the determinant of R is  x∈S µ(0,x), where µ is the M¨obius function of S. If t...

Ngày tải lên: 07/08/2014, 08:22

... b)-decomposition of G is a partition of the edge set of G such that each part is either a single edge or forms an H-subgraph. We assign a weight of b to each H-subgraph in the decompo- sition and a weight of ... of G is a partition of the edge set of G such that each part is either a single edg e or forms an H-subgraph, i.e., a graph isomorphic to H. We allow partitions on...

Ngày tải lên: 08/08/2014, 14:23

... definition of f(π) given in the paper of Dohmen et al [DPT03]. By abuse of notation, we write e ⊆ π to mean the vertex set of e is a subset of some block of π. Finally, let k 1 (π) denote the number of ... corresponding to the partition of V into singletons. It also has a maximum element, corresponding to partitioning V into the vertex sets of the components of H. Given...

Ngày tải lên: 08/08/2014, 14:23

... purpose of this paper is to present the properties of the meromorphic solutions of complex difference equations of the form  {J} α J z  j∈J fz  c j   Rz, fz,where{J} is a collection of ... with the basic notations of Nevanlinna’s value distribution theory; see 1–3. Recent interest in the problem of integrability of difference equations is a consequence of the e...

Ngày tải lên: 21/06/2014, 20:20

... distribution of two random elements of the set of partitions of m for some 0 ≤ m ≤ n, allowing a trivial partition of 0, (b) on the left side π n −{X n } denotes the random partition of n − X n derived ... points of a uniform sample into clusters. Given R, define an interval partition of [0, 1] comprised of gaps,thatisopenintervalcomponentsof[0, 1] \R,and of individ...

Ngày tải lên: 07/08/2014, 08:22

... consists of all edges of G with both ends in S. We will use the symbols π and σ to denote set partitions. The notation π  S means π is a set partition of the set S. The length (number of blocks) of ... denoted by (π). A set partition σ is called a refinement of a set partition π if every block of σ is contained in some block of π. To each graph G on V there correspond...

Ngày tải lên: 07/08/2014, 08:22

... states without proof that M(3, ∞) m = 0 for m ≥ 3 in ([11], Theorem 3.12). Our proof of this in Theorem 4.3 below illustrates the use of Case 1 of Corollary 2.13. This supplies a proof of J n,[d] for ... ([11], Proposition 3.11). Our proof of this in Theorem 4.4 below is different and provides a second example of Case 1 of Coro llar y 2.13. This supplies a proof of J n,[2] for...

Ngày tải lên: 07/08/2014, 21:21

... times to the sum of the entries of the i-th column o f A G 1 B P . Thus, the sum of the entries of the i-th column of A G 1 B P is kb i . Second, consider B P A G 2 . The sum of the entries in ... problem of determining the automorphisms Aut(G) of an undirected graph G. Recall that the elements of Aut(G) are those permutations of the vertices of G which preserve edge adj...

Ngày tải lên: 08/08/2014, 12:22