Expand all Collapse all | Results 51 - 59 of 59 |
51. CJM 1997 (vol 49 pp. 1281)
Pieri's formula via explicit rational equivalence Pieri's formula describes the intersection product of a Schubert
cycle by a special Schubert cycle on a Grassmannian.
We present a new geometric proof,
exhibiting an explicit chain of rational equivalences
from a suitable sum of distinct Schubert cycles
to the intersection of a Schubert cycle with a special
Schubert cycle. The geometry of these rational equivalences
indicates a link to a combinatorial proof of Pieri's formula using
Schensted insertion.
Keywords:Pieri's formula, rational equivalence, Grassmannian, Schensted insertion Categories:14M15, 05E10 |
52. CJM 1997 (vol 49 pp. 865)
Maps in locally orientable surfaces and integrals over real symmetric surfaces The genus series for maps is the generating series for the
number of rooted maps with a given number of vertices and
faces of each degree, and a given number of edges. It captures
topological information about surfaces, and appears in questions
arising in statistical mechanics, topology, group rings,
and certain aspects of free probability theory. An expression
has been given previously for the genus series for maps in
locally orientable surfaces in terms of zonal polynomials. The
purpose of this paper is to derive an integral representation
for the genus series. We then show how this can be used in
conjunction with integration techniques to determine the genus
series for monopoles in locally orientable surfaces. This
complements the analogous result for monopoles in orientable
surfaces previously obtained by Harer and Zagier. A conjecture,
subsequently proved by Okounkov, is given for the evaluation
of an expectation operator acting on the Jack symmetric function.
It specialises to known results for Schur functions and zonal
polynomials.
Categories:05C30, 05A15, 05E05, 15A52 |
53. CJM 1997 (vol 49 pp. 883)
Proof of a conjecture of Goulden and Jackson We prove an integration formula involving Jack polynomials
conjectured by I.~P.~Goulden and D.~M.~Jackson in connection with
enumeration of maps in surfaces.
Categories:05E05, 43A85, 57M15 |
54. CJM 1997 (vol 49 pp. 641)
Fine spectra and limit laws II First-order 0--1 laws. Using Feferman-Vaught techniques a condition on the fine
spectrum of an admissible class of structures is found
which leads to a first-order 0--1 law.
The condition presented is best possible in the
sense that if it is violated then one can find an admissible
class with the same fine spectrum which does not have
a first-order 0--1 law.
If the condition is satisfied (and hence we have a first-order %% 0--1 law)
Categories:03N45, 11N45, 11N80, 05A15, 05A16, 11M41, 11P81 |
55. CJM 1997 (vol 49 pp. 617)
On the zeros of some genus polynomials In the genus polynomial of the graph $G$, the coefficient of $x^k$
is the number of distinct embeddings of the graph $G$ on the
oriented surface of genus $k$. It is shown that for several
infinite families of graphs all the zeros of the genus polynomial
are real and negative. This implies that their coefficients, which
constitute the genus distribution of the graph, are log concave and
therefore also unimodal. The geometric distribution of the zeros
of some of these polynomials is also investigated and some new
genus polynomials are presented.
Categories:05C10, 05A15, 30C15, 26C10 |
56. CJM 1997 (vol 49 pp. 468)
Fine spectra and limit laws I. First-order laws Using Feferman-Vaught techniques we show a certain property of the fine
spectrum of an admissible class of structures leads to a first-order law.
The condition presented is best possible in the sense that if it is
violated then one can find an admissible class with the same fine
spectrum which does not have a first-order law. We present three
conditions for verifying that the above property actually holds.
The first condition is that the count function of an admissible class
has regular variation with a certain uniformity of convergence. This
applies to a wide range of admissible classes, including those
satisfying Knopfmacher's Axiom A, and those satisfying Bateman
and Diamond's condition.
The second condition is similar to the first condition, but designed
to handle the discrete case, {\it i.e.}, when the sizes of the structures
in an admissible class $K$ are all powers of a single integer. It applies
when either the class of indecomposables or the whole class satisfies
Knopfmacher's Axiom A$^\#$.
The third condition is also for the discrete case, when there is a
uniform bound on the number of $K$-indecomposables of any given size.
Keywords:First order limit laws, generalized number theory Categories:O3C13, 11N45, 11N80, 05A15, 05A16, 11M41, 11P81 |
57. CJM 1997 (vol 49 pp. 263)
Determinantal forms for symplectic and orthogonal Schur functions Symplectic and orthogonal Schur functions can be defined
combinatorially in a manner similar to the classical Schur functions.
This paper demonstrates that they can also be expressed as determinants.
These determinants are generated using planar decompositions of tableaux
into strips and the equivalence of these determinants to symplectic or
orthogonal Schur functions is established by Gessel-Viennot lattice path
techniques. Results for rational (also called {\it composite}) Schur functions
are also obtained.
Categories:05E05, 05E10, 20C33 |
58. CJM 1997 (vol 49 pp. 301)
On some alternative characterizations of Riordan arrays We give several new characterizations of Riordan Arrays, the most
important of which is: if $\{d_{n,k}\}_{n,k \in {\bf N}}$ is a lower
triangular array whose generic element $d_{n,k}$ linearly depends on
the elements in a well-defined though large area of the array, then
$\{d_{n,k}\}_{n,k \in {\bf N}}$ is Riordan. We also provide some
applications of these characterizations to the lattice path theory.
Categories:05A15, 05C38 |
59. CJM 1997 (vol 49 pp. 193)
Classifying PL $5$-manifolds by regular genus: the boundary case In the present paper, we face the problem of classifying classes of
orientable PL $5$-manifolds $M^5$ with $h \geq 1$ boundary components,
by making use of a combinatorial invariant called {\it regular genus}
${\cal G}(M^5)$. In particular, a complete classification up to
regular genus five is obtained:
$${\cal G}(M^5) = \gG \leq 5 \Longrightarrow M^5 \cong \#_{\varrho
- \gbG}(\bdo) \# \smo_{\gbG},$$
where $\gbG = {\cal G}(\partial M^5)$ denotes the regular genus of
the boundary $\partial M^5$ and $\smo_{\gbG}$ denotes the connected
sum of $h\geq 1$ orientable $5$-dimensional handlebodies
$\cmo_{\alpha_i}$ of genus $\alpha_i\geq 0$
($i=1,\ldots, h$), so that $\sum_{i=1}^h \alpha_i = \gbG.$
\par
Moreover, we give the following characterizations of orientable PL
$5$-manifolds $M^5$ with boundary satisfying particular conditions
related to the ``gap'' between ${\cal G}(M^5)$ and either
${\cal G}(\partial M^5)$ or the rank of their fundamental group
$\rk\bigl(\pi_1(M^5)\bigr)$:
$$\displaylines{{\cal G}(\partial M^5)= {\cal G}(M^5)
= \varrho \Longleftrightarrow M^5 \cong \smo_{\gG}\cr
{\cal G}(\partial M^5)= \gbG = {\cal G}(M^5)-1 \Longleftrightarrow
M^5 \cong (\bdo) \# \smo_{\gbG}\cr
{\cal G}(\partial M^5)= \gbG = {\cal G}(M^5)-2 \Longleftrightarrow
M^5 \cong \#_2 (\bdo) \# \smo_{\gbG}\cr
{\cal G}(M^5) = \rk\bigl(\pi_1(M^5)\bigr)= \varrho \Longleftrightarrow
M^5 \cong \#_{\gG - \gbG}(\bdo) \# \smo_{\gbG}.\cr}$$
\par
Further, the paper explains how the above results (together with
other known properties of regular genus of PL manifolds) may lead
to a combinatorial approach to $3$-dimensional Poincar\'e Conjecture.
Categories:57N15, 57Q15, 05C10 |