135 research outputs found
On algebraic structures of numerical integration on vector spaces and manifolds
Numerical analysis of time-integration algorithms has been applying advanced
algebraic techniques for more than fourty years. An explicit description of the
group of characters in the Butcher-Connes-Kreimer Hopf algebra first appeared
in Butcher's work on composition of integration methods in 1972. In more recent
years, the analysis of structure preserving algorithms, geometric integration
techniques and integration algorithms on manifolds have motivated the
incorporation of other algebraic structures in numerical analysis. In this
paper we will survey structures that have found applications within these
areas. This includes pre-Lie structures for the geometry of flat and torsion
free connections appearing in the analysis of numerical flows on vector spaces.
The much more recent post-Lie and D-algebras appear in the analysis of flows on
manifolds with flat connections with constant torsion. Dynkin and Eulerian
idempotents appear in the analysis of non-autonomous flows and in backward
error analysis. Non-commutative Bell polynomials and a non-commutative Fa\`a di
Bruno Hopf algebra are other examples of structures appearing naturally in the
numerical analysis of integration on manifolds.Comment: 42 pages, final versio
Hopf Algebras of m-permutations, (m+1)-ary trees, and m-parking functions
The m-Tamari lattice of F. Bergeron is an analogue of the clasical Tamari
order defined on objects counted by Fuss-Catalan numbers, such as m-Dyck paths
or (m+1)-ary trees. On another hand, the Tamari order is related to the product
in the Loday-Ronco Hopf algebra of planar binary trees. We introduce new
combinatorial Hopf algebras based on (m+1)-ary trees, whose structure is
described by the m-Tamari lattices.
In the same way as planar binary trees can be interpreted as sylvester
classes of permutations, we obtain (m+1)-ary trees as sylvester classes of what
we call m-permutations. These objects are no longer in bijection with
decreasing (m+1)-ary trees, and a finer congruence, called metasylvester,
allows us to build Hopf algebras based on these decreasing trees. At the
opposite, a coarser congruence, called hyposylvester, leads to Hopf algebras of
graded dimensions (m+1)^{n-1}, generalizing noncommutative symmetric functions
and quasi-symmetric functions in a natural way. Finally, the algebras of packed
words and parking functions also admit such m-analogues, and we present their
subalgebras and quotients induced by the various congruences.Comment: 51 page
Recommended from our members
Free Probability Theory
Free probability theory is a line of research which parallels aspects of classical probability, in a non-commutative context where tensor products are replaced by free products, and independent random variables are replaced by free random variables. The theory grew out of attempts to solve some longstanding problems about von Neumann algebras of free groups. In the almost twenty years since its creation, free probability has become a subject in its own right, with connections to several other parts of mathematics: operator algebras, the theory of random matrices, classical probability and the theory of large deviations, algebraic combinatorics, topology. Free probability also has connections with applied mathematics (wireless communication) and some mathematical models in theoretical physics. The Oberwolfach workshop on free probability brought together a very strong group of mathematicians representing the current directions of development in the area
Generalized bialgebras and triples of operads
We introduce the notion of generalized bialgebra, which includes the
classical notion of bialgebra (Hopf algebra) and many others. We prove that,
under some mild conditions, a connected generalized bialgebra is completely
determined by its primitive part. This structure theorem extends the classical
Poincar\'e-Birkhoff-Witt theorem and the Cartier-Milnor-Moore theorem, valid
for cocommutative bialgebras, to a large class of generalized bialgebras.
Technically we work in the theory of operads which permits us to give a
conceptual proof of our main theorem. It unifies several results, generalizing
PBW and CMM, scattered in the literature. We treat many explicit examples and
suggest a few conjectures.Comment: Slight modification of the quotient triple proposition (3.1.1). Typos
corrected. 110 page
Selected Papers in Combinatorics - a Volume Dedicated to R.G. Stanton
Professor Stanton has had a very illustrious career. His contributions to mathematics are varied and numerous. He has not only contributed to the mathematical literature as a prominent researcher but has fostered mathematics through his teaching and guidance of young people, his organizational skills and his publishing expertise. The following briefly addresses some of the areas where Ralph Stanton has made major contributions
The Partition Lattice in Many Guises
This dissertation is divided into four chapters. In Chapter 2 the equivariant homology groups of upper order ideals in the partition lattice are computed. The homology groups of these filters are written in terms of border strip Specht modules as well as in terms of links in an associated complex in the lattice of compositions. The classification is used to reproduce topological calculations of many well-studied subcomplexes of the partition lattice, including the d-divisible partition lattice and the Frobenius complex. In Chapter 3 the box polynomial B_{m,n}(x) is defined in terms of all integer partitions that fit in an m by n box. The real roots of the box polynomial are completely characterized, and an asymptotically tight bound on the norms of the complex roots is also given. An equivalent definition of the box polynomial is given via applications of the finite difference operator Delta to the monomial x^{m+n}. The box polynomials are also used to find identities counting set partitions with all even or odd blocks, respectively. Chapter 4 extends results from Chapter 3 to give combinatorial proofs for the ordinary generating function for set partitions with all even or all odd block sizes, respectively. This is achieved by looking at a multivariable generating function analog of the Stirling numbers of the second kind using restricted growth words. Chapter 5 introduces a colored variant of the ordered partition lattice, denoted Q_n^{\alpha}, as well an associated complex known as the alpha-colored permutahedron, whose face poset is Q_n^\alpha. Connections between the Eulerian polynomials and Stirling numbers of the second kind are developed via the fibers of a map from Q_n^{\alpha} to the symmetric group on n-element
Renormalization in tensor field theory and the melonic fixed point
This thesis focuses on renormalization of tensor field theories. Its first
part considers a quartic tensor model with symmetry and long-range
propagator. The existence of a non-perturbative fixed point in any at large
is established. We found four lines of fixed points parametrized by the
so-called tetrahedral coupling. One of them is infrared attractive, strongly
interacting and gives rise to a new kind of CFT, called melonic CFTs which are
then studied in more details. We first compute dimensions of bilinears and OPE
coefficients at the fixed point which are consistent with a unitary CFT at
large . We then compute corrections. At next-to-leading order, the
line of fixed points collapses to one fixed point. However, the corrections are
complex and unitarity is broken at NLO. Finally, we show that this model
respects the -theorem. The next part of the thesis investigates sextic
tensor field theories in rank and . In rank , we found two IR stable
real fixed points in short range and a line of IR stable real fixed points in
long range. Surprisingly, the only fixed point in rank is the Gaussian one.
For the rank model, in the short-range case, we still find two IR stable
fixed points at NLO. However, in the long-range case, the corrections to the
fixed points are non-perturbative and hence unreliable: we found no precursor
of the large fixed point. The last part of the thesis investigates the
class of model exhibiting a melonic large limit. We prove that models with
tensors in an irreducible representation of or in rank
indeed admit a large limit. This generalization relies on recursive bounds
derived from a detailed combinatorial analysis of Feynman graphs involved in
the perturbative expansion of our model.Comment: PhD thesis, 277 pages. Based on papers: arXiv:1903.03578,
arXiv:1909.07767, arXiv:1912.06641, arXiv:2007.04603, arXiv:2011.11276,
arXiv:2104.03665, arXiv:2109.08034, arXiv:2111.1179
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